2.1.1 A dual viewpoint

It is in fact sometimes useful to adopt a dual viewpoint, and to study the convergence of observables rather than probability densities. This amounts to analysing the convergence of $\text{e}^{t{\mathcal{L}}}{\it\varphi}$ to $\int _{{\mathcal{X}}}{\it\varphi}\,\text{d}{\it\pi}$. To make this remark precise, let us first note thatFootnote ³ $\text{e}^{t{\mathcal{L}}}$ indeed encodes the time evolution of observables since

(2.8)$$\begin{eqnarray}(\text{e}^{t{\mathcal{L}}}{\it\varphi})(x)=\mathbb{E}^{x}({\it\varphi}(x_{t})).\end{eqnarray}$$

To prove (2.8) we start from Itô’s formula: for a fixed time $t>0$, for a $C^{\infty }$ function ${\it\varphi}$, and $s\in [0,t]$, we have

$$\begin{eqnarray}\text{d}(\text{e}^{(t-s){\mathcal{L}}}{\it\varphi}(x_{s}))=(-\partial _{t}+{\mathcal{L}})(\text{e}^{(t-s){\mathcal{L}}}{\it\varphi})(x_{s})\,\text{d}s+{\rm\nabla}(\text{e}^{(t-s){\mathcal{L}}}{\it\varphi})\cdot {\it\sigma}(x_{s})\,\text{d}W_{s},\end{eqnarray}$$

so that

$$\begin{eqnarray}\mathbb{E}^{x}({\it\varphi}(x_{t}))-\text{e}^{t{\mathcal{L}}}{\it\varphi}(x)=\mathbb{E}^{x}\biggl(\int _{0}^{t}{\rm\nabla}(\text{e}^{(t-s){\mathcal{L}}}{\it\varphi})\cdot {\it\sigma}(x_{s})\,\text{d}W_{s}\biggr)=0.\end{eqnarray}$$

For a general stochastic differential equation, the exponential decay to equilibrium of $\text{e}^{t{\mathcal{L}}}$, seen as a bounded operator on a reflexive Banach space $E\subset L^{2}$, is equivalent to the exponential decay to equilibrium of $\text{e}^{t{\mathcal{L}}^{\dagger }}$ seen as a bounded operator on $E^{\prime }$, the dual of $E$. This equivalence is a consequence of the following equality: for $C^{\infty }$ compactly supported test functions ${\it\varphi}$ and ${\it\psi}_{0}$,

$$\begin{eqnarray}\mathbb{E}({\it\varphi}(x_{t}))=\int _{{\mathcal{X}}}\,\text{e}^{t{\mathcal{L}}}{\it\varphi}(x_{0}){\it\psi}_{0}(x_{0})\,\text{d}x_{0}=\int _{{\mathcal{X}}}{\it\varphi}(x)\,\text{e}^{t{\mathcal{L}}^{\dagger }}{\it\psi}_{0}(x)\,\text{d}x,\end{eqnarray}$$

where $(x_{t})_{t\geqslant 0}$ satisfies (1.11) and the law of $x_{0}$ is ${\it\psi}_{0}(x)\,\text{d}x$. In the particular case of overdamped Langevin and Langevin dynamics, the long-time behaviours of $\text{e}^{t{\mathcal{L}}}$ and $\text{e}^{t{\mathcal{L}}^{\dagger }}$ are in fact closely related thanks to simple algebraic identities: see respectively (2.15) and (2.36) below.

Note also that, upon replacing the observable ${\it\varphi}$ with ${\it\varphi}-\int _{{\mathcal{X}}}{\it\varphi}\,\text{d}{\it\pi}$, it suffices to understand the convergence to 0 of $\text{e}^{t{\mathcal{L}}}{\it\varphi}$ for functions ${\it\varphi}$ with average 0 with respect to ${\it\pi}$.

2.1.2 Exponential convergence of semigroups and definition of inverses

We introduce the operator norm on the vector space ${\mathcal{B}}(E)$ of bounded linear operators on a Banach space $E$, defined by

$$\begin{eqnarray}\Vert {\mathcal{A}}\Vert _{{\mathcal{B}}(E)}=\sup _{g\neq 0}\frac{\Vert {\mathcal{A}}g\Vert _{E}}{\Vert g\Vert _{E}}.\end{eqnarray}$$

Recall that ${\mathcal{B}}(E)$ equipped with this norm is a Banach space. As discussed at the beginning of this section, it is important in various contexts to obtain bounds on the inverse of the generator ${\mathcal{L}}$ on $E$. The Banach space $E$ is typically a subspace of functions with average 0 with respect to the invariant measure, otherwise $\text{e}^{t{\mathcal{L}}}{\it\varphi}$ does not converge to 0 as $t\rightarrow +\infty$.

The invertibility of the generator, and bounds on this inverse in ${\mathcal{B}}(E)$, can be deduced from decay estimates on the semigroup, as is made precise in the following result.

Proposition 2.1. Assume that

(2.9)$$\begin{eqnarray}\Vert \text{e}^{t{\mathcal{L}}}\Vert _{{\mathcal{B}}(E)}\leqslant C\,\text{e}^{-{\it\lambda}t}.\end{eqnarray}$$

Then the operator ${\mathcal{L}}$ is invertible on $E$, and

(2.10)$$\begin{eqnarray}\Vert {\mathcal{L}}^{-1}\Vert _{{\mathcal{B}}(E)}\leqslant \frac{C}{{\it\lambda}}.\end{eqnarray}$$

Proof. Introducing

$$\begin{eqnarray}U=-\int _{0}^{+\infty }\text{e}^{t{\mathcal{L}}}\,\text{d}t,\end{eqnarray}$$

a simple computation shows that, for ${\it\varphi}\in D({\mathcal{L}})=\{{\it\phi}\in E\mid {\mathcal{L}}{\it\phi}\in E\}$,

$$\begin{eqnarray}{\mathcal{L}}U{\it\varphi}=U{\mathcal{L}}{\it\varphi}=-\int _{0}^{+\infty }\frac{\text{d}}{\text{d}t}[\text{e}^{t{\mathcal{L}}}{\it\varphi}]={\it\varphi}.\end{eqnarray}$$

These equalities show that ${\mathcal{L}}$ is invertible on $E$, with $U={\mathcal{L}}^{-1}$. From the definition of $U$, it follows that

$$\begin{eqnarray}\Vert U\Vert _{{\mathcal{B}}(E)}\leqslant \int _{0}^{+\infty }\Vert \text{e}^{t{\mathcal{L}}}\Vert _{{\mathcal{B}}(E)}\,\text{d}t\leqslant C\int _{0}^{+\infty }\text{e}^{-{\it\lambda}t}\,\text{d}t=\frac{C}{{\it\lambda}},\end{eqnarray}$$

which proves the resolvent bound (2.10).

As a final remark, let us emphasize that the manipulations performed here for continuous-in-time dynamics can also be appropriately modified for discrete-in-time dynamics, such as encountered in the discretization of SDEs; see Corollary 2.26 below.

2.2.1 Poincaré inequalities and convergence in $L^{2}({\it\nu})$

We start by considering the Fokker–Planck equation reformulated as (2.15), for an initial condition $f_{0}={\it\psi}_{0}/{\it\nu}$ belonging to $L^{2}({\it\nu})$. Recall that

$$\begin{eqnarray}f_{0}\geqslant 0,\quad \int _{{\mathcal{D}}}f_{0}\,\text{d}{\it\nu}=1.\end{eqnarray}$$

The simplest setting to consider is when a so-called Poincaré inequality holds; see for instance Bakry et al. (Reference Bakry, Gentil and Ledoux2014, Section 4) for a very nice introduction to these inequalities. In this case, $f(t)=\text{e}^{t{\mathcal{L}}}f_{0}$ converges exponentially fast in $L^{2}({\it\nu})$ to the constant function $\mathbf{1}$. This can be rephrased as the exponential convergence of $\text{e}^{t{\mathcal{L}}}(f_{0}-\mathbf{1})$ to 0 in $L^{2}({\it\nu})$.

In order to state the convergence result, we introduce the following definition.

Definition 2.2 (Poincaré inequality).

Consider the functional spaces

$$\begin{eqnarray}L_{0}^{2}({\it\nu})=\biggl\{{\it\varphi}\in L^{2}({\it\nu})\,\bigg|\,\int _{{\mathcal{D}}}{\it\varphi}\,\text{d}{\it\nu}=0\biggr\},\end{eqnarray}$$

and

$$\begin{eqnarray}H^{1}({\it\nu})=\biggl\{{\it\varphi}\in L^{2}({\it\nu})\,\bigg|\,{\rm\nabla}{\it\varphi}\in (L^{2}({\it\nu}))^{d}\biggr\}.\end{eqnarray}$$

The measure ${\it\nu}$ is said to satisfy a Poincaré inequality with constant $R>0$ when

(2.17)$$\begin{eqnarray}\Vert {\it\varphi}\Vert _{L^{2}({\it\nu})}^{2}\leqslant \frac{1}{2R}\Vert {\rm\nabla}{\it\varphi}\Vert _{L^{2}({\it\nu})}^{2},\quad \text{for all }{\it\varphi}\in H^{1}({\it\nu})\cap L_{0}^{2}({\it\nu}).\end{eqnarray}$$

The constant $R>0$ depends on the potential $V$, the inverse temperature ${\it\beta}$ and the domain ${\mathcal{D}}$. We discuss below various sufficient conditions for ${\it\nu}$ to satisfy a Poincaré inequality.

The inequality (2.17) implies (and in fact is equivalent to) the exponential convergence to 0 of the semigroup $\text{e}^{t{\mathcal{L}}}$ considered as an operator on $L_{0}^{2}({\it\nu})$.

Proposition 2.3. The measure ${\it\nu}$ satisfies a Poincaré inequality with constant $R>0$ if and only if

(2.18)$$\begin{eqnarray}\Vert \text{e}^{t{\mathcal{L}}}\Vert _{{\mathcal{B}}(L_{0}^{2}({\it\nu}))}\leqslant \text{e}^{-2Rt/{\it\beta}}.\end{eqnarray}$$

Proof. Let us first assume that the measure ${\it\nu}$ satisfies a Poincaré inequality with constant $R>0$. In view of (2.16),

(2.19)$$\begin{eqnarray}-\langle {\mathcal{L}}{\it\varphi},{\it\varphi}\rangle _{L^{2}({\it\nu})}=\frac{1}{{\it\beta}}\Vert {\rm\nabla}{\it\varphi}\Vert _{L^{2}({\it\nu})}^{2}\geqslant \frac{2R}{{\it\beta}}\Vert {\it\varphi}\Vert _{L^{2}({\it\nu})}^{2},\quad \text{for all }{\it\varphi}\in L_{0}^{2}({\it\nu})\cap H^{1}({\it\nu}).\end{eqnarray}$$

Since 0 is an eigenvalue of the operator ${\mathcal{L}}$ (whose associated eigenvectors are constant functions), this inequality shows that the spectral gap of the self-adjoint operator $-{\mathcal{L}}$ on $L^{2}({\it\nu})$ is larger than or equal to $2R/{\it\beta}$ (see (2.82) and the discussion following this equality). The inequality (2.19) also gives the exponential decrease of the semigroup on $L_{0}^{2}({\it\nu})$ since

(2.20)$$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\biggl(\displaystyle \frac{1}{2}\Vert \text{e}^{t{\mathcal{L}}}{\it\varphi}\Vert _{L^{2}({\it\nu})}^{2}\biggr)=\langle \text{e}^{t{\mathcal{L}}}{\it\varphi},{\mathcal{L}}\,\text{e}^{t{\mathcal{L}}}{\it\varphi}\rangle _{L^{2}({\it\nu})}\leqslant -\frac{2R}{{\it\beta}}\Vert \text{e}^{t{\mathcal{L}}}{\it\varphi}\Vert _{L^{2}({\it\nu})}^{2}.\end{eqnarray}$$

By a Gronwall inequality, it follows that

$$\begin{eqnarray}\Vert \text{e}^{t{\mathcal{L}}}{\it\varphi}\Vert _{L_{0}^{2}({\it\nu})}\leqslant \text{e}^{-2Rt/{\it\beta}}\Vert {\it\varphi}\Vert _{L_{0}^{2}({\it\nu})},\quad \text{for any }{\it\varphi}\in L_{0}^{2}({\it\nu}).\end{eqnarray}$$

Assume now that $\Vert \text{e}^{t{\mathcal{L}}}\Vert _{{\mathcal{B}}(L_{0}^{2}({\it\nu}))}\leqslant \text{e}^{-2Rt/{\it\beta}}$. Then, for a given ${\it\varphi}\in L_{0}^{2}({\it\nu})$ and any $t>0$,

$$\begin{eqnarray}\frac{\Vert \text{e}^{t{\mathcal{L}}}{\it\varphi}\Vert _{L_{0}^{2}({\it\nu})}^{2}-\Vert {\it\varphi}\Vert _{L_{0}^{2}({\it\nu})}^{2}}{t}\leqslant \Vert {\it\varphi}\Vert _{L_{0}^{2}({\it\nu})}^{2}\,\frac{\text{e}^{-4Rt/{\it\beta}}-1}{t}.\end{eqnarray}$$

We next pass to the limit $t\rightarrow 0$, using the equalities in (2.19) and (2.20):

$$\begin{eqnarray}-\frac{2}{{\it\beta}}\Vert {\rm\nabla}{\it\varphi}\Vert _{L^{2}({\it\nu})}^{2}\leqslant -\frac{4R}{{\it\beta}}\Vert {\it\varphi}\Vert _{L_{0}^{2}({\it\nu})}^{2}.\end{eqnarray}$$

Since ${\it\varphi}\in L_{0}^{2}({\it\nu})$ is arbitrary, the Poincaré inequality (2.17) follows.

In view of Proposition 2.1, a useful corollary of the decay estimates on the semigroup is the following result.

Corollary 2.4. Assume that ${\it\nu}$ satisfies a Poincaré inequality with constant $R$. Then the operator ${\mathcal{L}}$ is invertible on $L_{0}^{2}({\it\nu})$, and the following equality holds in ${\mathcal{B}}(L_{0}^{2}({\it\nu}))$:

$$\begin{eqnarray}{\mathcal{L}}^{-1}=-\int _{0}^{+\infty }\,\text{e}^{t{\mathcal{L}}}\,\text{d}t.\end{eqnarray}$$

Moreover,

(2.21)$$\begin{eqnarray}\Vert {\mathcal{L}}^{-1}\Vert _{{\mathcal{B}}(L_{0}^{2}({\it\nu}))}\leqslant \frac{{\it\beta}}{2R}.\end{eqnarray}$$

Let us now give some sufficient conditions to obtain Poincaré inequalities. First, such inequalities are easily obtained for probability measures equivalent to the Lebesgue measure, on connected domains, as a consequence of the standard Poincaré–Wirtinger inequality. This is the case when ${\mathcal{D}}=(L\mathbb{T})^{d}$, for example. Indeed, consider a measure with density ${\it\rho}(q)$ with respect to Lebesgue measure, and such that there exist two positive constants ${\it\rho}_{\min }$ and ${\it\rho}_{\max }$ with $0<{\it\rho}_{\min }\leqslant {\it\rho}(q)\leqslant {\it\rho}_{\max }$ for all $q\in {\mathcal{D}}$. First, note that

$$\begin{eqnarray}\min _{c\in \mathbb{R}}\int _{{\mathcal{D}}}({\it\varphi}-c)^{2}\,\text{d}{\it\rho}=\int _{{\mathcal{D}}}({\it\varphi}-\mathbb{E}_{{\it\rho}}({\it\varphi}))^{2}\,\text{d}{\it\rho},\quad \text{for all }{\it\varphi}\in L^{2}({\it\rho}).\end{eqnarray}$$

Therefore,

$$\begin{eqnarray}\displaystyle \min _{c\in \mathbb{R}}\int _{{\mathcal{D}}}({\it\varphi}-c)^{2}\,\text{d}{\it\rho} & {\leqslant} & \displaystyle {\it\rho}_{\max }\min _{c\in \mathbb{R}}\int _{{\mathcal{D}}}({\it\varphi}-c)^{2}\leqslant \frac{1}{2R_{\text{PW}}}{\it\rho}_{\max }\int _{{\mathcal{D}}}|{\rm\nabla}{\it\varphi}|^{2}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{1}{2R_{\text{PW}}}\frac{{\it\rho}_{\max }}{{\it\rho}_{\min }}\int _{{\mathcal{D}}}|{\rm\nabla}{\it\varphi}|^{2}\,\text{d}{\it\rho},\nonumber\end{eqnarray}$$

where $R_{\text{PW}}$ is the constant of the standard Poincaré–Wirtinger inequality on the bounded, connected domain ${\mathcal{D}}$. This shows that the canonical measure ${\it\nu}$ defined in (2.13) satisfies a Poincaré inequality when the domain ${\mathcal{D}}$ is bounded, with the following upper bound for the Poincaré constant $R_{{\it\nu}}$:

$$\begin{eqnarray}R_{{\it\nu}}\leqslant R_{\text{PW}}\,\text{e}^{-{\it\beta}(V_{\max }-V_{\min })},\end{eqnarray}$$

where

$$\begin{eqnarray}V_{\max }=\max _{q\in {\mathcal{D}}}V(q),\quad V_{\min }=\min _{q\in {\mathcal{D}}}V(q).\end{eqnarray}$$

In fact, this argument can be made more general to show that Poincaré inequalities satisfy some stability property under bounded perturbations of the measure, similar to the result obtained in Theorem 2.11 below for logarithmic Sobolev inequalities.

On unbounded domains, some conditions on the potential $V$ are required to have a Poincaré inequality. For example, such an inequality holds when $V$ is uniformly convex (i.e., ${\rm\nabla}^{2}V\geqslant R\,\text{Id}_{d}$ with $R>0$), similar to the result stated in Theorem 2.10 below for logarithmic Sobolev inequalities. Another result is based on the growth at infinity of the potential: see Bakry, Barthe, Cattiaux and Guillin (Reference Bakry, Barthe, Cattiaux and Guillin2008, Corollary 1.6).

Theorem 2.5. Consider $V\in C^{2}(\mathbb{R}^{d})$. If there exists $a\in (0,1)$, $c>0$ and $R\geqslant 0$ such that

(2.22)$$\begin{eqnarray}a{\it\beta}|{\rm\nabla}V(q)|^{2}-{\rm\Delta}V(q)\geqslant c,\quad \text{for all }q\in \mathbb{R}^{d}\text{ such that }|q|\geqslant R,\end{eqnarray}$$

then ${\it\nu}$ satisfies a Poincaré inequality.

The above condition is satisfied, for instance, for potentials of the form

$$\begin{eqnarray}V(q)=K|q|^{n}+\widetilde{V}(q),\end{eqnarray}$$

where $n\geqslant 1$ and $\widetilde{V}$ is $C^{\infty }$ and such that

$$\begin{eqnarray}\lim _{|q|\rightarrow +\infty }\frac{|\widetilde{V}(q)|}{|q|^{n}}=0\quad \text{and}\quad \lim _{|q|\rightarrow +\infty }\frac{|{\rm\nabla}\widetilde{V}|^{2}+|{\rm\Delta}\widetilde{V}|}{|q|^{2(n-1)}}=0.\end{eqnarray}$$

The first condition implies that $\text{e}^{-{\it\beta}V}$ is integrable, while the second one ensures that (2.22) holds.

A useful result that we shall exploit below is that it is possible to deduce Poincaré inequalities by tensorization.

Proposition 2.6. If a probability measure is a product of $d$ probability measures satisfying Poincaré inequalities with constants $R_{i}$, then the product measure satisfies a Poincaré inequality with constant

$$\begin{eqnarray}\min (R_{1},\ldots ,R_{d}).\end{eqnarray}$$

Proof. To simplify the notation, we prove the result in the case $d=2$, for a probability measure ${\it\rho}(\text{d}q_{1}\,\text{d}q_{2})={\it\rho}_{1}(\text{d}q_{1})\,{\it\rho}_{2}(\text{d}q_{2})$, defined on ${\mathcal{D}}={\mathcal{D}}_{1}\times {\mathcal{D}}_{2}$. Consider a $C^{\infty }$ function ${\it\varphi}$ with compact support, and average 0 with respect to ${\it\rho}$. We first introduce the partial average

$$\begin{eqnarray}\overline{{\it\varphi}}(q_{2})=\int _{{\mathcal{D}}_{1}}{\it\varphi}(q_{1}^{\prime },q_{2})\,{\it\rho}_{1}(\text{d}q_{1}^{\prime }).\end{eqnarray}$$

Since ${\it\rho}_{1}$ satisfies a Poincaré inequality with constant $R_{1}$, and using the fact that the function $q_{1}\mapsto {\it\varphi}(q_{1},q_{2})-\overline{{\it\varphi}}(q_{2})$ has average 0 with respect to ${\it\rho}_{1}(\text{d}q_{1})$ for any value of $q_{2}$, we obtain

$$\begin{eqnarray}\int _{{\mathcal{D}}_{1}}|{\it\varphi}(q_{1},q_{2})-\overline{{\it\varphi}}(q_{2})|^{2}{\it\rho}_{1}(\text{d}q_{1})\leqslant \frac{1}{2R_{1}}\int _{{\mathcal{D}}_{1}}|{\rm\nabla}_{q_{1}}{\it\varphi}(q_{1},q_{2})|^{2}{\it\rho}_{1}(\text{d}q_{1}).\end{eqnarray}$$

This inequality can be rewritten as

(2.23)$$\begin{eqnarray}\int _{{\mathcal{D}}_{1}}|{\it\varphi}(q_{1},q_{2})|^{2}{\it\rho}_{1}(\text{d}q_{1})\leqslant |\overline{{\it\varphi}}(q_{2})|^{2}+\frac{1}{2R_{1}}\int _{{\mathcal{D}}_{1}}|{\rm\nabla}_{q_{1}}{\it\varphi}(q_{1},q_{2})|^{2}{\it\rho}_{1}(\text{d}q_{1}).\end{eqnarray}$$

Now, since $\overline{{\it\varphi}}$ has average 0 with respect to ${\it\rho}_{2}$ because ${\it\varphi}$ has average 0 with respect to ${\it\rho}$,

$$\begin{eqnarray}\displaystyle \int _{{\mathcal{D}}_{2}}|\overline{{\it\varphi}}(q_{2})|^{2}{\it\rho}_{2}(\text{d}q_{2}) & {\leqslant} & \displaystyle \frac{1}{2R_{2}}\int _{{\mathcal{D}}_{2}}|{\rm\nabla}_{q_{2}}\overline{{\it\varphi}}(q_{2})|^{2}{\it\rho}_{2}(\text{d}q_{2})\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{1}{2R_{2}}\int _{{\mathcal{D}}}|{\rm\nabla}_{q_{2}}{\it\varphi}(q_{1},q_{2})|^{2}{\it\rho}_{1}(\text{d}q_{1})\,{\it\rho}_{2}(\text{d}q_{2}),\nonumber\end{eqnarray}$$

where we used a Cauchy–Schwarz inequality in the last step. Integration of (2.23) with respect to ${\it\rho}_{2}(\text{d}q_{2})$ finally leads to

$$\begin{eqnarray}\displaystyle \int _{{\mathcal{D}}}|{\it\varphi}(q_{1},q_{2})|^{2}{\it\rho}(\text{d}q) & {\leqslant} & \displaystyle \frac{1}{2R_{1}}\int _{{\mathcal{D}}}|{\rm\nabla}_{q_{1}}{\it\varphi}(q_{1},q_{2})|^{2}{\it\rho}(\text{d}q)\nonumber\\ \displaystyle & & \displaystyle \quad +\frac{1}{2R_{2}}\int _{{\mathcal{D}}}|{\rm\nabla}_{q_{2}}{\it\varphi}(q_{1},q_{2})|^{2}{\it\rho}(\text{d}q)\nonumber\\ \displaystyle & & \displaystyle \leqslant \frac{1}{2\min (R_{1},R_{2})}\int _{{\mathcal{D}}}|{\rm\nabla}_{q}{\it\varphi}|^{2}\,\text{d}{\it\rho},\nonumber\end{eqnarray}$$

from which we deduce that ${\it\rho}$ satisfies a Poincaré inequality with constant $R=\min (R_{1},R_{2})$.

Finally, anticipating later discussion, we will see in Proposition 2.12 that the logarithmic Sobolev inequality implies the Poincaré inequality.

2.2.2 Logarithmic Sobolev inequalities and convergence in $L^{1}({\it\nu})$

Here we give some background on logarithmic Sobolev inequalities, which can be used to show convergence to the equilibrium state. More material can be read in the review papers by Guionnet and Zegarlinski (Reference Guionnet and Zegarlinski2003), Ledoux (Reference Ledoux2001) and Arnold, Markowich, Toscani and Unterreiter (Reference Arnold, Markowich, Toscani and Unterreiter2001), and in Bakry et al. (Reference Bakry, Gentil and Ledoux2014, Section 5). The latter two works have a PDE approach which may be more appealing to readers accustomed to analytical frameworks. Other useful introductory references include Ané et al. (Reference Ané, Blachère, Chafaï, Fougères, Gentil, Malrieu, Roberto and Scheffer2000) and Villani (Reference Villani2003).

The relative entropy and the Fisher information between two measures are defined as follows.

Definition 2.7 (entropy and Fisher information).

For two probability measures ${\it\pi}_{1}$ and ${\it\pi}_{2}$ defined on a space ${\mathcal{X}}$, and such that ${\it\pi}_{1}$ is absolutely continuous with respect to ${\it\pi}_{2}$ (denoted ${\it\pi}_{1}\ll {\it\pi}_{2}$ in the following), the entropy of ${\it\pi}_{1}$ with respect to ${\it\pi}_{2}$ is

(2.24)$$\begin{eqnarray}{\mathcal{H}}({\it\pi}_{1}\mid {\it\pi}_{2})=\int _{{\mathcal{X}}}\ln \biggl(\displaystyle \frac{\text{d}{\it\pi}_{1}}{\text{d}{\it\pi}_{2}}\biggr)\,\text{d}{\it\pi}_{1}.\end{eqnarray}$$

The Fisher information of ${\it\pi}_{1}$ with respect to ${\it\pi}_{2}$ is

(2.25)$$\begin{eqnarray}{\mathcal{I}}({\it\pi}_{1}\mid {\it\pi}_{2})=\int _{{\mathcal{X}}}\biggl|{\rm\nabla}\ln \biggl(\displaystyle \frac{\text{d}{\it\pi}_{1}}{\text{d}{\it\pi}_{2}}\biggr)\biggr|^{2}\,\text{d}{\it\pi}_{1}.\end{eqnarray}$$

Using strict convexity of $x\mapsto x\ln x$ and the fact that $x\ln x-x+1\geqslant 0$, it is easy to check that the entropy is non-negative. Moreover, it is zero if and only if the two probability measures are identical. This is also true for the Fisher information.

There are many possible definitions of the entropy besides the relative entropy ${\mathcal{H}}$ defined in (2.24). Some mathematical motivations for the use of the relative entropy ${\mathcal{H}}$ can be found in Markowich and Villani (Reference Markowich and Villani2000); see also Chafaï (Reference Chafaï2015) for a historical perspective. This particular entropy may be of interest for the following extensivity property. The relative entropy of the distribution of $N$ independent variables (or, in less probabilistic terms, the entropy of a tensorized measure) is the sum of the relative entropies of the distributions of each random variable. This suggests that the rate of convergence to equilibrium estimated with relative entropies for weakly dependent variables may remain stable when the number of variables becomes large (see for instance the paragraph on Kac’s spectral problem in Villani Reference Villani2002, Chapter 5). This extensivity is a consequence of the extensivity of the logarithm function involved in definition (2.24) (by which we mean that $\ln (f_{1}f_{2})=\ln (f_{1})+\ln (f_{2})$). This extensivity property makes the logarithmic Sobolev inequality and entropy approaches much more convenient for studying some nonlinear Fokker–Planck equations, as will become clear in Section 4.3.

Recall that the total variation between two measures, which reduces to the $L^{1}$-norm of the difference between the two densities when the two measures are absolutely continuous with respect to Lebesgue measure, can be bounded by the relative entropy. This is the Csiszár–Kullback inequality (see for example Ané et al. Reference Ané, Blachère, Chafaï, Fougères, Gentil, Malrieu, Roberto and Scheffer2000, Royer Reference Royer2007): introducing the total variation norm

(2.26)$$\begin{eqnarray}\Vert {\it\pi}_{1}-{\it\pi}_{2}\Vert _{\text{TV}}=\sup _{\Vert {\it\varphi}\Vert _{L^{\infty }}\leqslant 1}\int _{{\mathcal{X}}}{\it\varphi}(\text{d}{\it\pi}_{1}-\text{d}{\it\pi}_{2}),\end{eqnarray}$$

it holds that

(2.27)$$\begin{eqnarray}\Vert {\it\pi}_{1}-{\it\pi}_{2}\Vert _{\text{TV}}\leqslant \sqrt{2{\mathcal{H}}({\it\pi}_{1}\mid {\it\pi}_{2})}.\end{eqnarray}$$

In other words, an upper bound for the entropy between ${\it\pi}_{1}$ and ${\it\pi}_{2}$ yields an upper bound for a distance between ${\it\pi}_{1}$ and ${\it\pi}_{2}$ (even though the relative entropy is not a distance because it is not symmetric in its arguments).

We now present a way to obtain an estimate of the rate of convergence to zero of the entropy ${\mathcal{H}}({\it\psi}(t,\cdot )\mid {\it\nu})$ for the solutions of the Fokker–Planck equation (2.12). To this end we introduce the following functional inequality.

Definition 2.8 (logarithmic Sobolev inequality).

A probability measure ${\it\pi}_{2}$ satisfies a logarithmic Sobolev inequality with constant $R>0$ (or LSI($R$) for short) if, for all probability measures ${\it\pi}_{1}$ such that ${\it\pi}_{1}\ll {\it\pi}_{2}$,

(2.28)$$\begin{eqnarray}{\mathcal{H}}({\it\pi}_{1}\mid {\it\pi}_{2})\leqslant \frac{1}{2R}{\mathcal{I}}({\it\pi}_{1}\mid {\it\pi}_{2}).\end{eqnarray}$$

The exponential decay of ${\mathcal{H}}({\it\psi}(t,\cdot )\mid {\it\nu})$ to 0 can then be shown provided ${\it\nu}$ satisfies a so-called logarithmic Sobolev inequality (LSI). In fact, there is some equivalence between the two notions, as is made precise in the following result, which is the equivalent of Proposition 2.3 where a similar result is obtained for Poincaré inequalities and exponential decay in some $L^{2}$-space.

Proposition 2.9. The measure ${\it\nu}$ satisfies $\text{LSI}(R)$ if and only if, for any initial condition ${\it\psi}_{0}\geqslant 0$ with integral 1 and finite relative entropy with respect to ${\it\nu}$ (i.e. ${\mathcal{H}}({\it\psi}_{0}\mid {\it\nu})<+\infty$), the solution ${\it\psi}(t)$ of the Fokker–Planck equation (2.12) satisfies

(2.29)$$\begin{eqnarray}{\mathcal{H}}({\it\psi}(t)\mid {\it\nu})\leqslant {\mathcal{H}}({\it\psi}_{0}\mid {\it\nu})\exp (-2{\it\beta}^{-1}Rt),\quad \text{for all }t\geqslant 0.\end{eqnarray}$$

From the Csiszár–Kullback inequality (2.27), it is then possible to deduce the exponential convergence to zero with rate ${\it\beta}^{-1}R$ of the norm $\Vert {\it\psi}(t,\cdot )-{\it\nu}\Vert _{L^{1}}$.

Proof. Note first that the Fokker–Planck equation (2.12) can be rewritten as

$$\begin{eqnarray}\partial _{t}{\it\psi}={\mathcal{L}}^{\dagger }{\it\psi}=\frac{1}{{\it\beta}}\text{div}\biggl({\it\nu}{\rm\nabla}\biggl(\frac{{\it\psi}}{{\it\nu}}\biggr)\biggr).\end{eqnarray}$$

A straightforward computation shows that

$$\begin{eqnarray}\frac{\text{d}}{\text{d}t}({\mathcal{H}}({\it\psi}(t,\cdot )\mid {\it\nu}))=\int _{{\mathcal{D}}}\partial _{t}{\it\psi}\biggl(\ln \biggl(\displaystyle \frac{{\it\psi}}{{\it\nu}}\biggr)+1\biggr)=\int _{{\mathcal{D}}}\partial _{t}{\it\psi}\,\ln \biggl(\displaystyle \frac{{\it\psi}}{{\it\nu}}\biggr),\end{eqnarray}$$

since

$$\begin{eqnarray}\int _{{\mathcal{D}}}{\it\psi}(t)=1.\end{eqnarray}$$

Therefore,

$$\begin{eqnarray}\frac{\text{d}}{\text{d}t}({\mathcal{H}}({\it\psi}(t,\cdot )\mid {\it\nu}))=\int _{{\mathcal{D}}}({\mathcal{L}}^{\dagger }{\it\psi})\,\ln \biggl(\displaystyle \frac{{\it\psi}}{{\it\nu}}\biggr)=-\frac{1}{{\it\beta}}\int _{{\mathcal{D}}}{\rm\nabla}\biggl(\displaystyle \frac{{\it\psi}}{{\it\nu}}\biggr)\cdot {\rm\nabla}\biggl[\ln \biggl(\displaystyle \frac{{\it\psi}}{{\it\nu}}\biggr)\biggr]\,\text{d}{\it\nu},\end{eqnarray}$$

which can be summarized as

(2.30)$$\begin{eqnarray}\frac{\text{d}}{\text{d}t}({\mathcal{H}}({\it\psi}(t,\cdot )\mid {\it\nu}))=-\frac{1}{{\it\beta}}{\mathcal{I}}({\it\psi}(t,\cdot )\mid {\it\nu}).\end{eqnarray}$$

Assume now that ${\it\nu}$ satisfies an $\text{LSI}(R)$. The estimate (2.29) then directly follows from (2.30).

Assume conversely that (2.29) holds for any initial condition ${\it\psi}_{0}$ with finite relative entropy with respect to ${\it\nu}$. Then,

$$\begin{eqnarray}\frac{{\mathcal{H}}({\it\psi}(t)\mid {\it\nu})-{\mathcal{H}}({\it\psi}_{0}\mid {\it\nu})}{t}\leqslant {\mathcal{H}}({\it\psi}_{0}\mid {\it\nu})\frac{\exp (-2{\it\beta}^{-1}Rt)-1}{t}.\end{eqnarray}$$

In the limit $t\rightarrow 0$, one obtains, with (2.30),

$$\begin{eqnarray}-\frac{1}{{\it\beta}}{\mathcal{I}}({\it\psi}_{0}\mid {\it\nu})\leqslant -\frac{2R}{{\it\beta}}\,{\mathcal{H}}({\it\psi}_{0}\mid {\it\nu}),\end{eqnarray}$$

which indeed leads to (2.28).

We now give several ways to obtain LSIs for measures of the form (2.13).

1. (i) When the potential $V$ satisfies a strict convexity condition of the form $\text{Hess}(V)\geqslant R\,\text{Id}_{d}$ with $R>0$, then an LSI with constant ${\it\beta}R$ holds for ${\it\nu}$, as first shown in Bakry and Emery (Reference Bakry and Emery1985) (see Theorem 2.10 below).

2. (ii) If ${\it\psi}_{\infty }=\prod _{i=1}^{d}{\it\psi}_{\infty }^{i}$ and each measure ${\it\psi}_{\infty }^{i}(q)\,\text{d}q$ satisfies an LSI with constant $R_{i}$, then ${\it\psi}_{\infty }$ satisfies an LSI with constant $R=\min \{R_{1},\ldots ,R_{d}\}$ (see Gross Reference Gross1975). This is the equivalent of Proposition 2.6 for LSI.

3. (iii) When an LSI with constant $R$ is satisfied by ${\it\nu}(\text{d}q)=Z_{V}^{-1}\,\text{e}^{-V(q)}\,\text{d}q$, then the modified measure

   $$\begin{eqnarray}Z_{V+\widetilde{V}}^{-1}\,\text{e}^{-(V(q)+\widetilde{V}(q))}\,\text{d}q\end{eqnarray}$$
   (with $\widetilde{V}$ bounded) satisfies an LSI with constant $\widetilde{R}=R\,\text{e}^{\inf \widetilde{V}-\sup \widetilde{V}}$. This property expresses some stability with respect to bounded perturbations (see Holley and Stroock Reference Holley and Stroock1987 and Theorem 2.11 below).

4. (iv) There are also results on an LSI for the measure when a marginal law and the corresponding conditional laws satisfy an LSI (see Blower and Bolley Reference Blower and Bolley2006), or when all the conditional laws satisfy an LSI under some weak coupling assumption (see Otto and Reznikoff Reference Otto and Reznikoff2007). Such results can be extended to the non-linear setting, that is, in the case when the marginal distribution is obtained for some variable $z={\it\xi}(q)\in \mathbb{R}^{m}$ (with $m<d$): see Lelièvre (Reference Lelièvre2009).

Let us now state more precisely, and prove, two fundamental results ensuring that an LSI holds for canonical measures of the form (1.5).

Proof. The idea is to differentiate the Fisher information with respect to time, in order to prove its exponential convergence to 0, and then to insert it in (2.30). Introduce $f(t)={\it\psi}(t)/{\it\nu}$ for an arbitrary initial condition $f_{0}={\it\psi}_{0}/{\it\nu}$. In view of (2.15), this function evolves according to $\partial _{t}f={\mathcal{L}}f$, so that

$$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}[{\mathcal{I}}({\it\psi}(t)\mid {\it\nu})] & = & \displaystyle \frac{\text{d}}{\text{d}t}\biggl(\int _{{\mathcal{D}}}|{\rm\nabla}\ln f(t)|^{2}f(t)\,\text{d}{\it\nu}\biggr)\nonumber\\ \displaystyle & = & \displaystyle 2\int _{{\mathcal{D}}}{\rm\nabla}\biggl(\displaystyle \frac{\partial _{t}f(t)}{f(t)}\biggr)\cdot {\rm\nabla}f(t)\,\text{d}{\it\nu}+\int _{{\mathcal{D}}}|{\rm\nabla}\ln f(t)|^{2}\partial _{t}f(t)\,\text{d}{\it\nu}\nonumber\\ \displaystyle & = & \displaystyle 2\int _{{\mathcal{D}}}{\rm\nabla}\biggl(\displaystyle \frac{{\mathcal{L}}f(t)}{f(t)}\biggr)\cdot {\rm\nabla}f(t)\,\text{d}{\it\nu}+\int _{{\mathcal{D}}}|{\rm\nabla}\ln f(t)|^{2}{\mathcal{L}}f(t)\,\text{d}{\it\nu}.\nonumber\end{eqnarray}$$

Since

$$\begin{eqnarray}{\mathcal{L}}(\ln f)=\frac{{\mathcal{L}}f}{f}-\frac{|{\rm\nabla}f|^{2}}{{\it\beta}f^{2}}=\frac{{\mathcal{L}}f}{f}-\frac{1}{{\it\beta}}|{\rm\nabla}(\ln f)|^{2},\end{eqnarray}$$

we obtain, after some algebraic calculations,

$$\begin{eqnarray}\displaystyle \frac{1}{2}\frac{\text{d}}{\text{d}t}[{\mathcal{I}}({\it\psi}(t)\mid {\it\nu})] & = & \displaystyle \int _{{\mathcal{D}}}{\rm\nabla}[{\mathcal{L}}(\ln f(t))]\cdot {\rm\nabla}[\ln f(t)]\,f(t)\,\text{d}{\it\nu}\nonumber\\ \displaystyle & & \displaystyle \quad -\int _{{\mathcal{D}}}{\mathcal{L}}\biggl[\frac{1}{2}|{\rm\nabla}(\ln f(t))|^{2}\biggr]\,f(t)\,\text{d}{\it\nu}.\nonumber\end{eqnarray}$$

Now, using

$$\begin{eqnarray}{\rm\nabla}\biggl(\displaystyle \frac{1}{2}|{\rm\nabla}h|^{2}\biggr)={\rm\nabla}^{2}h\cdot {\rm\nabla}h,\quad {\rm\Delta}\biggl(\displaystyle \frac{1}{2}|{\rm\nabla}h|^{2}\biggr)={\rm\nabla}^{2}h:{\rm\nabla}^{2}h+{\rm\nabla}({\rm\Delta}h)\cdot {\rm\nabla}h,\end{eqnarray}$$

it follows that

$$\begin{eqnarray}\displaystyle {\mathcal{L}}\biggl(\displaystyle \frac{1}{2}|{\rm\nabla}h|^{2}\biggr) & = & \displaystyle \frac{1}{{\it\beta}}({\rm\nabla}^{2}h:{\rm\nabla}^{2}h+{\rm\nabla}({\rm\Delta}h)\cdot {\rm\nabla}h)-({\rm\nabla}V)^{T}({\rm\nabla}^{2}h){\rm\nabla}h\nonumber\\ \displaystyle & {\geqslant} & \displaystyle {\rm\nabla}({\mathcal{L}}h)\cdot {\rm\nabla}h+({\rm\nabla}h)^{T}({\rm\nabla}^{2}V){\rm\nabla}h\nonumber\\ \displaystyle & {\geqslant} & \displaystyle {\rm\nabla}({\mathcal{L}}h)\cdot {\rm\nabla}h+R|{\rm\nabla}h|^{2},\nonumber\end{eqnarray}$$

where we have used ${\rm\nabla}^{2}h:{\rm\nabla}^{2}h\geqslant 0$ in the second line, and the assumption on ${\rm\nabla}^{2}V$ in the third one. We next replace $h$ with $\ln f(t)$ and obtain

$$\begin{eqnarray}\frac{1}{2}\frac{\text{d}}{\text{d}t}[{\mathcal{I}}({\it\psi}(t)\mid {\it\nu})]\leqslant -R\int _{{\mathcal{D}}}|{\rm\nabla}(\ln f(t))|^{2}\,f(t)\,\text{d}{\it\nu}=-R\,{\mathcal{I}}({\it\psi}(t)\mid {\it\nu}).\end{eqnarray}$$

This shows that the Fisher information converges exponentially fast to zero:

(2.31)$$\begin{eqnarray}{\mathcal{I}}({\it\psi}(t)\mid {\it\nu})\leqslant \text{e}^{-2Rt}{\mathcal{I}}({\it\psi}(0)\mid {\it\nu}).\end{eqnarray}$$

The next step is to prove that ${\mathcal{H}}({\it\psi}(t,\cdot )\mid {\it\nu})$ converges to 0 as $t\rightarrow +\infty$. To this end, we use the argument provided in Arnold et al. (Reference Arnold, Markowich, Toscani and Unterreiter2001). Since the operator $-{\mathcal{L}}$ is self-adjoint on $L^{2}({\it\nu})$ and positive by (2.19), its spectral measure $P_{{\it\lambda}}$ is supported by $[0,+\infty )$; see Reed and Simon (Reference Reed and Simon1975a) and Dautray and Lions (Reference Dautray and Lions1990) for the definition of the spectral measure. In addition, (2.19) also shows that 0 is a non-degenerate eigenvalue whose associated eigenvectors are constant functions (recall that since ${\it\nu}$ satisfies LSI$({\it\beta}R)$, ${\it\nu}$ also satisfies a Poincaré inequality with the same constant: see Proposition 2.12 below). Therefore,

$$\begin{eqnarray}f(t)=\text{e}^{t{\mathcal{L}}}f_{0}=P_{0}f_{0}+\int _{(0,+\infty )}\text{e}^{-t{\it\lambda}}\,\text{d}(P_{{\it\lambda}}f_{0}).\end{eqnarray}$$

In fact, $P_{0}$ is the projector onto the eigenspace associated with the eigenvalue 0, so that $P_{0}f_{0}=\int _{{\mathcal{D}}}f_{0}\,\text{d}{\it\nu}=1$. Moreover,

$$\begin{eqnarray}\Vert f(t)-\mathbf{1}\Vert _{L^{2}({\it\nu})}^{2}=\int _{(0,+\infty )}\text{e}^{-2t{\it\lambda}}\,m_{f_{0}}(\text{d}{\it\lambda}),\quad m_{f_{0}}(\text{d}{\it\lambda})=\text{d}(\Vert P_{{\it\lambda}}f_{0}\Vert _{L^{2}({\it\nu})}^{2}).\end{eqnarray}$$

By dominated convergence, the integral on the right-hand side of the above equality converges to 0 as $t\rightarrow +\infty$.

The conclusion now follows by a time integration of (2.30) together with the decay estimate (2.31), which leads to

$$\begin{eqnarray}{\mathcal{H}}({\it\psi}_{0}\mid {\it\nu})-{\mathcal{H}}({\it\psi}(t,\cdot )\mid {\it\nu})\leqslant \frac{1}{{\it\beta}}\int _{0}^{t}{\mathcal{I}}({\it\psi}(s)\mid {\it\nu})\,\text{d}s\leqslant \frac{1}{2{\it\beta}R}{\mathcal{I}}({\it\psi}_{0}\mid {\it\nu}).\end{eqnarray}$$

In the limit $t\rightarrow +\infty$, this inequality becomes

$$\begin{eqnarray}{\mathcal{H}}({\it\psi}_{0}\mid {\it\nu})\leqslant \frac{1}{2{\it\beta}R}{\mathcal{I}}({\it\psi}_{0}\mid {\it\nu}),\end{eqnarray}$$

which gives the claimed LSI since ${\it\psi}_{0}$ is arbitrary.

Proof. Introduce ${\it\phi}(x)=x\ln x-1+x$. Consider any function $f\geqslant 0$ such that

(2.32)$$\begin{eqnarray}\int _{{\mathcal{D}}}f\,\text{d}\widetilde{{\it\nu}}=1,\end{eqnarray}$$

and let $\overline{f}$ denote the average of $f$ with respect to ${\it\nu}$:

(2.33)$$\begin{eqnarray}\overline{f}=\int _{{\mathcal{D}}}f\,\text{d}{\it\nu}.\end{eqnarray}$$

Then

$$\begin{eqnarray}\displaystyle {\mathcal{H}}(f\widetilde{{\it\nu}}\mid \widetilde{{\it\nu}}) & = & \displaystyle \int _{{\mathcal{D}}}f\ln f\,\text{d}\widetilde{{\it\nu}}=\int _{{\mathcal{D}}}{\it\phi}(f)\,\text{d}\widetilde{{\it\nu}}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \int _{{\mathcal{D}}}[{\it\phi}(f)-({\it\phi}(\overline{f})+{\it\phi}^{\prime }(\overline{f})(1-\overline{f}))]\,\text{d}\widetilde{{\it\nu}}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \int _{{\mathcal{D}}}[{\it\phi}(f)-{\it\phi}(\overline{f})+{\it\phi}^{\prime }(\overline{f})(\overline{f}-f)]\,\text{d}\widetilde{{\it\nu}},\nonumber\end{eqnarray}$$

since the convexity of ${\it\phi}$ implies that ${\it\phi}(1)=0\geqslant {\it\phi}(\overline{f})+{\it\phi}^{\prime }(\overline{f})(1-\overline{f})$, and where we have used (2.32) to obtain the last line. Note that ${\it\phi}(f(q))-{\it\phi}(\overline{f})+{\it\phi}^{\prime }(\overline{f})(\overline{f}-f(q))\geqslant 0$, still by the convexity of ${\it\phi}$. Since the integrand is non-negative, we can therefore reintroduce ${\it\nu}$ as follows:

$$\begin{eqnarray}\displaystyle {\mathcal{H}}(f\widetilde{{\it\nu}}\mid \widetilde{{\it\nu}}) & {\leqslant} & \displaystyle \frac{1}{\widetilde{Z}}\,\text{e}^{-\inf \widetilde{V}}\int _{{\mathcal{D}}}[{\it\phi}(f)-{\it\phi}(\overline{f})+{\it\phi}^{\prime }(\overline{f})(\overline{f}-f)]\,\text{d}{\it\nu}\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{\widetilde{Z}}\,\text{e}^{-\inf \widetilde{V}}\biggl(\int _{{\mathcal{D}}}{\it\phi}(f)\,\text{d}{\it\nu}-{\it\phi}(\overline{f})\biggr),\nonumber\end{eqnarray}$$

in view of (2.33). On the other hand,

$$\begin{eqnarray}\int _{{\mathcal{D}}}{\it\phi}(f)\,\text{d}{\it\nu}-{\it\phi}(\overline{f})=\int _{{\mathcal{D}}}f\ln f\,\text{d}{\it\nu}-\overline{f}\ln \overline{f}=\overline{f}\int _{{\mathcal{D}}}\frac{f}{\overline{f}}\ln \frac{f}{\overline{f}}\,\text{d}{\it\nu}=\overline{f}\int _{{\mathcal{D}}}{\it\phi}\biggl(\displaystyle \frac{f}{\overline{f}}\biggr)\,\text{d}{\it\nu}.\end{eqnarray}$$

Since ${\it\nu}$ satisfies LSI($R$), and $f/\overline{f}$ has integral 1 with respect to ${\it\nu}$,

$$\begin{eqnarray}\int _{{\mathcal{D}}}{\it\phi}\biggl(\displaystyle \frac{f}{\overline{f}}\biggr)\,\text{d}{\it\nu}\leqslant \frac{1}{2R}\int _{{\mathcal{D}}}\biggl|{\rm\nabla}\ln \biggl(\displaystyle \frac{f}{\overline{f}}\biggr)\biggr|^{2}\biggl(\displaystyle \frac{f}{\overline{f}}\biggr)\,\text{d}{\it\nu}=\frac{1}{2R\overline{f}}\int _{{\mathcal{D}}}|{\rm\nabla}(\ln f)|^{2}f\,\text{d}{\it\nu}.\end{eqnarray}$$

In conclusion,

$$\begin{eqnarray}\displaystyle {\mathcal{H}}(f\widetilde{{\it\nu}}\mid \widetilde{{\it\nu}}) & {\leqslant} & \displaystyle \frac{1}{2R\widetilde{Z}}\,\text{e}^{-\inf \widetilde{V}}\int _{{\mathcal{D}}}|{\rm\nabla}(\ln f)|^{2}f\,\text{d}{\it\nu}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{1}{2R}\,\text{e}^{\sup \widetilde{V}-\inf \widetilde{V}}\int _{{\mathcal{D}}}|{\rm\nabla}(\ln f)|^{2}f\,\text{d}\widetilde{{\it\nu}},\nonumber\end{eqnarray}$$

which gives the claimed statement.

The previous result gives lower bounds on the LSI constants. It suggests that the LSI constant, and hence the rate of convergence to equilibrium of the overdamped Langevin dynamics, decreases exponentially as the temperature decreases and/or energetic barriers increase. Consider for instance the low-temperature regime ${\it\beta}\rightarrow +\infty$, for a reference convex potential $V_{0}(q)=a|q|^{2}/2$ with $a>0$, perturbed by a bounded potential $\widetilde{V}$, for instance $\widetilde{V}(q)=b\exp (-|q|^{2})$ with $b>0$. In this case, Theorem 2.10 shows that the canonical measure associated with $V$, at inverse temperature ${\it\beta}$, satisfies LSI(${\it\beta}a$), while the canonical measure associated with $V+\widetilde{V}$, at inverse temperature ${\it\beta}$, satisfies LSI(${\it\beta}a\,\text{e}^{-{\it\beta}b}$) by Theorem 2.11. The LSI constants obtained by the Holley–Stroock perturbative argument are of course lower bounds, but it is indeed expected that the optimal constant decreases exponentially fast as the temperature decreases when the potential exhibits energetic barriers. This can be rigorously proved in one-dimensional systems for example; see Section 2.4 in Menz and Schlichting (Reference Menz and Schlichting2014).

2.2.3 Relationship between logarithmic Sobolev inequalities and Poincaré inequalities

Initial conditions ${\it\psi}_{0}$ belonging to $L^{2}({\it\nu})$ also have a finite relative entropy with respect to ${\it\nu}$. Indeed, using the inequality

$$\begin{eqnarray}x\ln x-x+1\leqslant (x-1)^{2},\quad \text{for all }x\geqslant 0,\end{eqnarray}$$

it holds that

$$\begin{eqnarray}{\mathcal{H}}({\it\psi}_{0}\,|\,{\it\nu})\leqslant \biggl\|\frac{{\it\psi}_{0}}{{\it\nu}}-\mathbf{1}\biggr\|_{L^{2}({\it\nu})}^{2}.\end{eqnarray}$$

This shows that LSIs allow us to cover a larger set of initial conditions than Poincaré inequalities. Unsurprisingly, the following result shows that LSIs are stronger than Poincaré inequalities.

Proof. The idea of the proof is that the Poincaré inequality can be seen as a linearization of the logarithmic Sobolev inequality. Start from the LSI

$$\begin{eqnarray}\int _{{\mathcal{D}}}{\it\phi}\ln {\it\phi}\,\text{d}{\it\nu}\leqslant \frac{1}{2R}\int _{{\mathcal{D}}}\frac{|{\rm\nabla}{\it\phi}|^{2}}{{\it\phi}}\,\text{d}{\it\nu},\end{eqnarray}$$

where ${\it\phi}\geqslant 0$ has average 1 with respect to ${\it\nu}$. The function ${\it\phi}$ can be chosen of the form ${\it\phi}=1+{\it\varepsilon}{\it\varphi}$ where ${\it\varphi}\in L^{\infty }({\mathcal{D}})$ has average 0 with respect to ${\it\nu}$. For $0<{\it\varepsilon}\leqslant 1/(2\Vert {\it\varphi}\Vert _{L^{\infty }})$,

$$\begin{eqnarray}\int _{{\mathcal{D}}}(1+{\it\varepsilon}{\it\varphi})\ln (1+{\it\varepsilon}{\it\varphi})\,\text{d}{\it\nu}\leqslant \frac{{\it\varepsilon}^{2}}{2R}\int _{{\mathcal{D}}}\frac{|{\rm\nabla}{\it\varphi}|^{2}}{1+{\it\varepsilon}{\it\varphi}}\,\text{d}{\it\nu}=\frac{{\it\varepsilon}^{2}}{2R}\biggl(\int _{{\mathcal{D}}}|{\rm\nabla}{\it\varphi}|^{2}\,\text{d}{\it\nu}+\text{O}({\it\varepsilon})\biggr).\end{eqnarray}$$

Since $(1+{\it\varepsilon}{\it\varphi})\ln (1+{\it\varepsilon}{\it\varphi})={\it\varepsilon}{\it\varphi}+({\it\varepsilon}{\it\varphi})^{2}+\text{O}({\it\varepsilon}^{3})$, and using the fact that ${\it\varphi}$ has average 0 with respect to ${\it\nu}$,

$$\begin{eqnarray}\int _{{\mathcal{D}}}(1+{\it\varepsilon}{\it\varphi})\ln (1+{\it\varepsilon}{\it\varphi})\,\text{d}{\it\nu}={\it\varepsilon}^{2}\biggl(\int _{{\mathcal{D}}}{\it\varphi}^{2}\,\text{d}{\it\nu}+\text{O}({\it\varepsilon})\biggr).\end{eqnarray}$$

The conclusion follows for bounded functions ${\it\varphi}$ by taking the limit ${\it\varepsilon}\rightarrow 0$. The general case can finally be deduced by a density argument.

2.3.1 Hypoellipticity

Although the generator of the Langevin dynamics and its adjoint are not elliptic, they can be shown to be hypoelliptic, following the work of Hörmander Reference Hörmander1967, Reference Hörmander1985. To this end we introduce the commutator of two operators $A,B$:

$$\begin{eqnarray}[A,B]=AB-BA.\end{eqnarray}$$

We can then state Hörmander’s theorem.

Theorem 2.13. Consider $C^{\infty }$ vector fields on the $D$-dimensional space ${\mathcal{Y}}$,

$$\begin{eqnarray}A_{j}=\mathop{\sum }_{i=1}^{D}A_{j,i}(y)\partial _{y_{i}},\end{eqnarray}$$

and introduce the operator

$$\begin{eqnarray}{\mathcal{A}}=A_{0}+\mathop{\sum }_{j=1}^{J}A_{j}^{\dagger }A_{j},\end{eqnarray}$$

where $A_{j}^{\dagger }$ is the (formal) adjoint of $A_{j}$ on $L^{2}({\mathcal{Y}})$. Assume that the Lie algebra spanned by

$$\begin{eqnarray}\{A_{j}\}_{j=0,\ldots ,J},\quad \{[A_{j},A_{k}]\}_{j,k=0,\ldots ,J},\quad \{[[A_{j},A_{k}],A_{l}]\}_{j,k,l=0,\ldots ,J},\quad \ldots\end{eqnarray}$$

has maximal rank $D$ at every point $y\in {\mathcal{Y}}$. Then ${\mathcal{A}}$ is hypoelliptic, namely there exists ${\it\varepsilon}>0$ such that ${\mathcal{A}}f\in H_{\text{loc}}^{s}$ implies $f\in H_{\text{loc}}^{s+{\it\varepsilon}}$.

In particular, solutions $f$ of the equation ${\mathcal{A}}f=0$ are $C^{\infty }$.

In order to use this result for Langevin dynamics, we consider ${\mathcal{Y}}={\mathcal{E}}$, $D=2d$, $J=d$, and rewrite the generator ${\mathcal{L}}$ as

$$\begin{eqnarray}{\mathcal{L}}=A_{0}+\mathop{\sum }_{i=1}^{d}A_{i}^{\dagger }A_{i},\end{eqnarray}$$

with

$$\begin{eqnarray}A_{0}={\mathcal{L}}_{\text{ham}}-{\it\gamma}p^{T}M^{-1}{\rm\nabla}_{p},\quad A_{i}=\sqrt{\frac{{\it\gamma}}{{\it\beta}}}\partial _{p_{i}}\quad \text{for }1\leqslant i\leqslant d,\end{eqnarray}$$

where

$$\begin{eqnarray}{\mathcal{L}}_{\text{ham}}=p^{T}M^{-1}{\rm\nabla}_{q}-{\rm\nabla}V^{T}{\rm\nabla}_{p}\end{eqnarray}$$

is the generator of the Hamiltonian part of the dynamics. Assuming for simplicity that $M$ is diagonal with entries $m_{i}$ (although the computations can be extended to account for general positive definite mass matrices), we obtain

$$\begin{eqnarray}[A_{i},A_{0}]{\it\varphi}=\sqrt{\frac{{\it\gamma}}{{\it\beta}}}\frac{1}{m_{i}}(\partial _{q_{i}}-{\it\gamma}\partial _{p_{i}}){\it\varphi}\quad \text{for all }i\in \{1,\ldots ,d\}.\end{eqnarray}$$

Therefore, the Lie algebra generated by $\{A_{i}\}_{i=1,\ldots ,d}$ and $\{[A_{i},A_{0}]\}_{i=1,\ldots ,d}$ has maximal rank $2d$ at every configuration $(q,p)\in {\mathcal{E}}$. This proves that ${\mathcal{L}}$ is hypoelliptic. Similar computations show that ${\mathcal{L}}^{\dagger }$ is also hypoelliptic.

Note also that Theorem 2.13 can be used to prove that $\partial _{t}-{\mathcal{L}}$ and $\partial _{t}-{\mathcal{L}}^{\dagger }$ are hypoelliptic on ${\mathcal{Y}}=\mathbb{R}\times {\mathcal{E}}$. This ensures that the transition kernel associated with the Langevin dynamics admits a $C^{\infty }$ density (see Section 2.4).

2.3.2 Lack of coercivity on $L_{0}^{2}({\it\mu})$

We recall that we let ${\mathcal{A}}^{\ast }$ denote the adjoint of a closed operator ${\mathcal{A}}$ with respect to the scalar product on $L^{2}({\it\mu})$ (see (2.7)). It turns out to be convenient to rewrite the generator (2.34) as

$$\begin{eqnarray}{\mathcal{L}}={\mathcal{L}}_{\text{ham}}-\frac{{\it\gamma}}{{\it\beta}}\mathop{\sum }_{i=1}^{d}\partial _{p_{i}}^{\ast }\partial _{p_{i}},\end{eqnarray}$$

where $\partial _{p_{i}}^{\ast }={\it\beta}(M^{-1}p)_{i}-\partial _{p_{i}}$. This decomposition highlights an important structural property of Langevin generators: they are the sum of a Hamiltonian part, and a symmetric dissipation operator which is degenerate, however, since it acts only on the momenta. A simple computation shows that the Hamiltonian part is antisymmetric on $L^{2}({\it\mu})$, so that the adjoint of ${\mathcal{L}}$ on $L^{2}({\it\mu})$ reads

$$\begin{eqnarray}{\mathcal{L}}^{\ast }=-{\mathcal{L}}_{\text{ham}}-\frac{{\it\gamma}}{{\it\beta}}\mathop{\sum }_{i=1}^{d}\partial _{p_{i}}^{\ast }\partial _{p_{i}}.\end{eqnarray}$$

The Fokker–Planck equation (2.1)–(2.3) can be reformulated in terms of this operator, upon writing ${\it\psi}(t)=f(t)\,{\it\mu}$, as

(2.36)$$\begin{eqnarray}\partial _{t}f={\mathcal{L}}^{\ast }f,\end{eqnarray}$$

where the initial condition $f(0)=f_{0}={\it\psi}_{0}/{\it\mu}$ satisfies

$$\begin{eqnarray}f_{0}\geqslant 0,\quad \int _{{\mathcal{E}}}f_{0}\,\text{d}{\it\mu}=1.\end{eqnarray}$$

Note that there is a slight inconsistency in this rewriting: the initial condition $f_{0}$ should be considered in $L^{1}({\it\mu})$, whereas the functional framework considered here requires the stronger integrability condition $f_{0}\in L^{2}({\it\mu})$. This situation can be improved by using relative entropies: see Proposition 2.22 below.

In any case, $\text{e}^{t{\mathcal{L}}^{\ast }}=(\text{e}^{t{\mathcal{L}}})^{\ast }$, so decay estimates obtained for the bounded operator $\text{e}^{t{\mathcal{L}}}$ immediately transfer to its adjoint. For simplicity of notation, the convergence results in Theorems 2.15, 2.18 and Proposition 2.20 are stated for the semigroup $\text{e}^{t{\mathcal{L}}}$, but the reader should bear in mind that the results equally apply to the semigroup $\text{e}^{t{\mathcal{L}}^{\ast }}$, which then yields results on the long-time behaviour of the Fokker–Planck equation by (2.36).

Note also that solutions to the Fokker–Planck equation (2.36) are expected to converge to the constant function $\mathbf{1}$. Upon subtracting this constant function from the initial condition $f_{0}$, the convergence of the law amounts to the convergence to 0 of $\text{e}^{t{\mathcal{L}}^{\ast }}(f_{0}-\mathbf{1})$. This motivates the introduction of the following functional space:

$$\begin{eqnarray}L_{0}^{2}({\it\mu})=\biggl\{{\it\varphi}\in L^{2}({\it\mu})\,\bigg|\,\int _{{\mathcal{E}}}{\it\varphi}\,\text{d}{\it\mu}=0\biggr\}.\end{eqnarray}$$

The Langevin generator ${\mathcal{L}}$ defined in (2.34), however, fails to be coercive on $L_{0}^{2}({\it\mu})$ since second derivatives in $q$ are missing. In fact, for $C^{\infty }$ and compactly supported test functions ${\it\varphi}$,

(2.37)$$\begin{eqnarray}-\langle {\mathcal{L}}{\it\varphi},{\it\varphi}\rangle _{L^{2}({\it\mu})}=\frac{1}{{\it\beta}}\Vert {\rm\nabla}_{p}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2},\end{eqnarray}$$

which should be compared to (2.19) for overdamped Langevin dynamics. The key idea of hypocoercivity is to introduce some mixed derivatives in $q$ and $p$ in a modified scalar product in order to retrieve some dissipation in $q$ through some commutator identities. This motivates the name for the technique, in view of the analogy with hypoellipticity, which allows us, via certain commutator identities, to state results for non-elliptic operators that are similar to regularity results obtained for elliptic operators.

More precisely, coercivity is obtained in the Hilbert space $H^{1}({\it\mu})\cap L_{0}^{2}({\it\mu})$, where

$$\begin{eqnarray}H^{1}({\it\mu})=\{{\it\varphi}\in L^{2}({\it\mu})\mid {\rm\nabla}_{p}{\it\varphi},{\rm\nabla}_{q}{\it\varphi}\in (L^{2}({\it\mu}))^{d}\}\end{eqnarray}$$

is endowed with a scalar product different from (but equivalent to) the canonical one:

$$\begin{eqnarray}\langle {\it\varphi}_{1},{\it\varphi}_{2}\rangle _{H^{1}({\it\mu})}=\langle {\it\varphi}_{1},{\it\varphi}_{2}\rangle _{L^{2}({\it\mu})}+\langle {\rm\nabla}_{q}{\it\varphi}_{1},{\rm\nabla}_{q}{\it\varphi}_{2}\rangle _{L^{2}({\it\mu})}+\langle {\rm\nabla}_{p}{\it\varphi}_{1},{\rm\nabla}_{p}{\it\varphi}_{2}\rangle _{L^{2}({\it\mu})}.\end{eqnarray}$$

It is then possible to resort to some Gronwall estimates and deduce the longtime convergence of $f(t)$ when a Poincaré inequality holds for ${\it\mu}$ (as in the proof of Proposition 2.3).

The idea of using mixed derivatives was already present in the computations performed in Talay (Reference Talay2002, Section 3), and was later generalized in Villani (Reference Villani2009). A careful application of the general hypocoercivity framework to Langevin dynamics can be read in Hairer and Pavliotis (Reference Hairer and Pavliotis2008), where some emphasis is laid on the limiting regime ${\it\gamma}\rightarrow 0$; see also Leimkuhler, Matthews and Stoltz (Reference Leimkuhler, Matthews and Stoltz2015) for the regime ${\it\gamma}\rightarrow +\infty$.

2.3.3 Estimates in $H^{1}({\it\mu})$

Although hypocoercivity eventually provides decay estimates in $H^{1}({\it\mu})$, it turns out to be convenient, for the proof, to work with a specific scalar product equivalent to the canonical scalar product on $H^{1}({\it\mu})$. We introduce the following scalar product:

(2.38)$$\begin{eqnarray}\langle \!\langle u,v\rangle \!\rangle =\langle u,v\rangle +a\langle {\rm\nabla}_{p}u,{\rm\nabla}_{p}v\rangle -b\langle {\rm\nabla}_{p}u,{\rm\nabla}_{q}v\rangle -b\langle {\rm\nabla}_{q}u,{\rm\nabla}_{p}v\rangle +c\langle {\rm\nabla}_{q}u,{\rm\nabla}_{q}v\rangle ,\end{eqnarray}$$

where, for simplicity of notation, we let $\langle \cdot ,\cdot \rangle$ denote the standard scalar product on $L^{2}({\it\mu})$. In order for the above scalar product to be equivalent to the canonical scalar product on $H^{1}({\it\mu})$, we assume in the remainder of this section that

(2.39)$$\begin{eqnarray}a,c>0\quad \text{and}\quad ac-b^{2}>0.\end{eqnarray}$$

Proof. Of course,

$$\begin{eqnarray}|\langle \!\langle u,u\rangle \!\rangle |\leqslant \max (1,a+|b|,c+|b|)\Vert u\Vert _{H^{1}({\it\mu})}^{2}.\end{eqnarray}$$

It therefore remains to prove that $\Vert u\Vert _{H^{1}({\it\mu})}^{2}$ can be controlled by $\langle \!\langle u,u\rangle \!\rangle$. Note first that, by a Cauchy–Schwarz inequality,

$$\begin{eqnarray}\displaystyle & & \displaystyle a\langle {\rm\nabla}_{p}u,{\rm\nabla}_{p}u\rangle -b\langle {\rm\nabla}_{p}u,{\rm\nabla}_{q}u\rangle -b\langle {\rm\nabla}_{q}u,{\rm\nabla}_{p}u\rangle +c\langle {\rm\nabla}_{q}u,{\rm\nabla}_{q}u\rangle \nonumber\\ \displaystyle & & \displaystyle \qquad \geqslant \left(\begin{array}{@{}c@{}}\Vert {\rm\nabla}_{p}u\Vert _{L^{2}({\it\mu})}\\ \Vert {\rm\nabla}_{q}u\Vert _{L^{2}({\it\mu})}\end{array}\right)^{T}\left(\begin{array}{@{}cc@{}}a & -|b|\\ -|b| & c\end{array}\right)\left(\begin{array}{@{}c@{}}\Vert {\rm\nabla}_{p}u\Vert _{L^{2}({\it\mu})}\\ \Vert {\rm\nabla}_{q}u\Vert _{L^{2}({\it\mu})}\end{array}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad \geqslant {\it\alpha}(\Vert {\rm\nabla}_{p}u\Vert _{L^{2}({\it\mu})}^{2}+\Vert {\rm\nabla}_{q}u\Vert _{L^{2}({\it\mu})}^{2}),\nonumber\end{eqnarray}$$

where

(2.40)$$\begin{eqnarray}{\it\alpha}=\frac{1}{2}(a+c-\sqrt{(a-c)^{2}+4b^{2}})=\displaystyle \frac{2(ac-b^{2})}{a+c+\sqrt{(a-c)^{2}+4b^{2}}}>0.\end{eqnarray}$$

This shows that

$$\begin{eqnarray}\langle \!\langle u,u\rangle \!\rangle \geqslant \min (1,{\it\alpha})\Vert u\Vert _{H^{1}({\it\mu})}^{2},\end{eqnarray}$$

which allows us to conclude the proof.

As mentioned above, for notational simplicity, we study the convergence to 0 of $\text{e}^{t{\mathcal{L}}}{\it\varphi}$ for ${\it\varphi}\in L_{0}^{2}({\it\mu})$ rather than the convergence to 0 of $\text{e}^{t{\mathcal{L}}^{\ast }}(f_{0}-1)$ for $f_{0}\in L^{2}({\it\mu})$. However, the results of this section can be straightforwardly extended to the latter case by changing the sign of the antisymmetric part of the operator (which is handled by changing the sign of $b$ in the definition of the scalar product $\langle \!\langle \cdot ,\cdot \rangle \!\rangle$). The following result can then be stated.

Theorem 2.15 (hypocoercivity).

Fix ${\it\gamma}>0$, and assume either that the domain ${\mathcal{D}}$ is bounded, or that there exists ${\it\rho}>0$ such that

(2.41)$$\begin{eqnarray}|{\rm\nabla}^{2}V(q)|\leqslant {\it\rho}(1+|{\rm\nabla}V(q)|),\quad \text{for all }q\in {\mathcal{D}},\end{eqnarray}$$

when ${\mathcal{D}}$ is not bounded. Then there exist $a,b,c\in \mathbb{R}$ satisfying (2.39) and $K>0$ such that, for any ${\it\varphi}\in H^{1}({\it\mu})\cap L_{0}^{2}({\it\mu})$ and any $t\geqslant 0$,

(2.42)$$\begin{eqnarray}\frac{\text{d}}{\text{d}t}[\langle \!\langle \text{e}^{t{\mathcal{L}}}{\it\varphi},\,\text{e}^{t{\mathcal{L}}}{\it\varphi}\rangle \!\rangle ]\leqslant -K(\Vert {\rm\nabla}_{p}\text{e}^{t{\mathcal{L}}}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}+\Vert {\rm\nabla}_{q}\text{e}^{t{\mathcal{L}}}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}).\end{eqnarray}$$

Further, if in addition a Poincaré inequality holds for the measure ${\it\nu}(\text{d}q)=Z_{{\it\nu}}^{-1}\,\text{e}^{-{\it\beta}V(q)}\,\text{d}q$, then there exists ${\it\kappa}>0$ such that, for any $t\geqslant 0$,

(2.43)$$\begin{eqnarray}\langle \!\langle \text{e}^{t{\mathcal{L}}}{\it\varphi},\,\text{e}^{t{\mathcal{L}}}{\it\varphi}\rangle \!\rangle \leqslant \text{e}^{-2{\it\kappa}t}\langle \!\langle {\it\varphi},{\it\varphi}\rangle \!\rangle ,\quad \text{for all }{\it\varphi}\in H^{1}({\it\mu})\cap L_{0}^{2}({\it\mu}).\end{eqnarray}$$

As a consequence, there exists $C\geqslant 1$ such that

(2.44)$$\begin{eqnarray}\Vert \text{e}^{t{\mathcal{L}}}\Vert _{{\mathcal{B}}(H^{1}({\it\mu})\cap L_{0}^{2}({\it\mu}))}\leqslant C\,\text{e}^{-{\it\kappa}t},\quad \text{for all }t\geqslant 0.\end{eqnarray}$$

In the last inequality, $H^{1}({\it\mu})$ is endowed with the canonical scalar product. The operator bound (2.44) is obtained from (2.43) by taking the supremum over functions ${\it\varphi}\in H^{1}({\it\mu})\cap L_{0}^{2}({\it\mu})$ and using the equivalence of norms provided by Lemma 2.14. Note that this implies that $C\geqslant 1$.

An immediate consequence of the above convergence result is the following corollary (obtained from Proposition 2.1 with $E=H^{1}({\it\mu})\cap L_{0}^{2}({\it\mu})$).

Corollary 2.16. Under the same assumptions as in Theorem 2.15, the operator ${\mathcal{L}}$ is invertible on $H^{1}({\it\mu})\cap L_{0}^{2}({\it\mu})$, and the following equality holds in ${\mathcal{B}}(H^{1}({\it\mu})\cap L_{0}^{2}({\it\mu}))$:

$$\begin{eqnarray}{\mathcal{L}}^{-1}=-\int _{0}^{+\infty }\,\text{e}^{t{\mathcal{L}}}\,\text{d}t.\end{eqnarray}$$

Moreover,

(2.45)$$\begin{eqnarray}\Vert {\mathcal{L}}^{-1}\Vert _{{\mathcal{B}}(H^{1}({\it\mu})\cap L_{0}^{2}({\it\mu}))}\leqslant \frac{C}{{\it\kappa}},\end{eqnarray}$$

where $C$ and ${\it\kappa}$ are the same constants as in (2.44).

Let us now present the proof of Theorem 2.15.

Proof. Fix a $C^{\infty }$ function ${\it\varphi}\in H^{1}({\it\mu})\cap L_{0}^{2}({\it\mu})$ (the general case being obtained by a density argument). Note that

$$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\biggl(\displaystyle \frac{1}{2}\langle \!\langle \text{e}^{t{\mathcal{L}}}{\it\varphi},\,\text{e}^{t{\mathcal{L}}}{\it\varphi}\rangle \!\rangle \biggr)=\langle \!\langle \text{e}^{t{\mathcal{L}}}{\it\varphi},{\mathcal{L}}\,\text{e}^{t{\mathcal{L}}}{\it\varphi}\rangle \!\rangle .\end{eqnarray}$$

Our aim is to find non-negative constants $a,b,c$ such that (2.39) holds and, for all $C^{\infty }$ functions ${\it\varphi}$ with compact support,

$$\begin{eqnarray}\langle \!\langle {\it\varphi},{\mathcal{L}}{\it\varphi}\rangle \!\rangle \leqslant -{\it\kappa}\langle \!\langle {\it\varphi},{\it\varphi}\rangle \!\rangle .\end{eqnarray}$$

This allows us to obtain (2.42) and thus the desired exponential decrease (2.43), using the Poincaré inequality and the Gronwall lemma. The idea of hypocoercivity is thus to find a scalar product equivalent to the canonical $H^{1}({\it\mu})$ scalar product and such that $-{\mathcal{L}}$ is coercive with respect to this scalar product (while it fails to be coercive with respect to the canonical $L^{2}({\it\mu})$ scalar product (see (2.37)), as well as with respect to the canonical $H^{1}({\it\mu})$ scalar product).

Let us first compute the various terms in $\langle \!\langle {\it\varphi},{\mathcal{L}}{\it\varphi}\rangle \!\rangle$. It is useful to first establish some commutator identities:

$$\begin{eqnarray}\displaystyle [\partial _{p_{i}},{\mathcal{L}}]{\it\varphi} & = & \displaystyle \frac{1}{m_{i}}(\partial _{q_{i}}-{\it\gamma}\partial _{p_{i}}){\it\varphi},\nonumber\\ \displaystyle [\partial _{q_{i}},{\mathcal{L}}]{\it\varphi} & = & \displaystyle -{\rm\nabla}(\partial _{q_{i}}V)^{T}{\rm\nabla}_{p}{\it\varphi}.\nonumber\end{eqnarray}$$

Some straightforward computations then show that

$$\begin{eqnarray}\displaystyle & & \displaystyle \langle \!\langle {\it\varphi},{\mathcal{L}}{\it\varphi}\rangle \!\rangle =-\frac{{\it\gamma}}{{\it\beta}}\Vert {\rm\nabla}_{p}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}\nonumber\\ \displaystyle & & \displaystyle \qquad +a\mathop{\sum }_{i=1}^{d}\biggl(-\frac{{\it\gamma}}{{\it\beta}}\Vert {\rm\nabla}_{p}(\partial _{p_{i}}{\it\varphi})\Vert _{L^{2}({\it\mu})}^{2}-\frac{{\it\gamma}}{m_{i}}\Vert \partial _{p_{i}}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}+\frac{1}{m_{i}}\langle \partial _{p_{i}}{\it\varphi},\partial _{q_{i}}{\it\varphi}\rangle \biggr)\nonumber\\ \displaystyle & & \displaystyle \qquad +c\mathop{\sum }_{i=1}^{d}\biggl(-\frac{{\it\gamma}}{{\it\beta}}\Vert {\rm\nabla}_{p}(\partial _{q_{i}}{\it\varphi})\Vert _{L^{2}({\it\mu})}^{2}-\langle \partial _{q_{i}}{\it\varphi},{\rm\nabla}(\partial _{q_{i}}V)^{T}{\rm\nabla}_{p}{\it\varphi}\rangle \biggr)\nonumber\\ \displaystyle & & \displaystyle \qquad -b\mathop{\sum }_{i=1}^{d}\biggl(-\frac{{\it\gamma}}{{\it\beta}}\langle {\rm\nabla}_{p}(\partial _{p_{i}}{\it\varphi}),{\rm\nabla}_{p}(\partial _{q_{i}}{\it\varphi})\rangle -\langle \partial _{p_{i}}{\it\varphi},{\rm\nabla}(\partial _{q_{i}}V)^{T}{\rm\nabla}_{p}{\it\varphi}\rangle \biggr)\nonumber\\ \displaystyle & & \displaystyle \qquad -b\mathop{\sum }_{i=1}^{d}\biggl(-\frac{{\it\gamma}}{{\it\beta}}\langle {\rm\nabla}_{p}(\partial _{q_{i}}{\it\varphi}),{\rm\nabla}_{p}(\partial _{p_{i}}{\it\varphi})\rangle +\frac{1}{m_{i}}\Vert \partial _{q_{i}}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}-\frac{{\it\gamma}}{m_{i}}\langle \partial _{q_{i}}{\it\varphi},\partial _{p_{i}}{\it\varphi}\rangle \biggr).\nonumber\end{eqnarray}$$

Note in particular that dissipation terms $\Vert \partial _{q_{i}}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}$ appear in the last line. This motivates choosing the parameter $b$ positive.Footnote ⁶ The next step is to bound the right-hand side of the previous equality from above using Cauchy–Schwarz inequalities, by (recall that we assume $b\geqslant 0$)

$$\begin{eqnarray}\displaystyle \langle \!\langle {\it\varphi},{\mathcal{L}}{\it\varphi}\rangle \!\rangle & {\leqslant} & \displaystyle -{\it\gamma}\biggl(\displaystyle \frac{1}{{\it\beta}}+\frac{a}{m_{+}}\biggr)\Vert {\rm\nabla}_{p}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}-\frac{b}{m_{+}}\Vert {\rm\nabla}_{q}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}\nonumber\\ \displaystyle & & \displaystyle +\frac{a+b{\it\gamma}}{m_{-}}\Vert {\rm\nabla}_{p}{\it\varphi}\Vert _{L^{2}({\it\mu})}\Vert {\rm\nabla}_{q}{\it\varphi}\Vert _{L^{2}({\it\mu})}\nonumber\\ \displaystyle & & \displaystyle -\frac{a{\it\gamma}}{{\it\beta}}\mathop{\sum }_{i=1}^{d}\Vert {\rm\nabla}_{p}(\partial _{p_{i}}{\it\varphi})\Vert _{L^{2}({\it\mu})}^{2}-\frac{c{\it\gamma}}{{\it\beta}}\mathop{\sum }_{i=1}^{d}\Vert {\rm\nabla}_{p}(\partial _{q_{i}}{\it\varphi})\Vert _{L^{2}({\it\mu})}^{2}\nonumber\\ \displaystyle & & \displaystyle +\frac{2b{\it\gamma}}{{\it\beta}}\mathop{\sum }_{i=1}^{d}\Vert {\rm\nabla}_{p}(\partial _{p_{i}}{\it\varphi})\Vert _{L^{2}({\it\mu})}\Vert {\rm\nabla}_{p}(\partial _{q_{i}}{\it\varphi})\Vert _{L^{2}({\it\mu})}\nonumber\\ \displaystyle & & \displaystyle +\langle (b{\rm\nabla}_{p}-c{\rm\nabla}_{q}){\it\varphi},({\rm\nabla}^{2}V){\rm\nabla}_{p}{\it\varphi}\rangle ,\nonumber\end{eqnarray}$$

where

(2.46)$$\begin{eqnarray}m_{+}=\max (m_{1},\ldots ,m_{d}),\quad m_{-}=\min (m_{1},\ldots ,m_{d}).\end{eqnarray}$$

Condition (2.39) shows that (by a computation similar to that performed in the proof of Lemma 2.14)

(2.47)$$\begin{eqnarray}\displaystyle & & \displaystyle a\Vert {\rm\nabla}_{p}(\partial _{p_{i}}{\it\varphi})\Vert _{L^{2}({\it\mu})}^{2}+c\Vert {\rm\nabla}_{p}(\partial _{q_{i}}{\it\varphi})\Vert _{L^{2}({\it\mu})}^{2}-2b\Vert {\rm\nabla}_{p}(\partial _{p_{i}}{\it\varphi})\Vert _{L^{2}({\it\mu})}\Vert {\rm\nabla}_{p}(\partial _{q_{i}}{\it\varphi})\Vert _{L^{2}({\it\mu})}\nonumber\\ \displaystyle & & \displaystyle \quad \;\geqslant {\it\alpha}(\Vert {\rm\nabla}_{p}(\partial _{p_{i}}{\it\varphi})\Vert _{L^{2}({\it\mu})}^{2}+\Vert {\rm\nabla}_{p}(\partial _{q_{i}}{\it\varphi})\Vert _{L^{2}({\it\mu})}^{2}),\end{eqnarray}$$

with ${\it\alpha}>0$ defined in (2.40). Therefore, the upper bound on $\langle \!\langle {\it\varphi},{\mathcal{L}}{\it\varphi}\rangle \!\rangle$ simplifies as

(2.48)$$\begin{eqnarray}\displaystyle \langle \!\langle {\it\varphi},{\mathcal{L}}{\it\varphi}\rangle \!\rangle & {\leqslant} & \displaystyle -{\it\gamma}\biggl(\displaystyle \frac{1}{{\it\beta}}+\frac{a}{m_{+}}\biggr)\Vert {\rm\nabla}_{p}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}-\frac{b}{m_{+}}\Vert {\rm\nabla}_{q}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}\nonumber\\ \displaystyle & & \displaystyle \;\;+\frac{a+b{\it\gamma}}{m_{-}}\Vert {\rm\nabla}_{p}{\it\varphi}\Vert _{L^{2}({\it\mu})}\Vert {\rm\nabla}_{q}{\it\varphi}\Vert _{L^{2}({\it\mu})}\nonumber\\ \displaystyle & & \displaystyle \;\;-\frac{{\it\alpha}{\it\gamma}}{{\it\beta}}\mathop{\sum }_{i=1}^{d}(\Vert {\rm\nabla}_{p}(\partial _{p_{i}}{\it\varphi})\Vert _{L^{2}({\it\mu})}^{2}+\Vert {\rm\nabla}_{p}(\partial _{q_{i}}{\it\varphi})\Vert _{L^{2}({\it\mu})}^{2})\nonumber\\ \displaystyle & & \displaystyle \;\;+\langle (b{\rm\nabla}_{p}-c{\rm\nabla}_{q}){\it\varphi},({\rm\nabla}^{2}V){\rm\nabla}_{p}{\it\varphi}\rangle .\end{eqnarray}$$

In order to control the term $\langle (b{\rm\nabla}_{p}-c{\rm\nabla}_{q}){\it\varphi},({\rm\nabla}^{2}V){\rm\nabla}_{p}{\it\varphi}\rangle$, two cases have to be distinguished.

1. (i) The position space ${\mathcal{D}}$ is compact. Here ${\rm\nabla}^{2}V$ is uniformly bounded, and there exists a constant $C_{V}>0$ such that

   $$\begin{eqnarray}\displaystyle & & \displaystyle |\langle (b{\rm\nabla}_{p}-c{\rm\nabla}_{q}){\it\varphi},({\rm\nabla}^{2}V){\rm\nabla}_{p}{\it\varphi}\rangle |\leqslant C_{V}\Vert {\rm\nabla}_{p}{\it\varphi}\Vert _{L^{2}({\it\mu})}\Vert (b{\rm\nabla}_{p}-c{\rm\nabla}_{q}){\it\varphi}\Vert _{L^{2}({\it\mu})}\nonumber\\ \displaystyle & & \displaystyle \qquad \leqslant bC_{V}\Vert {\rm\nabla}_{p}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}+cC_{V}\Vert {\rm\nabla}_{p}{\it\varphi}\Vert _{L^{2}({\it\mu})}\Vert {\rm\nabla}_{q}{\it\varphi}\Vert _{L^{2}({\it\mu})}.\nonumber\end{eqnarray}$$

2. (ii) The position space is not compact. Conditions must be imposed on the potential energy function in order to control the growth of the Hessian at infinity. One possible condition is that ${\rm\nabla}^{2}V$ (considered as a multiplication operator acting on vectors) is relatively bounded by ${\rm\nabla}_{q}$ on $L^{2}({\it\mu})$, that is, there exist $A_{V},B_{V}\geqslant 0$ such that

   $$\begin{eqnarray}\forall {\it\phi}\in H^{1}({\it\mu}),\quad \Vert {\it\phi}{\rm\nabla}^{2}V\Vert _{L^{2}({\it\mu})}\leqslant A_{V}\Vert {\it\phi}\Vert _{L^{2}({\it\mu})}+B_{V}\Vert {\rm\nabla}_{q}{\it\phi}\Vert _{L^{2}({\it\mu})}.\end{eqnarray}$$
   This condition is satisfied when (2.41) holds (see Lemma A.24 of Villani Reference Villani2009). In this case,
   $$\begin{eqnarray}\displaystyle & & \displaystyle |\langle (b{\rm\nabla}_{p}-c{\rm\nabla}_{q}){\it\varphi},({\rm\nabla}^{2}V){\rm\nabla}_{p}{\it\varphi}\rangle |\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant (A_{V}\Vert {\rm\nabla}_{p}{\it\varphi}\Vert _{L^{2}({\it\mu})}+B_{V}\Vert {\rm\nabla}_{p}{\rm\nabla}_{q}{\it\varphi}\Vert _{L^{2}({\it\mu})})\Vert (b{\rm\nabla}_{p}-c{\rm\nabla}_{q}){\it\varphi}\Vert _{L^{2}({\it\mu})}.\nonumber\end{eqnarray}$$
   The term involving derivatives in both $q$ and $p$ can be controlled by the dissipative terms $-\frac{{\it\alpha}{\it\gamma}}{{\it\beta}}\Vert \partial _{p_{j}}\partial _{q_{i}}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}$ in (2.48).

For simplicity, we consider the case when ${\mathcal{D}}$ is compact. Then,

$$\begin{eqnarray}\displaystyle \langle \!\langle {\it\varphi},{\mathcal{L}}{\it\varphi}\rangle \!\rangle & {\leqslant} & \displaystyle \biggl[bC_{V}-{\it\gamma}\biggl(\displaystyle \frac{1}{{\it\beta}}+\frac{a}{m_{+}}\biggr)\biggr]\Vert {\rm\nabla}_{p}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}-\frac{b}{m_{+}}\Vert {\rm\nabla}_{q}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}\nonumber\\ \displaystyle & & \displaystyle \quad +\biggl(\displaystyle \frac{a+b{\it\gamma}}{m_{-}}+cC_{V}\biggr)\Vert {\rm\nabla}_{p}{\it\varphi}\Vert _{L^{2}({\it\mu})}\Vert {\rm\nabla}_{q}{\it\varphi}\Vert _{L^{2}({\it\mu})}\nonumber\\ \displaystyle & = & \displaystyle -X^{T}{\mathcal{M}}X,\nonumber\end{eqnarray}$$

where

$$\begin{eqnarray}X=\left(\begin{array}{@{}c@{}}\Vert {\rm\nabla}_{p}{\it\varphi}\Vert _{L^{2}({\it\mu})}\\ \Vert {\rm\nabla}_{q}{\it\varphi}\Vert _{L^{2}({\it\mu})}\end{array}\right)\!,\quad {\mathcal{M}}=\left(\begin{array}{@{}cc@{}}A & C/2\\ C/2 & B\end{array}\right)\!,\end{eqnarray}$$

with

$$\begin{eqnarray}A={\it\gamma}\biggl(\displaystyle \frac{1}{{\it\beta}}+\frac{a}{m_{+}}\biggr)-bC_{V},\quad B=\frac{b}{m_{+}},\quad C=\frac{a+b{\it\gamma}}{m_{-}}+cC_{V}.\end{eqnarray}$$

In addition to (2.39), the values of $a,b,c$ should be such that

(2.49)$$\begin{eqnarray}4AB>C^{2},\end{eqnarray}$$

in order for ${\mathcal{M}}$ to be positive definite. If this is the case, (2.42) follows. The condition (2.49) is satisfied for instance when

(2.50)$$\begin{eqnarray}a=c={\it\varepsilon},\quad b={\it\varepsilon}^{1+{\it\delta}},\quad {\it\delta}\in (0,1),\end{eqnarray}$$

for ${\it\varepsilon}>0$ sufficiently small. Note also that this choice is compatible with (2.39) when ${\it\varepsilon}<1$.

Let us now explain how to get (2.43) from (2.42), assuming that ${\it\nu}$ satisfies a Poincaré inequality. This requires us to retrieve some control on the $L^{2}({\it\mu})$-norm of ${\it\varphi}$ from the norms of the gradient. We use to this end the fact that, by the tensorization argument stated in Proposition 2.6, the canonical measure ${\it\mu}$ satisfies a Poincaré inequality, whose constant we denote by $R$. Therefore, $X^{T}SX\geqslant \langle \!\langle {\it\varphi},{\it\varphi}\rangle \!\rangle$ with

$$\begin{eqnarray}S=\left(\begin{array}{@{}cc@{}}a+1/(2R) & -b\\ -b & c+1/(2R)\end{array}\right)\!.\end{eqnarray}$$

We finally define ${\it\kappa}$ as the largest positive constant such that $X^{T}{\mathcal{M}}X\geqslant {\it\kappa}X^{T}SX$. In fact, ${\it\kappa}$ is the smallest eigenvalue of $S^{-1/2}{\mathcal{M}}S^{-1/2}$ (or equivalently of ${\mathcal{M}}S^{-1}$). Since $S$ is symmetric positive definite, this shows that ${\it\kappa}>0$ since ${\mathcal{M}}$ is also positive definite thanks to (2.50). This concludes the proof of (2.43).

One interest of the hypocoercive approach is that the constants ${\it\kappa}$ and $C$ in (2.44) can be made quite explicit in terms of the various factors (related to the potential such as the bound on the Hessian $C_{V}$ and the Poincaré constant $R$, or to the masses). Let us illustrate this for various asymptotic cases.

1. (i) ${\it\beta}\rightarrow +\infty$. The constant $R$ typically decreases exponentially with the temperature, with a lower bound scaling as $\text{e}^{{\it\beta}(\inf \widetilde{V}-\sup \widetilde{V})}$ when $V=V_{\text{convex}}+\widetilde{V}$. In this decomposition, the potential is separated into a strongly convex part $V_{\text{convex}}$ whose Hessian is uniformly lower bounded by a positive constant while $\widetilde{V}$ is some bounded perturbation. On the other hand, upon rescaling the values $a,b,c$ in the definition of the scalar product by a factor $1/{\it\beta}$ (which amounts to considering $a=\overline{a}/{\it\beta}$, etc.), it can be shown that the smallest eigenvalue ${\it\alpha}$ of ${\mathcal{M}}$ is of order $1/{\it\beta}$. Therefore, the smallest eigenvalue ${\it\kappa}$ of $S^{-1/2}{\mathcal{M}}S^{-1/2}$ admits a lower bound which decreases exponentially with ${\it\beta}$.

2. (ii) Hamiltonian limit ${\it\gamma}\rightarrow 0$ or overdamped limit ${\it\gamma}\rightarrow +\infty$. The results stated in Theorem 2.15 are obtained for a given value of ${\it\gamma}$. On the other hand, the Langevin dynamics becomes singular in the limit ${\it\gamma}\rightarrow 0$, where it reduces to the Hamiltonian dynamics (which is not ergodic with respect to the canonical measure), and in the limit ${\it\gamma}\rightarrow +\infty$, where it converges to the overdamped Langevin dynamics (see the discussion after (1.9)). It is therefore expected that the convergence rate to equilibrium of the Langevin dynamics degrades as ${\it\gamma}\rightarrow 0$ or ${\it\gamma}\rightarrow +\infty$.

   Hypocoercivity allows us to quantify the degradation of the convergence rate. Let ${\it\kappa}_{{\it\gamma}}$ denote the exponential decay rate given by Theorem 2.15 for a given value of ${\it\gamma}$. Upon choosing $(a,b,c)={\it\gamma}(\overline{a},\overline{b},\overline{c})$ when ${\it\gamma}\rightarrow 0$ and $(a,b,c)=(\overline{a},\overline{b},\overline{c})/{\it\gamma}$ when ${\it\gamma}\rightarrow +\infty$, a careful analysis of the scaling of ${\it\kappa}_{{\it\gamma}}$ shows that there exists $\overline{{\it\kappa}}>0$ such that

   (2.51)$$\begin{eqnarray}{\it\kappa}_{{\it\gamma}}\geqslant \min \biggl({\it\gamma},\frac{1}{{\it\gamma}}\biggr)\overline{{\it\kappa}}.\end{eqnarray}$$
   The decay rate therefore becomes singular both in the Hamiltonian and overdamped limits. In both cases, the decay is apparent only at long time scales, of order $t/{\it\gamma}$ as ${\it\gamma}\rightarrow 0$ (the fluctuation/dissipation is so small that energy diffusion is only observed at long times: see Hairer and Pavliotis Reference Hairer and Pavliotis2008 for a precise statement) and ${\it\gamma}t$ as ${\it\gamma}\rightarrow +\infty$ (the fluctuation/dissipation is so large that the momenta are continuously randomized, which leads to some effective Brownian motion on the positions over long times). Note however that, for the limits considered here, the constant $C$ in (2.44) becomes singular. A more convenient formulation of the scaling limits ${\it\gamma}\rightarrow 0$ or ${\it\gamma}\rightarrow +\infty$ is provided by Proposition 2.20 below.

Remark 2.17 (degenerate scalar product).

The standard hypocoercive approach relies on estimates in $H^{1}({\it\mu})$, obtained under the non-degeneracy condition (2.39). However, in the situation when ${\rm\nabla}^{2}V$ is bounded, it is possible with a slight modification of the above argumentFootnote ⁷ to state an exponential convergence in the degenerate case $a=b=c$, for which the associated squared norm is

$$\begin{eqnarray}\Vert f\Vert _{L^{2}({\it\mu})}^{2}+a\Vert ({\rm\nabla}_{p}-{\rm\nabla}_{q})f\Vert _{L^{2}({\it\mu})}^{2}.\end{eqnarray}$$

2.3.4 Estimates in $L^{2}({\it\mu})$

The passage from bounds in $H^{1}({\it\mu})$ to bounds in $L^{2}({\it\mu})$ follows from hypoelliptic regularization results. Such results are presented in Villani (Reference Villani2009, Theorem A.8) or Hairer and Pavliotis (Reference Hairer and Pavliotis2008, Section 6.1). We follow the latter approach, which is more straightforward, although the so-obtained results are not as strong as the results presented in Villani (Reference Villani2009).

Theorem 2.18 (hypoelliptic regularization).

Assume ${\rm\nabla}^{2}V\in L^{\infty }({\mathcal{D}})$ or that (2.41) holds. Then there exists $K>0$ such that, for any ${\it\varphi}\in L^{2}({\it\mu})$,

$$\begin{eqnarray}\Vert {\rm\nabla}_{p}\text{e}^{t{\mathcal{L}}}{\it\varphi}\Vert _{L^{2}({\it\mu})}+\Vert {\rm\nabla}_{q}\text{e}^{t{\mathcal{L}}}{\it\varphi}\Vert _{L^{2}({\it\mu})}\leqslant \frac{K}{t^{3/2}}\Vert {\it\varphi}\Vert _{L^{2}({\it\mu})},\quad \text{for all }0<t\leqslant 1.\end{eqnarray}$$

Combining this inequality with $t=1$ and Theorem 2.15, we can conclude that, for $t\geqslant 1$ and ${\it\varphi}\in L_{0}^{2}({\it\mu})$,

$$\begin{eqnarray}\displaystyle \Vert \text{e}^{t{\mathcal{L}}}{\it\varphi}\Vert _{L^{2}({\it\mu})}^{2} & {\leqslant} & \displaystyle \langle \!\langle \text{e}^{t{\mathcal{L}}}{\it\varphi},\,\text{e}^{t{\mathcal{L}}}{\it\varphi}\rangle \!\rangle \nonumber\\ \displaystyle & {\leqslant} & \displaystyle \text{e}^{-2{\it\kappa}(t-1)}\langle \!\langle \text{e}^{{\mathcal{L}}}{\it\varphi},\,\text{e}^{{\mathcal{L}}}{\it\varphi}\rangle \!\rangle \nonumber\\ \displaystyle & {\leqslant} & \displaystyle \widetilde{C}\,\text{e}^{-2{\it\kappa}t}\Vert {\it\varphi}\Vert _{L^{2}({\it\mu})}^{2},\nonumber\end{eqnarray}$$

which gives an exponential decay in $L^{2}({\it\mu})$. For completeness, let us recall the proof of Theorem 2.18, as presented in Hairer and Pavliotis (Reference Hairer and Pavliotis2008).

Proof. As at the end of the proof of Theorem 2.15, we consider for simplicity the case when ${\rm\nabla}^{2}V$ is bounded. We denote $\text{e}^{t{\mathcal{L}}}{\it\varphi}$ by ${\it\varphi}(t)$ in this proof. Define

(2.52)$$\begin{eqnarray}\displaystyle N_{{\it\varphi}}(t) & = & \displaystyle \frac{1}{2}[\Vert {\it\varphi}(t)\Vert _{L^{2}({\it\mu})}^{2}+c_{1}t\Vert {\rm\nabla}_{p}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}^{2}\nonumber\\ \displaystyle & & \displaystyle \quad -c_{2}t^{2}\langle {\rm\nabla}_{q}{\it\varphi}(t),{\rm\nabla}_{p}{\it\varphi}(t)\rangle _{L^{2}({\it\mu})}+c_{3}t^{3}\Vert {\rm\nabla}_{q}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}^{2}],\end{eqnarray}$$

for some positive constants $c_{1},c_{2},c_{3}$ to be determined later on. Note that $2N_{{\it\varphi}}(t)$ corresponds to the norm induced by the scalar product (2.38) for time-dependent coefficients $a(t)=c_{1}t$, $2b(t)=c_{2}t^{2}$ and $c(t)=c_{3}t^{3}$. This choice is motivated in Remark 2.19 below. The time derivative of this quantity reads

$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{d}N_{{\it\varphi}}(t)}{\text{d}t}=\langle {\it\varphi}(t),{\mathcal{L}}{\it\varphi}(t)\rangle +\frac{c_{1}}{2}\Vert {\rm\nabla}_{p}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}^{2}+c_{1}t\mathop{\sum }_{i=1}^{d}\langle \partial _{p_{i}}{\it\varphi}(t),\partial _{p_{i}}{\mathcal{L}}{\it\varphi}(t)\rangle \nonumber\\ \displaystyle & & \displaystyle \qquad -c_{2}t\langle {\rm\nabla}_{q}{\it\varphi}(t),{\rm\nabla}_{p}{\it\varphi}(t)\rangle \nonumber\\ \displaystyle & & \displaystyle \qquad -\frac{c_{2}t^{2}}{2}\mathop{\sum }_{i=1}^{d}[\langle \partial _{q_{i}}{\it\varphi}(t),\partial _{p_{i}}{\mathcal{L}}{\it\varphi}(t)\rangle +\langle \partial _{p_{i}}{\it\varphi}(t),\partial _{q_{i}}{\mathcal{L}}{\it\varphi}(t)\rangle ]\nonumber\\ \displaystyle & & \displaystyle \qquad +\frac{3c_{3}t^{2}}{2}\Vert {\rm\nabla}_{q}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}^{2}+c_{3}t^{3}\mathop{\sum }_{i=1}^{d}\langle \partial _{q_{i}}{\it\varphi}(t),\partial _{q_{i}}{\mathcal{L}}{\it\varphi}(t)\rangle .\nonumber\end{eqnarray}$$

Using computations similar to those in the proof of Theorem 2.15, we obtain

(2.53)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{d}N_{{\it\varphi}}(t)}{\text{d}t}=-\biggl(\displaystyle \frac{{\it\gamma}}{{\it\beta}}-\frac{c_{1}}{2}\biggr)\Vert {\rm\nabla}_{p}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}^{2}\nonumber\\ \displaystyle & & \displaystyle \qquad -c_{1}t\mathop{\sum }_{i=1}^{d}\biggl[\frac{{\it\gamma}}{{\it\beta}}\Vert {\rm\nabla}_{p}(\partial _{p_{i}}{\it\varphi}(t))\Vert _{L^{2}({\it\mu})}^{2}+\frac{{\it\gamma}}{m_{i}}\Vert \partial _{p_{i}}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}^{2}\biggr]\nonumber\\ \displaystyle & & \displaystyle \qquad +\mathop{\sum }_{i=1}^{d}\biggl[t\biggl(\displaystyle \frac{c_{1}}{m_{i}}-c_{2}\biggr)+\frac{{\it\gamma}c_{2}t^{2}}{2m_{i}}\biggr]\langle \partial _{p_{i}}{\it\varphi}(t),\partial _{q_{i}}{\it\varphi}(t)\rangle \nonumber\\ \displaystyle & & \displaystyle \qquad +\frac{c_{2}t^{2}}{2}\mathop{\sum }_{i=1}^{d}\frac{2{\it\gamma}}{{\it\beta}}\langle {\rm\nabla}_{p}(\partial _{p_{i}}{\it\varphi}(t)),{\rm\nabla}_{p}(\partial _{q_{i}}{\it\varphi}(t))\rangle +\langle \partial _{p_{i}}{\it\varphi}(t),{\rm\nabla}(\partial _{q_{i}}V)^{T}{\rm\nabla}_{p}{\it\varphi}(t)\rangle \nonumber\\ \displaystyle & & \displaystyle \qquad +t^{2}\mathop{\sum }_{i=1}^{d}\biggl(\displaystyle \frac{3c_{3}}{2}-\frac{c_{2}}{m_{i}}\biggr)\Vert \partial _{q_{i}}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}^{2}\nonumber\\ \displaystyle & & \displaystyle \qquad -c_{3}t^{3}\mathop{\sum }_{i=1}^{d}\biggl[\frac{{\it\gamma}}{{\it\beta}}\Vert {\rm\nabla}_{p}(\partial _{q_{i}}{\it\varphi}(t))\Vert _{L^{2}({\it\mu})}^{2}+\langle \partial _{q_{i}}{\it\varphi}(t),{\rm\nabla}(\partial _{q_{i}}V)^{T}{\rm\nabla}_{p}{\it\varphi}(t)\rangle \biggr].\end{eqnarray}$$

We now choose $c_{1},c_{2},c_{3}$ such that $\text{d}N_{{\it\varphi}}/\text{d}t\leqslant 0$. First, we restrict ourselves to coefficients such that

$$\begin{eqnarray}c_{2}t^{2}xy\leqslant c_{1}tx^{2}+c_{3}t^{3}y^{2},\quad \text{for all }(x,y)\in \mathbb{R}^{2},\end{eqnarray}$$

which is satisfied once

(2.54)$$\begin{eqnarray}c_{1},c_{3}\geqslant 0,\quad 4c_{1}c_{3}\geqslant c_{2}^{2}.\end{eqnarray}$$

Then,

$$\begin{eqnarray}\displaystyle -c_{1}t\mathop{\sum }_{i=1}^{d}\frac{{\it\gamma}}{{\it\beta}}\Vert {\rm\nabla}_{p}(\partial _{p_{i}}{\it\varphi}(t))\Vert _{L^{2}({\it\mu})}^{2}+\frac{c_{2}t^{2}}{2}\mathop{\sum }_{i=1}^{d}\frac{2{\it\gamma}}{{\it\beta}}\langle {\rm\nabla}_{p}(\partial _{p_{i}}{\it\varphi}(t)),{\rm\nabla}_{p}(\partial _{q_{i}}{\it\varphi}(t))\rangle & & \displaystyle \nonumber\\ \displaystyle -c_{3}t^{3}\mathop{\sum }_{i=1}^{d}\frac{{\it\gamma}}{{\it\beta}}\Vert {\rm\nabla}_{p}(\partial _{q_{i}}{\it\varphi}(t))\Vert _{L^{2}({\it\mu})}^{2}\leqslant 0, & & \displaystyle \nonumber\end{eqnarray}$$

so that, with $C_{V}=\Vert {\rm\nabla}^{2}V\Vert _{L^{\infty }}$,

$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{d}N_{{\it\varphi}}(t)}{\text{d}t}\leqslant -\biggl(\displaystyle \frac{{\it\gamma}}{{\it\beta}}+\frac{{\it\gamma}c_{1}t}{m_{+}}-\frac{c_{1}}{2}-C_{V}\frac{c_{2}t^{2}}{2}\biggr)\Vert {\rm\nabla}_{p}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}^{2}\nonumber\\ \displaystyle & & \displaystyle \qquad -t^{2}\biggl(\displaystyle \frac{c_{2}}{m_{+}}-\frac{3c_{3}}{2}\biggr)\Vert {\rm\nabla}_{q}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}^{2}\nonumber\\ \displaystyle & & \displaystyle \qquad +\biggl[t\biggl(\displaystyle \frac{c_{1}}{m_{-}}-c_{2}\biggr)+\frac{{\it\gamma}c_{2}t^{2}}{2m_{-}}+C_{V}c_{3}t^{3}\biggr]\Vert {\rm\nabla}_{p}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}\Vert {\rm\nabla}_{q}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}.\nonumber\end{eqnarray}$$

On the time interval $[0,1]$, we finally obtain the following upper bound:

$$\begin{eqnarray}\displaystyle \frac{\text{d}N_{{\it\varphi}}(t)}{\text{d}t} & {\leqslant} & \displaystyle -\biggl(\displaystyle \frac{{\it\gamma}}{{\it\beta}}-\frac{c_{1}}{2}-C_{V}\frac{c_{2}}{2}\biggr)\Vert {\rm\nabla}_{p}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}^{2}\nonumber\\ \displaystyle & & \displaystyle \quad -t^{2}\biggl(\displaystyle \frac{c_{2}}{m_{+}}-\frac{3c_{3}}{2}\biggr)\Vert {\rm\nabla}_{q}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}^{2}\nonumber\\ \displaystyle & & \displaystyle \quad +t\biggl[\frac{c_{1}}{m_{-}}+\frac{{\it\gamma}c_{2}}{2m_{-}}+C_{V}c_{3}\biggr]\Vert {\rm\nabla}_{p}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}\Vert {\rm\nabla}_{q}{\it\varphi}(t)\Vert _{L^{2}({\it\mu})}.\nonumber\end{eqnarray}$$

We next consider coefficients $c_{3}\ll c_{2}\ll c_{1}\ll 1$ satisfying (2.54) and $c_{1}^{2}\ll c_{2}$. For instance, $c_{1}={\it\varepsilon}$, $c_{2}={\it\varepsilon}^{1+{\it\delta}}$ and $c_{3}={\it\varepsilon}^{1+{\it\delta}+{\it\eta}}$, with $0<{\it\eta}<{\it\delta}<1$ and for ${\it\varepsilon}>0$ sufficiently small. We can then deduce that $\text{d}N_{{\it\varphi}}(t)/\text{d}t\leqslant 0$, which implies $N_{{\it\varphi}}(t)\leqslant N_{{\it\varphi}}(0)=\Vert {\it\varphi}\Vert _{L^{2}({\it\mu})}^{2}/2$. The desired conclusion immediately follows.

By tracking the dependency of the constant $K$ on ${\it\gamma}$, Theorem 2.18 combined with Theorem 2.15 and the estimate (2.51) leads to the following result.

Proposition 2.20. Assume that ${\rm\nabla}^{2}V\in L^{\infty }({\mathcal{D}})$ and that a Poincaré inequality holds for ${\it\nu}$. Then there exist $C,\overline{{\it\kappa}}>0$ such that, for all ${\it\gamma}>0$, for all ${\it\varphi}\in L_{0}^{2}({\it\mu})$,

(2.55)$$\begin{eqnarray}\Vert \text{e}^{t{\mathcal{L}}}{\it\varphi}\Vert _{L^{2}({\it\mu})}\leqslant C\,\text{e}^{-\overline{{\it\kappa}}\min ({\it\gamma},{\it\gamma}^{-1})t}\Vert {\it\varphi}\Vert _{L^{2}({\it\mu})}.\end{eqnarray}$$

This result again emphasizes the fact that, in the limiting regimes of very low or very large frictions, the system should be considered over long times $\max ({\it\gamma},1/{\it\gamma})t$ in order to display a non-trivial behaviour.

Similarly to Corollary 2.16, the following result can be stated.

Corollary 2.21. Under the same assumptions as in Proposition 2.20, the operator ${\mathcal{L}}$ is invertible on $L_{0}^{2}({\it\mu})$, and the equality

$$\begin{eqnarray}{\mathcal{L}}^{-1}=-\int _{0}^{+\infty }\,\text{e}^{t{\mathcal{L}}}\,\text{d}t\end{eqnarray}$$

holds in ${\mathcal{B}}(L_{0}^{2}({\it\mu}))$. Moreover,

(2.56)$$\begin{eqnarray}\Vert {\mathcal{L}}^{-1}\Vert _{{\mathcal{B}}(L_{0}^{2}({\it\mu}))}\leqslant \frac{C}{\overline{{\it\kappa}}\min ({\it\gamma},{\it\gamma}^{-1})},\end{eqnarray}$$

where $C$ and $\overline{{\it\kappa}}$ are the same constants as in (2.55).

2.3.5 Hypocoercivity in the entropic sense

Here we show how the computations performed in $H^{1}({\it\mu})$ can be modified to prove the decay of some logarithmic entropy, similar to the one used for LSIs. The interest of this approach is that initial conditions in larger functional spaces can be considered. The general strategy is presented in Villani (Reference Villani2009, Section 6). These computations can be simplified for Langevin-type dynamics: see Olla and Letizia (Reference Olla and Letizia2015, Appendix D).

We state here convergence results for the function $f(t)$, which is the Radon–Nikodym derivative of the law ${\it\psi}(t)$ of the process at time $t$ with respect to the density of the invariant measure ${\it\mu}$. Starting from $f_{0}\geqslant 0$ with $\int _{{\mathcal{E}}}f_{0}\,\text{d}{\it\mu}=1$, the function $f(t)$ therefore evolves according to the Fokker–Planck equation (2.36), that is, $f(t)=\text{e}^{t{\mathcal{L}}^{\ast }}f_{0}$. It is expected that $f(t)$ converges to the constant function $\mathbf{1}$.

To quantify the convergence rate, we introduce the entropy functional

$$\begin{eqnarray}\mathscr{E}(f)=\int _{{\mathcal{E}}}f\,\ln f\,\text{d}{\it\mu}+\int _{{\mathcal{E}}}\frac{{\rm\nabla}f^{T}S{\rm\nabla}f}{f}\,\text{d}{\it\mu},\end{eqnarray}$$

where $S\in \mathbb{R}^{2d\times 2d}$ is a non-negative symmetric matrix. Note that the functional $\mathscr{E}$ combines the relative entropy and a generalization of the Fisher information introduced in Definition 2.7. It is sufficient for our purposes to restrict ourselves to the case when $S$ is constant (but see Villani Reference Villani2009, Remark 29 for a context where a dependence on the position $q$ may be useful). Finally, note that $\mathscr{E}(\mathbf{1})=0$, so that the entropy is expected to converge to 0 as $t\rightarrow +\infty$.

Proposition 2.22. Assume ${\rm\nabla}^{2}V\in L^{\infty }({\mathcal{D}})$. Then there exist a non-negative symmetric matrix $S\in \mathbb{R}^{2d\times 2d}$ and ${\it\alpha}>0$ such that

(2.57)$$\begin{eqnarray}\frac{\text{d}}{\text{d}t}[\mathscr{E}(f(t))]\leqslant -{\it\alpha}\int _{{\mathcal{E}}}\frac{|{\rm\nabla}f(t)|^{2}}{f(t)}\,\text{d}{\it\mu},\quad \text{for all }t\geqslant 0.\end{eqnarray}$$

If in addition ${\it\nu}(\text{d}q)$ satisfies a logarithmic Sobolev inequality, then there exists ${\it\kappa}>0$ such that

(2.58)$$\begin{eqnarray}\frac{\text{d}}{\text{d}t}[\mathscr{E}(f(t))]\leqslant -{\it\kappa}\,\mathscr{E}(f(t)),\quad \text{for all }t\geqslant 0.\end{eqnarray}$$

In particular, $0\leqslant \mathscr{E}(f(t))\leqslant \mathscr{E}(f_{0})\,\text{e}^{-{\it\kappa}t}$ for any $t\geqslant 0$.

Notice that we do not require the matrix $S$ to be definite for the result to hold.

Proof. One of the main ideas in the proof is to rewrite the time derivative of the second term in the expression of $\mathscr{E}$ as a sum of terms similar to the ones appearing in the proof of Theorem 2.15. To this end, consider $g(t)=\sqrt{f(t)}$, so that

(2.59)$$\begin{eqnarray}\int _{{\mathcal{E}}}\frac{{\rm\nabla}f(t)^{T}S{\rm\nabla}f(t)}{f(t)}\,\text{d}{\it\mu}=4\int _{{\mathcal{E}}}{\rm\nabla}g(t)^{T}S{\rm\nabla}g(t)\,\text{d}{\it\mu}.\end{eqnarray}$$

In order to determine the time evolution of this quantity, we first write the time evolution for $g(t)$. Simple computations show that

$$\begin{eqnarray}\displaystyle \partial _{t}g(t) & = & \displaystyle \frac{\partial _{t}f(t)}{2\sqrt{f(t)}}=\frac{{\mathcal{L}}^{\ast }f(t)}{2g(t)},\nonumber\\ \displaystyle {\rm\Delta}_{p}g(t) & = & \displaystyle \frac{{\rm\Delta}_{p}f(t)}{2\sqrt{f(t)}}-\frac{|{\rm\nabla}_{p}f(t)|^{2}}{4f(t)^{3/2}}=\frac{{\rm\Delta}_{p}f(t)}{2g(t)}-\frac{|{\rm\nabla}_{p}g(t)|^{2}}{g(t)},\nonumber\end{eqnarray}$$

so that

$$\begin{eqnarray}\partial _{t}g(t)={\mathcal{L}}^{\ast }g(t)+\frac{{\it\gamma}}{{\it\beta}}\frac{|{\rm\nabla}_{p}g(t)|^{2}}{g(t)}.\end{eqnarray}$$

We next compute the time derivatives of the various terms in $\mathscr{E}(f(t))$, relying on the reformulation (2.59). This leads to expressions very similar to those encountered in the proof of Theorem 2.15. First,

$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{d}}{\text{d}t}\biggl(\int _{{\mathcal{E}}}f(t)\,\ln f(t)\,\text{d}{\it\mu}\biggr)=\int _{{\mathcal{E}}}(1+\ln f(t)){\mathcal{L}}^{\ast }f(t)\,\text{d}{\it\mu}\nonumber\\ \displaystyle & & \displaystyle \qquad =\int _{{\mathcal{E}}}{\mathcal{L}}_{\text{ham}}(1+\ln f(t))f(t)\,\text{d}{\it\mu}-\frac{{\it\gamma}}{{\it\beta}}\int _{{\mathcal{E}}}{\rm\nabla}_{p}f(t)\cdot {\rm\nabla}_{p}(\ln f(t))\,\text{d}{\it\mu}\nonumber\\ \displaystyle & & \displaystyle \qquad =\int _{{\mathcal{E}}}{\mathcal{L}}_{\text{ham}}f(t)\,\text{d}{\it\mu}-\frac{{\it\gamma}}{{\it\beta}}\int _{{\mathcal{E}}}\frac{|{\rm\nabla}_{p}f(t)|^{2}}{f(t)}\,\text{d}{\it\mu}=-\frac{4{\it\gamma}}{{\it\beta}}\int _{{\mathcal{E}}}|{\rm\nabla}_{p}g(t)|^{2}\,\text{d}{\it\mu},\nonumber\end{eqnarray}$$

in view of the invariance of ${\it\mu}$ by ${\mathcal{L}}_{\text{ham}}$. Next,

$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{d}}{\text{d}t}\biggl(\int _{{\mathcal{E}}}|\partial _{p_{i}}g(t)|^{2}\,\text{d}{\it\mu}\biggr)=2\int _{{\mathcal{E}}}\partial _{p_{i}}g(t)\,\partial _{p_{i}}({\mathcal{L}}^{\ast }g(t))\,\text{d}{\it\mu}\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad +\frac{2{\it\gamma}}{{\it\beta}}\!\int _{{\mathcal{E}}}\partial _{p_{i}}g(t)\,\partial _{p_{i}}\biggl(\displaystyle \frac{|{\rm\nabla}_{p}g(t)|^{2}}{g(t)}\biggr)\,\text{d}{\it\mu}\nonumber\\ \displaystyle & & \displaystyle \qquad =2\int _{{\mathcal{E}}}\partial _{p_{i}}g(t)\,{\mathcal{L}}^{\ast }(\partial _{p_{i}}g(t))\,\text{d}{\it\mu}-\frac{2}{m_{i}}\int _{{\mathcal{E}}}\partial _{p_{i}}g(t)\,(\partial _{q_{i}}+{\it\gamma}\partial _{p_{i}})g(t)\,\text{d}{\it\mu}\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad +\frac{2{\it\gamma}}{{\it\beta}}\int _{{\mathcal{E}}}\partial _{p_{i}}g(t)\biggl(\displaystyle \frac{2{\rm\nabla}_{p}(\partial _{p_{i}}g(t)){\rm\nabla}_{p}g(t)}{g(t)}-\frac{|{\rm\nabla}_{p}g(t)|^{2}}{g(t)^{2}}\partial _{p_{i}}g(t)\biggr)\,\text{d}{\it\mu}\nonumber\\ \displaystyle & & \displaystyle \qquad =-\frac{2}{m_{i}}\int _{{\mathcal{E}}}\partial _{p_{i}}g(t)\,(\partial _{q_{i}}+{\it\gamma}\partial _{p_{i}})g(t)\,\text{d}{\it\mu}\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad -\frac{2{\it\gamma}}{{\it\beta}}\int _{{\mathcal{E}}}|{\rm\nabla}_{p}\partial _{p_{i}}g(t)|^{2}-2{\rm\nabla}_{p}(\partial _{p_{i}}g(t))\cdot \frac{(\partial _{p_{i}}g(t)){\rm\nabla}_{p}g(t)}{g(t)}\,\text{d}{\it\mu}\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad -\,\displaystyle \frac{2{\it\gamma}}{{\it\beta}}\int _{{\mathcal{E}}}\frac{|{\rm\nabla}_{p}g(t)|^{2}(\partial _{p_{i}}g(t))^{2}}{g(t)^{2}}\,\text{d}{\it\mu},\nonumber\\ \displaystyle & & \displaystyle \nonumber\end{eqnarray}$$

where we have used

$$\begin{eqnarray}\displaystyle \int _{{\mathcal{E}}}\partial _{p_{i}}g(t)\,{\mathcal{L}}^{\ast }(\partial _{p_{i}}g(t))\,\text{d}{\it\mu} & = & \displaystyle -\int _{{\mathcal{E}}}\partial _{p_{i}}g(t)\,{\mathcal{L}}_{\text{ham}}(\partial _{p_{i}}g(t))\,\text{d}{\it\mu}\nonumber\\ \displaystyle & & \displaystyle \quad -\frac{2{\it\gamma}}{{\it\beta}}\mathop{\sum }_{j=1}^{d}\int _{{\mathcal{E}}}\partial _{p_{i}}g(t)\,\partial _{p_{j}}^{\ast }\partial _{p_{j}}(\partial _{p_{i}}g(t))\,\text{d}{\it\mu},\nonumber\end{eqnarray}$$

and the first integral on the right-hand side vanishes since ${\mathcal{L}}_{\text{ham}}$ is antisymmetric. Therefore,

$$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}\biggl(\int _{{\mathcal{E}}}|\partial _{p_{i}}g(t)|^{2}\,\text{d}{\it\mu}\biggr) & = & \displaystyle -\frac{2}{m_{i}}\int _{{\mathcal{E}}}\partial _{p_{i}}g(t)\,(\partial _{q_{i}}+{\it\gamma}\partial _{p_{i}})g(t)\,\text{d}{\it\mu}\nonumber\\ \displaystyle & & \displaystyle \quad -\frac{2{\it\gamma}}{{\it\beta}}\int _{{\mathcal{E}}}\biggl|{\rm\nabla}_{p}\partial _{p_{i}}g(t)-\frac{(\partial _{p_{i}}g(t)){\rm\nabla}_{p}g(t)}{g(t)}\biggr|^{2}\,\text{d}{\it\mu}.\nonumber\end{eqnarray}$$

Similar computations give

$$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}\biggl(\int _{{\mathcal{E}}}|\partial _{q_{i}}g(t)|^{2}\,\text{d}{\it\mu}\biggr) & = & \displaystyle 2\int _{{\mathcal{E}}}\partial _{q_{i}}g(t)\,({\rm\nabla}\partial _{q_{i}}V)\cdot {\rm\nabla}_{p}g(t)\,\text{d}{\it\mu}\nonumber\\ \displaystyle & & \displaystyle \quad -\frac{2{\it\gamma}}{{\it\beta}}\int _{{\mathcal{E}}}\biggl|{\rm\nabla}_{p}\partial _{q_{i}}g(t)-\frac{(\partial _{q_{i}}g(t)){\rm\nabla}_{p}g(t)}{g(t)}\biggr|^{2}\,\text{d}{\it\mu},\nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{d}}{\text{d}t}\biggl(\int _{{\mathcal{E}}}\partial _{p_{i}}g(t)\,\partial _{q_{i}}g(t)\,\text{d}{\it\mu}\biggr)=\int _{{\mathcal{E}}}\partial _{p_{i}}g(t)({\rm\nabla}\partial _{q_{i}}V)\cdot {\rm\nabla}_{p}g(t)\,\text{d}{\it\mu}\nonumber\\ \displaystyle & & \displaystyle -\frac{1}{m_{i}}\int _{{\mathcal{E}}}\partial _{q_{i}}g(t)\,(\partial _{q_{i}}+{\it\gamma}\partial _{p_{i}})g(t)\,\text{d}{\it\mu}\nonumber\\ \displaystyle & & \displaystyle -\frac{2{\it\gamma}}{{\it\beta}}\int _{{\mathcal{E}}}\biggl(\!{\rm\nabla}_{p}\partial _{q_{i}}g(t)-\frac{(\partial _{q_{i}}g(t)){\rm\nabla}_{p}g(t)}{g(t)}\!\biggr)\cdot \biggl(\!{\rm\nabla}_{p}\partial _{p_{i}}g(t)-\frac{(\partial _{p_{i}}g(t)){\rm\nabla}_{p}g(t)}{g(t)}\!\biggr)\text{d}{\it\mu}.\nonumber\end{eqnarray}$$

Considering

$$\begin{eqnarray}S=\left(\begin{array}{@{}cc@{}}a\,\text{Id}_{d} & b\,\text{Id}_{d}\\ b\,\text{Id}_{d} & c\,\text{Id}_{d}\end{array}\right)\!,\end{eqnarray}$$

it follows that

$$\begin{eqnarray}\frac{\text{d}}{\text{d}t}(\mathscr{E}(f(t)))=-\frac{4{\it\gamma}}{{\it\beta}}\int _{{\mathcal{E}}}|{\rm\nabla}_{p}g(t)|^{2}\,\text{d}{\it\mu}+\mathop{\sum }_{i=1}^{d}\int _{{\it\varepsilon}}U_{i}(t)\,\text{d}{\it\mu},\end{eqnarray}$$

where

$$\begin{eqnarray}\displaystyle & & \displaystyle U_{i}(t)=8a\partial _{q_{i}}g(t)\,({\rm\nabla}\partial _{q_{i}}V)\cdot {\rm\nabla}_{p}g(t)\nonumber\\ \displaystyle & & \displaystyle \;\;-8b\biggl(-\partial _{p_{i}}g(t)\,({\rm\nabla}\partial _{q_{i}}V)\cdot {\rm\nabla}_{p}g(t)+\frac{1}{m_{i}}\partial _{q_{i}}g(t)\,(\partial _{q_{i}}+{\it\gamma}\partial _{p_{i}})g(t)\biggr)\nonumber\\ \displaystyle & & \displaystyle \;\;-\frac{8c}{m_{i}}\partial _{p_{i}}g(t)\,(\partial _{q_{i}}+{\it\gamma}\partial _{p_{i}})g(t)\nonumber\\ \displaystyle & & \displaystyle -\frac{8{\it\gamma}a}{{\it\beta}}\biggl|{\rm\nabla}_{p}\partial _{q_{i}}g(t)-\frac{(\partial _{q_{i}}g(t)){\rm\nabla}_{p}g(t)}{g(t)}\biggr|^{2}\nonumber\\ \displaystyle & & \displaystyle \;\;-\frac{16{\it\gamma}b}{{\it\beta}}\biggl({\rm\nabla}_{p}\partial _{q_{i}}g(t)-\frac{(\partial _{q_{i}}g(t)){\rm\nabla}_{p}g(t)}{g(t)}\biggr)\cdot \biggl({\rm\nabla}_{p}\partial _{p_{i}}g(t)-\frac{(\partial _{p_{i}}g(t)){\rm\nabla}_{p}g(t)}{g(t)}\biggr)\nonumber\\ \displaystyle & & \displaystyle \;\;-\frac{8{\it\gamma}c}{{\it\beta}}\biggl|{\rm\nabla}_{p}\partial _{p_{i}}g(t)-\frac{(\partial _{p_{i}}g(t)){\rm\nabla}_{p}g(t)}{g(t)}\biggr|^{2}.\nonumber\end{eqnarray}$$

Note that the sum of the last three terms in $U_{i}(t)$, integrated against ${\it\mu}$, can be bounded from above by

(2.60)$$\begin{eqnarray}-\frac{8{\it\gamma}}{{\it\beta}}\mathop{\sum }_{i=1}^{d}(aX_{i}^{2}-2bX_{i}Y_{i}+cY_{i}^{2}),\end{eqnarray}$$

with

$$\begin{eqnarray}\displaystyle X_{i} & = & \displaystyle \biggl\|{\rm\nabla}_{p}\partial _{q_{i}}g(t)-\displaystyle \frac{(\partial _{q_{i}}g(t)){\rm\nabla}_{p}g(t)}{g(t)}\biggr\|_{L^{2}({\it\mu})},\nonumber\\ \displaystyle Y_{i} & = & \displaystyle \biggl\|{\rm\nabla}_{p}\partial _{p_{i}}g(t)-\displaystyle \frac{(\partial _{p_{i}}g(t)){\rm\nabla}_{p}g(t)}{g(t)}\biggr\|_{L^{2}({\it\mu})}.\nonumber\end{eqnarray}$$

We assume in the following

$$\begin{eqnarray}ac-b^{2}\geqslant 0,\end{eqnarray}$$

which ensures that the matrix $S$ is non-negative and that (2.60) is non-positive. Then, recalling definition (2.46) for $m_{+}$ and $m_{-}$,

$$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}}{\text{d}t}(\mathscr{E}(f(t))) & {\leqslant} & \displaystyle -\,4{\it\gamma}\biggl(\frac{1}{{\it\beta}}+\frac{2c}{m_{+}}\biggr)\Vert {\rm\nabla}_{p}g(t)\Vert _{L^{2}({\it\mu})}^{2}-\frac{8b}{m_{+}}\Vert {\rm\nabla}_{q}g(t)\Vert _{L^{2}({\it\mu})}^{2}\nonumber\\ \displaystyle & & \displaystyle \;\;+8\mathop{\sum }_{i=1}^{d}\int _{{\mathcal{E}}}\,(a\partial _{q_{i}}g(t)+b\partial _{p_{i}}g(t))\,({\rm\nabla}\partial _{q_{i}}V)\cdot {\rm\nabla}_{p}g(t)\,\text{d}{\it\mu}\nonumber\\ \displaystyle & & \displaystyle \;\;-\,8({\it\gamma}b+c)\mathop{\sum }_{i=1}^{d}\frac{1}{m_{i}}\int _{{\mathcal{E}}}\partial _{q_{i}}g(t)\,\partial _{p_{i}}g(t)\,\text{d}{\it\mu}.\nonumber\end{eqnarray}$$

Since ${\rm\nabla}^{2}V$ is assumed to be bounded and

$$\begin{eqnarray}\Vert {\rm\nabla}_{p}g(t)\Vert _{L^{2}({\it\mu})}^{2}+\Vert {\rm\nabla}_{q}g(t)\Vert _{L^{2}({\it\mu})}^{2}=\frac{1}{4}\int _{{\mathcal{E}}}\frac{|{\rm\nabla}f(t)|^{2}}{f(t)}\,\text{d}{\it\mu},\end{eqnarray}$$

inequality (2.57) follows by choosing appropriate values for the coefficients $a,b,c$ in $S$. For instance, $a=b=c={\it\varepsilon}>0$ with ${\it\varepsilon}>0$ sufficiently small.

To deduce (2.58), we note that, by tensorization of LSI, there exists $R>0$ such that, for any $C^{\infty }$ function $h\geqslant 0$ satisfying

$$\begin{eqnarray}\int _{{\mathcal{E}}}h\,\text{d}{\it\mu}=1,\end{eqnarray}$$

the following inequality holds:

$$\begin{eqnarray}\int _{{\mathcal{E}}}h\ln h\,\text{d}{\it\mu}\leqslant \frac{1}{2R}\int _{{\mathcal{E}}}\frac{|{\rm\nabla}_{p}h|^{2}+|{\rm\nabla}_{q}h|^{2}}{h}\,\text{d}{\it\mu}.\end{eqnarray}$$

In addition,

$$\begin{eqnarray}\int _{{\mathcal{E}}}\frac{({\rm\nabla}h)^{T}S{\rm\nabla}h}{h}\,\text{d}{\it\mu}\leqslant {\it\lambda}_{S}\int _{{\mathcal{E}}}\frac{|{\rm\nabla}h|^{2}}{h}\,\text{d}{\it\mu},\end{eqnarray}$$

where ${\it\lambda}_{S}$ is the largest eigenvalue of $S$. Combining these two estimates, we obtain the existence of ${\it\kappa}>0$ such that

$$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}(\mathscr{E}(f(t))) & {\leqslant} & \displaystyle -{\it\kappa}\biggl(\int _{{\mathcal{E}}}f(t)\ln f(t)\,\text{d}{\it\mu}+\int _{{\mathcal{E}}}\frac{({\rm\nabla}f(t))^{T}S{\rm\nabla}f(t)}{f(t)}\,\text{d}{\it\mu}\biggr)\nonumber\\ \displaystyle & = & \displaystyle -{\it\kappa}\mathscr{E}(f(t)),\nonumber\end{eqnarray}$$

which is indeed (2.58). The exponential decay follows by a Gronwall inequality.

Here again, the final constant ${\it\kappa}$ can be made quite explicit in terms of the various parameters in the model (e.g. LSI constant $R$, bounds on ${\rm\nabla}^{2}V$).

2.3.6 Subelliptic estimates

To conclude this section on Langevin dynamics, let us briefly mention another technique to prove decay estimates for $\text{e}^{t{\mathcal{L}}}$ and obtain finer spectral properties of the generator ${\mathcal{L}}$ of Langevin dynamics or its adjoint, namely subelliptic estimates as provided in Eckmann and Hairer (Reference Eckmann and Hairer2003) and Hérau and Nier (Reference Hérau and Nier2004). The techniques for proving these estimates are based on generalizations of Hörmander’s techniques. One output is that ${\mathcal{L}}$ is an operator with a compact resolvent on $L^{2}({\it\mu})$ (in the sense that $({\it\lambda}-{\mathcal{L}})^{-1}$ is compact for any ${\it\lambda}\not \in {\it\sigma}({\mathcal{L}})$), so that its spectrum is discrete. Moreover, it can be proved that the spectrum is contained in some cusp-shaped region: there exist $C,{\it\alpha}>0$ such that

$$\begin{eqnarray}{\it\sigma}({\mathcal{L}})\subset \{z\in \mathbb{C}\mid \text{Re}(z)\geqslant 0,\;|\text{Im}(z)|\leqslant C(1+\text{Re}(z))^{{\it\alpha}}\}.\end{eqnarray}$$

2.4.1 General framework

Consider a stochastic evolution described by its evolution operator $P$, with associated transition kernel $P(x,\text{d}x^{\prime })$. This kernel induces an operator on bounded measurable functions as follows: for any ${\it\varphi}\in L^{\infty }({\mathcal{X}})$,

$$\begin{eqnarray}P{\it\varphi}(x)=\int _{{\mathcal{X}}}{\it\varphi}(x^{\prime })P(x,\text{d}x^{\prime }).\end{eqnarray}$$

For example, a natural way to define a transition kernel for the general SDE (1.11) is the following for a given time $t_{0}>0$: for any measurable set $S$,

(2.61)$$\begin{eqnarray}P(x,S)=\mathbb{E}^{x}(\mathbf{1}_{S}(x_{t_{0}})).\end{eqnarray}$$

Here $\mathbb{E}^{x}$ is the expectation over the realizations of (1.11) starting from $x_{0}=x$, and $\mathbf{1}_{S}$ is the indicator function of the set $S$. In view of (2.8), the operator associated with the integral kernel (2.61) is $P=\text{e}^{t_{0}{\mathcal{L}}}$.

We make the following assumptions.

Assumption 2.23 (Lyapunov condition).

There exists a function ${\mathcal{K}}:{\mathcal{X}}\rightarrow [1,+\infty )$ and constants $R\geqslant 0$ and ${\it\alpha}\in (0,1)$ such that

(2.62)$$\begin{eqnarray}(P{\mathcal{K}})(x)\leqslant {\it\alpha}{\mathcal{K}}(x)+R,\quad \text{for all }x\in {\mathcal{X}}.\end{eqnarray}$$

Let us first insist on the fact that we require ${\mathcal{K}}\geqslant 1$. The Lyapunov condition implies that the dynamics returns to the region of the configuration space where the values of ${\mathcal{K}}$ are not too large. Note that it is trivially satisfied when the configuration space ${\mathcal{X}}$ is bounded (in which case ${\mathcal{K}}=\mathbf{1}$ is a possible choice). For unbounded spaces, one typically chooses functions ${\mathcal{K}}$ which go to infinity at infinity, so that the Lyapunov condition ensures that the dynamics returns to a compact region around the origin.

Assumption 2.24 (minorization condition).

There exists a constant ${\it\eta}\in (0,1)$ and a probability measure ${\it\lambda}$ such that

$$\begin{eqnarray}\inf _{x\in {\mathcal{C}}}P(x,\text{d}y)\geqslant {\it\eta}\,{\it\lambda}(\text{d}y),\end{eqnarray}$$

where ${\mathcal{C}}=\{x\in {\mathcal{X}}\mid {\mathcal{K}}(x)\leqslant {\mathcal{K}}_{\max }\}$ for some ${\mathcal{K}}_{\max }>1+2R/(1-{\it\alpha})$, where ${\it\alpha},R$ are introduced in Assumption 2.23.

This condition ensures that there is a sufficiently strong coupling of the evolution in the region where the Lyapunov function is bounded by ${\mathcal{K}}_{\max }$. When the configuration space ${\mathcal{X}}$ is compact, or when the set ${\mathcal{C}}$ is compact, this condition is easy to prove for SDEs, since it is implied by a combination of the following two conditions.

1. (i) Some irreducibility property of the following form: for a given time $t_{0}>0$, and for any $x\in {\mathcal{C}}$ and Borel set $S\subset {\mathcal{X}}$ of positive Lebesgue measure, we have $\mathbb{P}^{x}(x_{t_{0}},S)>0$.

2. (ii) Some regularity results on the density of the transition kernel, as provided by hypoellipticity (see Section 2.3.1), for instance.

A typical choice for ${\it\lambda}(\text{d}y)$ is then the Lebesgue measure restricted to ${\mathcal{C}}$, namely ${\it\lambda}(\text{d}y)=\mathbf{1}_{{\mathcal{C}}}(y)\,\text{d}y$. For further details, see the application of the general framework provided in Section 2.4.2 to Langevin and overdamped Langevin dynamics.

We next introduce the functional space used to measure convergence, which we denote by $L_{{\mathcal{K}}}^{\infty }({\mathcal{X}})$. It is the Banach space of measurable functions ${\it\varphi}$ such that

(2.63)$$\begin{eqnarray}\Vert {\it\varphi}\Vert _{L_{{\mathcal{K}}}^{\infty }}=\biggl\|\frac{{\it\varphi}}{{\mathcal{K}}}\biggr\|_{L^{\infty }}<+\infty .\end{eqnarray}$$

Theorem 2.25 (exponential convergence in weighted $L^{\infty }$-spaces).

Suppose that Assumptions 2.23 and 2.24 hold. Then $P$ admits a unique invariant probability measure ${\it\pi}$. This measure is such that

(2.64)$$\begin{eqnarray}\int _{{\mathcal{X}}}{\mathcal{K}}\,\text{d}{\it\pi}<+\infty .\end{eqnarray}$$

Moreover, there exist $C>0$ and $r\in (0,1)$ such that, for any ${\it\varphi}\in L_{{\mathcal{K}}}^{\infty }({\mathcal{X}})$ and any $n\in \mathbb{N}$,

(2.65)$$\begin{eqnarray}\biggl\|P^{n}{\it\varphi}-\int _{{\mathcal{X}}}{\it\varphi}\,\text{d}{\it\pi}\biggr\|_{L_{{\mathcal{K}}}^{\infty }}\leqslant Cr^{n}\biggl\|{\it\varphi}-\int _{{\mathcal{X}}}{\it\varphi}\,\text{d}{\it\pi}\biggr\|_{L_{{\mathcal{K}}}^{\infty }}.\end{eqnarray}$$

The proof of this result can be read in Hairer and Mattingly (Reference Hairer and Mattingly2011). Let us summarize the main steps. The first idea is to introduce a family of distances $d_{a}$ on ${\mathcal{X}}$ (with $a>0$), and to consider the induced Lipschitz seminorms $\Vert \cdot \Vert _{a}$ on real-valued functions defined on ${\mathcal{X}}$. These seminorms turn out to be equivalent to the $L_{{\mathcal{K}}}^{\infty }({\mathcal{X}})$-norm (up to the addition of constants to the functions under consideration). Moreover, it is possible to prove a contraction principle of the form

(2.66)$$\begin{eqnarray}|P{\it\varphi}(x)-P{\it\varphi}(y)|\leqslant rd_{a}(x,y)\Vert {\it\varphi}\Vert _{a}\end{eqnarray}$$

for some constant $r\in (0,1)$, which implies a contraction principle for the operator $P$ considered on $L_{{\mathcal{K}}}^{\infty }({\mathcal{X}})$ endowed with the seminorm $\Vert \cdot \Vert _{a}$. The existence of the invariant measure is finally obtained by a fixed-point theorem, and (2.65) is deduced from the contraction inequality upon starting from $y$ distributed according to the invariant measure ${\it\pi}$. The prefactor $C\geqslant 1$ is related to the equivalence of the $\Vert \cdot \Vert _{L_{{\mathcal{K}}}^{\infty }}$ and $\Vert \cdot \Vert _{a}$ norms (as for the proof of hypocoercivity; see the comment after Theorem 2.15).

The key point in the proof is therefore to obtain the contraction inequality (2.66). This is done by distinguishing the cases ${\mathcal{K}}(x)+{\mathcal{K}}(y)\geqslant R$ (in this situation, the Lyapunov condition is used) or ${\mathcal{K}}(x)+{\mathcal{K}}(y)\leqslant R$ (the contraction then relies on the coupling ensured by the minorization condition).

Let us now present a more compact reformulation of (2.65) in terms of operators. Introduce

$$\begin{eqnarray}L_{{\mathcal{K}},0}^{\infty }({\mathcal{X}})=\biggl\{{\it\varphi}\in L_{{\mathcal{K}}}^{\infty }({\mathcal{X}})\,\bigg|\,\int _{{\mathcal{X}}}{\it\varphi}\,\text{d}{\it\pi}=0\biggr\}.\end{eqnarray}$$

Then (2.65) can be equivalently written as an inequality for bounded operators on $L_{{\mathcal{K}},0}^{\infty }({\mathcal{X}})$:

(2.67)$$\begin{eqnarray}\Vert P^{n}\Vert _{{\mathcal{B}}(L_{{\mathcal{K}},0}^{\infty })}\leqslant Cr^{n}.\end{eqnarray}$$

The exponential convergence allows us to obtain bounds on the resolvent $\text{Id}-P$, which will prove useful in various contexts (for error estimates on time discretizations or for the definition of the asymptotic variance of Markov chains). The following result is the version of Proposition 2.1 obtained in the context of this section.

Corollary 2.26 (invertibility of $\text{Id}-P$).

Let Assumptions 2.23 and 2.24 hold. Then the bounded operator $\text{Id}-P$ is invertible on $L_{{\mathcal{K}},0}^{\infty }({\mathcal{X}})$, and

$$\begin{eqnarray}\Vert (\text{Id}-P)^{-1}\Vert _{{\mathcal{B}}(L_{{\mathcal{K}},0}^{\infty })}\leqslant \frac{C}{1-r}.\end{eqnarray}$$

Proof. Note that the sum $\sum _{n\geqslant 0}P^{n}$ is convergent in ${\mathcal{B}}(L_{{\mathcal{K}},0}^{\infty })$ in view of (2.67). In addition, a simple computation gives

$$\begin{eqnarray}(\text{Id}-P)\mathop{\sum }_{n=0}^{+\infty }P^{n}=\biggl(\mathop{\sum }_{n=0}^{+\infty }P^{n}\biggr)(\text{Id}-P)=\text{Id}.\end{eqnarray}$$

This shows that $\text{Id}-P$ is invertible and

(2.68)$$\begin{eqnarray}(\text{Id}-P)^{-1}=\mathop{\sum }_{n=0}^{+\infty }P^{n}.\end{eqnarray}$$

Moreover,

$$\begin{eqnarray}\Vert (\text{Id}-P)^{-1}\Vert _{{\mathcal{B}}(L_{{\mathcal{K}},0}^{\infty })}\leqslant \mathop{\sum }_{n=0}^{+\infty }\Vert P^{n}\Vert _{{\mathcal{B}}(L_{{\mathcal{K}},0}^{\infty })}\leqslant C\mathop{\sum }_{n=0}^{+\infty }r^{n}=\frac{C}{1-r},\end{eqnarray}$$

which gives the claimed upper bound.

2.4.2 Application to Langevin and overdamped Langevin dynamics

To apply the convergence results to the stochastic evolutions typically encountered in molecular dynamics, we fix some time $t_{0}>0$ and consider the evolution operator

$$\begin{eqnarray}P=\text{e}^{t_{0}{\mathcal{L}}},\end{eqnarray}$$

where ${\mathcal{L}}$ is the generator of the dynamics under consideration. For simplicity, we consider the dynamics (1.8) and (1.9), for which the force $-{\rm\nabla}V(q)$ is a gradient field. The results obtained below, however, can easily be extended to more general force fields. This is one of the main interests of weighted $L^{\infty }$ estimates: they do not require structural assumptions on the type of dynamics considered.

We first discuss how to establish the Lyapunov condition (2.62) in Assumption 2.23 for Langevin and overdamped Langevin dynamics. Let us start by recalling that, as discussed after Assumption 2.23, it is trivial to satisfy a Lyapunov condition when the state space is compact, by choosing ${\mathcal{K}}=\mathbf{1}$. This situation is encountered for overdamped Langevin dynamics when ${\mathcal{D}}$ is compact. For overdamped Langevin dynamics in unbounded position spaces, or Langevin dynamics (for which the momentum space is always unbounded), the Lyapunov condition (2.62) can be established on the basis of a Lyapunov condition stated in terms of the generator of the dynamics. Indeed, assume that the following inequality is true for some $a>0$ and $b\geqslant 0$, and a given Lyapunov function ${\mathcal{K}}\geqslant 1$:

(2.69)$$\begin{eqnarray}{\mathcal{L}}{\mathcal{K}}\leqslant -a{\mathcal{K}}+b.\end{eqnarray}$$

This leads, by a time integration on the interval $[0,s]$, to

(2.70)$$\begin{eqnarray}\text{e}^{s{\mathcal{L}}}{\mathcal{K}}\leqslant \text{e}^{-as}{\mathcal{K}}+\frac{b}{a}(1-\text{e}^{-as})\leqslant \text{e}^{-as}{\mathcal{K}}+\frac{b}{a}.\end{eqnarray}$$

In particular, for $s=t_{0}$, we obtain

(2.71)$$\begin{eqnarray}P{\mathcal{K}}\leqslant \text{e}^{-at_{0}}{\mathcal{K}}+\frac{b}{a},\end{eqnarray}$$

which is (2.62) with ${\it\alpha}=\text{e}^{-at_{0}}\in (0,1)$ and $R=b/a\geqslant 0$. We therefore choose ${\mathcal{K}}$ in order for (2.69) to be satisfied. For overdamped Langevin dynamics in unbounded spaces, a typical choice is ${\mathcal{K}}_{n}(q)=|q|^{n}$ for $n\geqslant 2$. In this case, for all $q\in {\mathcal{D}}$,

$$\begin{eqnarray}{\mathcal{L}}{\mathcal{K}}_{n}(q)=n\biggl(-{\rm\nabla}V(q)\cdot q+\frac{d+n-2}{{\it\beta}}\biggr)|q|^{n-2}.\end{eqnarray}$$

Some assumptions on the asymptotic behaviour of the drift are needed at this stage, such as

(2.72)$$\begin{eqnarray}q\cdot {\rm\nabla}V(q)\geqslant A|q|^{2}-B,\quad \text{for all }q\in {\mathcal{D}},\end{eqnarray}$$

for some constant $A>0$ and $B\in \mathbb{R}$. This condition is satisfied for potentials $V(q)$ behaving at infinity as $|q|^{k}$ with $k\geqslant 2$. With (2.72), the Lyapunov condition (2.69) indeed holds since, for all $q\in {\mathcal{D}}$,

$$\begin{eqnarray}{\mathcal{L}}{\mathcal{K}}_{n}(q)\leqslant -An{\mathcal{K}}_{n}(q)+\frac{n(d+n-2)}{{\it\beta}}|q|^{n-2}+Bn,\end{eqnarray}$$

so that

$$\begin{eqnarray}\limsup _{|q|\rightarrow +\infty }\,\frac{{\mathcal{L}}{\mathcal{K}}_{n}(q)}{{\mathcal{K}}_{n}(q)}\leqslant -An.\end{eqnarray}$$

There exist therefore $b_{n}\geqslant 0$ such that

$$\begin{eqnarray}{\mathcal{L}}{\mathcal{K}}_{n}\leqslant -\frac{An}{2}\,{\mathcal{K}}_{n}+b_{n}.\end{eqnarray}$$

For Langevin dynamics, the choice ${\mathcal{K}}_{n}(q,p)=1+|p|^{n}$ with $n\geqslant 2$ is appropriate when the position space is compact. Indeed, with $m_{+}$ defined in (2.46), for all $(q,p)\in {\mathcal{E}}$,

$$\begin{eqnarray}\displaystyle {\mathcal{L}}{\mathcal{K}}_{n}(q,p) & = & \displaystyle -np^{T}{\rm\nabla}V(q)|p|^{n-2}-{\it\gamma}n\biggl(p^{T}M^{-1}p-\frac{n+d-2}{{\it\beta}}\biggr)|p|^{n-2}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle -\frac{{\it\gamma}n}{m_{+}}|p|^{n}+n\Vert {\rm\nabla}V\Vert _{L^{\infty }}|p|^{n-1}+\frac{{\it\gamma}n(n+d-2)}{{\it\beta}}|p|^{n-2},\nonumber\end{eqnarray}$$

so that there exists $b_{n}\geqslant 0$ such that

$$\begin{eqnarray}{\mathcal{L}}{\mathcal{K}}_{n}\leqslant -\frac{{\it\gamma}n}{2m_{+}}\,{\mathcal{K}}_{n}+b_{n}.\end{eqnarray}$$

For Langevin dynamics in an unbounded position space ${\mathcal{D}}$, a possible choice is (see Mattingly, Stuart and Higham Reference Mattingly, Stuart and Higham2002)

(2.73)$$\begin{eqnarray}{\mathcal{K}}_{n}(q,p)=\biggl(1+H(q,p)-V_{-}+\frac{{\it\gamma}}{2}p^{T}M^{-1}q+\frac{{\it\gamma}^{2}}{4}q^{T}M^{-1}q\biggr)^{n},~~\text{for all }(q,p)\in {\mathcal{E}},\end{eqnarray}$$

where we assume that the potential energy function $V(q)$ is bounded from below by $V_{-}>-\infty$ and that there exist $A,B>0$ and $C\in \mathbb{R}$ such that

(2.74)$$\begin{eqnarray}q^{T}M^{-1}{\rm\nabla}V(q)\geqslant AV(q)+Bq^{T}M^{-1}q+C,\quad \text{for all }q\in {\mathcal{D}}.\end{eqnarray}$$

This condition is satisfied for instance when $V(q)$ behaves at infinity as $|q|^{k}$ with $k\geqslant 2$. Then, for any $n\geqslant 1$, we have ${\mathcal{K}}_{n}(q,p)\geqslant 1$ and ${\mathcal{K}}_{n}(q,p)\rightarrow +\infty$ as $|(q,p)|\rightarrow +\infty$. In addition, there exist $a>0$ and $b_{n}\in \mathbb{R}$ such that

$$\begin{eqnarray}{\mathcal{L}}{\mathcal{K}}_{n}\leqslant -na{\mathcal{K}}_{n}+b_{n}.\end{eqnarray}$$

Note that the Lyapunov function has a component $p^{T}M^{-1}q$ which allows us to retrieve some dissipation in the $q$-direction from the dissipation in the momenta. This feature is reminiscent of the cross-derivative used in the scalar product introduced for the proof of hypocoercivity in Section 2.3 (see the terms proportional to $b$ in (2.38)).

Let us now discuss how to obtain the minorization condition in Assumption 2.24. Note first that this condition can be rewritten as: for any Borel set $S\subset {\mathcal{X}}$,

$$\begin{eqnarray}P(x,S)\geqslant {\it\eta}{\it\lambda}(S).\end{eqnarray}$$

A convenient way to obtain such a lower bound is to first establish that the transition kernel can be rewritten as

(2.75)$$\begin{eqnarray}P(x,S)=\int _{S}p_{t_{0}}(x,y)\,\text{d}y,\end{eqnarray}$$

where the density $p_{t_{0}}(x,y)$ of the transition kernel is a $C^{\infty }$ function of $(x,y)\in {\mathcal{X}}^{2}$. This is the case, appealing to parabolic regularity for overdamped Langevin dynamics and hypoellipticity for Langevin dynamics (see for instance the discussion in Section 8 of Rey-Bellet Reference Rey-Bellet, Attal, Joye and Pillet2006). The second step is to prove that, for any open set ${\mathcal{O}}\subset {\mathcal{X}}$ and any $x\in {\mathcal{X}}$, we have

(2.76)$$\begin{eqnarray}P(x,{\mathcal{O}})>0.\end{eqnarray}$$

This is proved by controllability arguments, as reviewed in Section 7 of Rey-Bellet (Reference Rey-Bellet, Attal, Joye and Pillet2006), where the cases of Langevin and overdamped Langevin are explicitly treated; see also Mattingly, Stuart and Higham (Reference Mattingly, Stuart and Higham2002) and Talay (Reference Talay2002) for Langevin dynamics. The idea is to choose some end configuration $y\in {\mathcal{O}}$, and to construct a realization of the Brownian motion such that the dynamics started at $x$ ends up at $y$ at time $t_{0}$. The continuity of the solutions of SDEs with respect to the realization of the Brownian motion then leads to (2.76). In view of (2.75), the condition (2.76) in fact shows that, for any compact set ${\mathcal{C}}\subset {\mathcal{X}}$,

$$\begin{eqnarray}p_{t_{0}}(x,y)>0,\quad \text{for all }(x,y)\in {\mathcal{C}}^{2}.\end{eqnarray}$$

Since $(x,y)\mapsto p_{t_{0}}(x,y)$ is continuous, this proves the existence of ${\it\delta}_{{\mathcal{C}}}>0$ such that $p_{t_{0}}(x,y)\geqslant {\it\delta}_{{\mathcal{C}}}$ for all $(x,y)\in {\mathcal{C}}^{2}$. The minorization condition then follows by considering the probability measure

$$\begin{eqnarray}{\it\lambda}(\text{d}y)=\frac{1}{|{\mathcal{C}}|}\mathbf{1}_{{\mathcal{C}}}(y)\,\text{d}y,\end{eqnarray}$$

and setting ${\it\eta}={\it\delta}_{{\mathcal{C}}}|{\mathcal{C}}|$.

At this stage, we have discussed how to obtain Assumptions 2.23 and 2.24. This already implies the decay estimates (2.65) by Theorem 2.25. These bounds correspond to evolutions observed at integer multiples of the reference time $t_{0}$. Let us now finally show how to deduce an exponential convergence result at all times $t\geqslant 0$. To this end, we decompose the time $t$ as $t=kt_{0}+{\it\theta}$ with $k\in \mathbb{N}$ and ${\it\theta}\in [0,t_{0})$. Then, for any ${\it\varphi}\in L_{{\mathcal{K}},0}^{\infty }({\mathcal{X}})$ we obtain

(2.77)$$\begin{eqnarray}\Vert \text{e}^{t{\mathcal{L}}}{\it\varphi}\Vert _{L_{{\mathcal{K}}}^{\infty }}\leqslant Cr^{k}\Vert \text{e}^{{\it\theta}{\mathcal{L}}}{\it\varphi}\Vert _{L_{{\mathcal{K}}}^{\infty }}\leqslant Cr^{k}\sup _{0\leqslant s\leqslant t_{0}}\Vert \text{e}^{s{\mathcal{L}}}{\it\varphi}\Vert _{L_{{\mathcal{K}}}^{\infty }}.\end{eqnarray}$$

Now, the inequality

$$\begin{eqnarray}|{\it\varphi}(x)|\leqslant \Vert {\it\varphi}\Vert _{L_{{\mathcal{K}}}^{\infty }}{\mathcal{K}}(x),\quad \text{for all }x\in {\mathcal{X}},\end{eqnarray}$$

leads to

$$\begin{eqnarray}\Vert \text{e}^{s{\mathcal{L}}}{\it\varphi}\Vert _{L_{{\mathcal{K}}}^{\infty }}\leqslant \Vert {\it\varphi}\Vert _{L_{{\mathcal{K}}}^{\infty }}\biggl\|\frac{\text{e}^{s{\mathcal{L}}}{\mathcal{K}}}{{\mathcal{K}}}\biggr\|_{L^{\infty }}.\end{eqnarray}$$

In view of the inequality (2.70), for any $s\geqslant 0$ we obtain

$$\begin{eqnarray}\biggl\|\frac{\text{e}^{s{\mathcal{L}}}{\mathcal{K}}}{{\mathcal{K}}}\biggr\|_{L^{\infty }}\leqslant \text{e}^{-as}+\frac{b}{a}\biggl\|\frac{1}{{\mathcal{K}}}\biggr\|_{L^{\infty }}\leqslant 1+\frac{b}{a}.\end{eqnarray}$$

Upon introducing ${\it\kappa}=-\log (r)/t_{0}>0$, (2.77) implies

(2.78)$$\begin{eqnarray}\Vert \text{e}^{t{\mathcal{L}}}{\it\varphi}\Vert _{L_{{\mathcal{K}}}^{\infty }}\leqslant C\biggl(1+\frac{b}{a}\biggr)\text{e}^{-{\it\kappa}kt_{0}}\Vert {\it\varphi}\Vert _{L_{{\mathcal{K}}}^{\infty }}\leqslant \widetilde{C}\text{e}^{-{\it\kappa}t}\Vert {\it\varphi}\Vert _{L_{{\mathcal{K}}}^{\infty }},\end{eqnarray}$$

with

$$\begin{eqnarray}\widetilde{C}=C\biggl(1+\frac{b}{a}\biggr)\text{e}^{{\it\kappa}t_{0}}.\end{eqnarray}$$

The exponential convergence result (2.78) leads to the following bounds on the resolvent.

Corollary 2.27. Consider one of the following situations:

1. (1) For overdamped Langevin dynamics, set ${\mathcal{X}}={\mathcal{D}}$ and ${\it\pi}={\it\nu}$, and assume either that the position space is compact, in which case ${\mathcal{K}}=\mathbf{1}$, or that (2.72) holds, in which case ${\mathcal{K}}(q)=1+|q|^{n}$ for some $n\geqslant 2$.

2. (2) For Langevin dynamics, set ${\mathcal{X}}={\mathcal{E}}$ and ${\it\pi}={\it\mu}$, and assume either that the position space is compact, in which case ${\mathcal{K}}(q,p)=1+|p|^{n}$ for some $n\geqslant 2$, or that (2.74) holds, in which case ${\mathcal{K}}$ is given by (2.73) for some $n\geqslant 1$.

Then the operator ${\mathcal{L}}$ is invertible on $L_{{\mathcal{K}},0}^{\infty }({\mathcal{X}})$, and the following equality holds in ${\mathcal{B}}(L_{{\mathcal{K}},0}^{\infty })$:

$$\begin{eqnarray}{\mathcal{L}}^{-1}=-\int _{0}^{+\infty }\,\text{e}^{t{\mathcal{L}}}\,\text{d}t.\end{eqnarray}$$

Moreover,

(2.79)$$\begin{eqnarray}\Vert {\mathcal{L}}^{-1}\Vert _{{\mathcal{B}}(L_{{\mathcal{K}},0}^{\infty })}\leqslant \frac{\widetilde{C}}{{\it\kappa}},\end{eqnarray}$$

where $\widetilde{C}$ and ${\it\kappa}$ are the same constants as in (2.78).

Although it is possible to keep track of the values of the various constants until (2.78), the final rate of convergence in weighted $L^{\infty }$-spaces is often not very sharp. Generally, the difficulty is in controlling the constant ${\it\eta}$ in the minorization condition of Assumption 2.24.