3.1 First decomposition

The core idea used to prove Theorem 2.1 is the decomposition of the solution of equation (2.1) into the sum of different processes of increasing regularity. In equation (2.1), the smoothness of solutions is dictated solely by the stochastic convolution term, namely the last term on the right-hand side of equation (2.2).

We begin by taking the stochastic convolution as the first level in our decomposition. This first level will be fed through the integrated nonlinearity, namely the first term on the right-hand side of equation (2.2). We will then keep only the roughest component and use it to force the next level in our hierarchy. At each level, we will include a stochastic forcing term that, although smoother than the forcing at the previous level, will be sufficiently rough to generate a stochastic convolution that is less smooth than the forcing generated by the previous level through the nonlinearity. Eventually, we will reach a level where the terms in the equation can be handled by classical methods and the expansion terminated.

More concretely, fixing the number of levels n, $n \in \mathbb {N}$ , we begin by positing the existence of process $X^{(0)}, \dots , X^{(n)}$ and remainder term $R^{(n)}$ so that

(3.1) $$ \begin{align} u_t \overset{\tiny{\mbox{dist}}}{=} \sum_{i=0}^n X^{(i)}_t + R^{(n)}_t,\\[-18pt]\nonumber \end{align} $$

where $\overset {\tiny{\mbox{dist}}}{=}$ denotes equality in law. We define the terms in this expansion by

(3.2) $$ \begin{align} \left\{ \begin{aligned} \mathscr{L} X^{(0)}_t &= Q_0 dW^{(0)}_t, \\ \mathscr{L}X^{(1)}_t &= B(X^{(0)}_t)dt + Q_1 dW_t^{(1)}, \\ \mathscr{L} X^{(2)}_t &= \big(B(X^{(0,1)}_t) - B(X^{(0,0)}_t)\big)dt + Q_2 dW_t^{(2)}, \\ \mathscr{L} X^{(3)}_t &= \big(B(X^{(0,2)}_t) - B(X^{(0,1)}_t)\big)dt + Q_3 dW_t^{(3)}, \\ \vdots\quad&\qquad \qquad\vdots\quad \qquad \qquad\vdots \qquad\qquad \qquad\vdots \\ \mathscr{L} X^{(n)}_t &= \big(B(X^{(0,n-1)}_t) - B(X^{(0,n-2)}_t)\big)dt + Q_n dW_t^{(n)}, \\ \mathscr{L} R^{(n)}_t &= \big(B(X^{(0,n)}_t+ R^{(n)}_t) - B(X^{(0,n-1)}_t)\big)dt + \widetilde{Q}_n d\widetilde{W} _t^{(n)}, \end{aligned} \right.\\[-18pt]\nonumber \end{align} $$

where the $Q_1,\dots ,Q_n, \widetilde {Q}_n$ are a collection of linear operators and the $W_t^{(0)}, \dots , W_t^{(n)}, \widetilde {W} _t^{(n)}$ are a collection of standard, independent, cylindrical Wiener processes and

$$ \begin{align*} X^{(m,n)}_t = \sum_{i=m}^n X^{(i)}_t, \end{align*} $$

with $X^{(0, -1)}_t = X^{(0, -2)}_t = 0$ . If we choose $X_0^{(0)}=\cdots =X_0^{(n)}=0$ and $R_0^{(n)}=u_0$ and require

(3.3) $$ \begin{align} Q_0 Q_0^* + Q_1 Q_1^* + \cdots + Q_n Q_n^* + \widetilde{Q}_n\widetilde{Q}_n^* = QQ^*, \end{align} $$

then

(3.4) $$ \begin{align} Q W_t \overset{\tiny{\mbox{dist}}}{=} \widetilde{Q}_n \widetilde{W} _t^{(n)} + \sum_{k=0}^n Q_k W_t^{(k)}, \end{align} $$

and, at least formally, the condition given in equation (3.1) holds. To make the argument complete, we need to demonstrate that each of the equations in (3.2) is well defined and has at least local solutions. The number of levels n will be chosen as a function of $\alpha $ . The closer $\alpha $ is to one, the more levels are required.

Notice that because of equation (3.4), which followed from equation (3.3), the stochastic convolution from equation (2.2) satisfies

(3.5) $$ \begin{align} z_t = e^{-t A} z_0+ \int_0^t e^{-(t-s) A} Q dW_s \overset{\tiny{\mbox{dist}}}{=} \widetilde{Z}_t^{(n)} +\sum_{k=0}^n Z_t^{(k)}\,, \end{align} $$

where

(3.6) $$ \begin{align} \mathscr{L} Z_t^{(k)} = Q_{k} dW_t^{(k)}\quad\text{and}\quad \mathscr{L} \widetilde{Z}_t^{(n)} = \widetilde{Q}_{n} d\widetilde{W}_t^{(n)}\, \end{align} $$

with initial conditions $Z_0^{(0)}= \cdots =Z_0^{(n)}=0$ and $\widetilde {Z}_0^{(n)}=z_0$ . It is worth noting that $ Z_t^{(0)}=X_t^{(0)}$ and that all equations but $R_t^{(n)}$ are ‘feed-forward’ in the sense that the forcing drift $B(X^{(0,k-1)}_t) - B(X^{(0,k-2)}_t)$ in the kth level is adapted to the filtration $\mathcal {F}_t^{(k-1)} = \sigma ( W_s^{(j)}: j \leq k-1, s \leq t)$ . In this sense, conditioned on $\mathcal {F}_t^{(k-1)}$ , $X^{(k)}_t$ is a forced linear equation with both stochastic and (conditionally) deterministic forcing. This in turn implies that conditioned on $\mathcal {F}_t^{(k-1)}$ , $X^{(k)}_t$ is a Gaussian random variable.

We will prove Theorem 2.1 by showing that for any fixed $t>0$ and all $k=1,\dots ,n$ ,Footnote ³

(3.7) $$ \begin{align} \begin{aligned} \mathrm{Law}(X_t^{(k)}\mid \mathcal{F}_t^{(k-1)}) &\sim \mathrm{Law}(Z_t^{(k)})\quad\text{a.s.,}\qquad \text{and}\\ \mathrm{Law}(R^{(n)}_t \mid \tau_\infty>t, \mathcal{F}_t^{(n)}) &\ll \mathrm{Law}(\widetilde{Z}_t^{(n)})\quad\text{a.s.}\, \end{aligned} \end{align} $$

We will see in Section 6.1 that the random existence time of $R^{(n)}$ is almost surely equal to that of u, and hence we will use $\tau _\infty $ in both settings. We will show in Section 6.2 how this sequence of statements about the conditional laws combined with the structure of equation (3.2) will imply Theorem 2.1.

3.2 Second decomposition

Accepting the result of equation (3.7) from the previous section, it might seem reasonable to replace the instances of $X_t^{(j)}$ in equation (3.1) with $Z_t^{(j)}$ . This would have a number of advantages. One is that the $Z_t^{(j)}$ are explicit Gaussian processes that will simplify the rigorous definition of some of the more singular terms in the decomposition. Additionally, it will emphasise the relationship between the $X_t^{(j)}$ construction in equation (3.1) and the tree constructions developed in the analysis of singular SPDEs (such as [Reference HairerHai13, Reference Mourrat, Weber and XuMWX15, Reference Gubinelli and PerkowskiGP17]), which is driven by isolating the singular objects in the solution.

Motivated by this discussion, we now consider the expansion

(3.8) $$ \begin{align} u_t \overset{\tiny{\mbox{dist}}}{=} \sum_{i=0}^n Y^{(i)}_t + S^{(n)}_t , \end{align} $$

where

(3.9) $$ \begin{align} \left\{ \begin{aligned} \mathscr{L} Y^{(0)}_t &= Q_0 dW^{(0)}_t, \\ \mathscr{L} Y^{(1)}_t &= B(Z^{(0)}_t)dt + Q_1 dW_t^{(1)}, \\ \mathscr{L} Y^{(2)}_t &= (B(Z^{(0,1)}_t) - B(Z^{(0,0)}_t))dt + Q_2 dW_t^{(2)}, \\ \mathscr{L} Y^{(3)}_t &= (B(Z^{(0,2)}_t) - B(Z^{(0,1)}_t))dt + Q_3 dW_t^{(3)}, \\ \vdots\quad&\qquad \qquad\vdots\quad \qquad \qquad\vdots \qquad\qquad \qquad\vdots \\ \mathscr{L} Y^{(n)}_t &= (B(Z^{(0,n-1)}_t) - B(Z^{(0,n-2)}_t))dt + Q_n dW_t^{(n)}, \\ \mathscr{L} S^{(n)}_t &= (B(Y^{(0,n)}_t+ S^{(n)}_t) - B(Z^{(0,n-1)}_t))dt + \widetilde{Q}_n d\widetilde{W} _t^{(n)}, \end{aligned} \right. \end{align} $$

and the Qs are again as in equation (3.3) and

$$ \begin{align*} Z^{(m,n)}_t = \sum_{i=m}^n Z_t^{(i)}, \end{align*} $$

with $Z_t^{(1)}, \dots , Z_t^{(n)}$ again defined by equation (3.6). Again, we take $Y_0^{(0)}=\cdots =Y_0^{(n)}=0$ and $S_0^{(n)}=u_0$ , where $u_0$ was the initial condition of equation (2.1).

Although in many ways we find the X expansion in equation (3.9) more intuitive and better motivated, we will find it easier to prove Theorem 2.1 for the Y expansion in equation (3.9) first. We will then use it to deduce Theorem 2.1 for the X expansion in equation (3.2).

More concretely, we will begin by proving that for each $k \leq n$ ,

(3.10) $$ \begin{align} \begin{aligned} \mathrm{Law}(Y_t^{(k)}\mid \mathcal{F}_t^{(k-1)}) &\sim \mathrm{Law}(Z_t^{(k)})\quad\text{a.s.,}\qquad\text{and}\\ \mathrm{Law}(S^{(n)}_t\mid t < \tau_\infty,\mathcal{F}_t^{(n)} ) &\ll \mathrm{Law}(\widetilde{Z}_t^{(n)}) \quad\text{a.s.}\,. \end{aligned} \end{align} $$

Then we will deduce that for $k=1,\dots ,n$ ,

(3.11) $$ \begin{align} \begin{aligned} \mathrm{Law}(X_t^{(k)}\mid\mathcal{F}_t^{(k-1)} ) &\sim \mathrm{Law}(Y_t^{(k)}\mid\mathcal{F}_t^{(k-1)})\quad\text{a.s.,}\qquad\text{and}\\ \mathrm{Law}(R^{(n)}_t\mid \tau_\infty>t, \mathcal{F}_t^{(n)}) &\ll \mathrm{Law}(\widetilde{Z}_t^{(n)} ) \quad\text{a.s.}\,. \end{aligned} \end{align} $$

By combining equation (3.10) with equation (3.11), we can deduce that equation (3.7) holds.

There is no fundamental obstruction to proving equation (3.7) directly, as it essentially requires the same calculation as proving equation (3.11). Similarly, we could have directly proven equation (3.7) before equation (3.10); however, along the way we would have collected most of the estimates needed to prove equation (3.10). We hope that proving all three statements, namely equations (3.7), (3.11) and (3.10), will help show the relationship between different ideas around singular SPDEs.

3.3 Cameron-Martin Theorem and Time-Shifted Girsanov Method

The Cameron-Martin Theorem and the closely related Girsanov Theorem are the classical tools for proving two stochastic processes are absolutely continuous. Both describe when a ‘shift’ in the drift can be absorbed into a stochastic forcing term while keeping the law of the resulting random variable or stochastic process absolutely continuous with respect to the original law.

More concretely, to prove equation (3.10), we will absorb the drift terms on the right-hand side of the equations for $Y_t^{(1)}, \dots , Y_t^{(n)}, S_t^{(n)}$ into the stochastic forcing term on the right-hand side of the the same equation. In the case of the Cameron-Martin Theorem, this absorption is done using the integrated, mild form of the equation, analogous to equation (2.2). The resulting expressions are identical in form to the analogous Z expressions implied by equation (3.6). The Girsanov Theorem proceeds similarly to the Cameron-Martin Theorem, but the drift is removed instantaneously at the level of the driving equation and not in an integrated form as in the Cameron-Martin Theorem. We will see that this leads to both stronger conclusions and a need for stronger assumptions in order to apply the Girsanov Theorem.

In [Reference Mattingly and SuidanMS05, Reference Mattingly and SuidanMS08, Reference WatkinsWat10], the Girsanov Theorem was recast by shifting the infinitesimal perturbation injected by the drift at one instance of time to a later instance. By shifting the drift perturbation forward in time by the flow $e^{-tA}$ , it is regularised in space. This regularised, time-shifted drift can then be compensated by the noise at the later moment in time, thereby extending the applicability of Girsanov’s Theorem. This extended applicability will be critical to our results. The price of the extended applicability is that only a modified form of trajectory-level absolute continuity is proven, which nonetheless is sufficient to deduce absolute continuity at the terminal time of the path. We have dubbed this approach the Time-Shifted Girsanov Method. A full discussion with all of the details is provided in Section 5.3.

3.4 Gaussian regularity

When applying either the Cameron-Martin Theorem or the Time-Shifted Girsanov Method, there is a tension between the roughness of the stochastic forcing, set by $Q\approx A^{\frac {\alpha }2}$ , and the roughness of the drift term on the right-hand side of the kth equations. The gap between these regularities cannot be too big. Hence, it is critical to understand the regularity of the solutions and the drift term involving the nonlinearity B.

When $\alpha < \frac 12$ , we will see that the Time-Shifted Girsanov Method can be applied directly to equation (2.1) to obtain the desired result following the general outline of [Reference Mattingly and SuidanMS08, Reference WatkinsWat10]. When $\alpha \in [\frac {1}{2},1)$ , the multilevel decomposition from equation (3.2) and equation (3.9) will be required to make sure the jump in regularity between the stochastic forcing in an equation and the drift to be removed is not too large.

The reason for this change at $\alpha =\frac 12$ is fundamental to our discussion. When $\alpha < \frac 12$ , the product of all the spatial functions f and g contained in $B(f,g)$ is well-defined classically as the functions will have positive Hölder regularity. Hence, pointwise multiplication is well-defined. When $\alpha \geq \frac 12$ , we must leverage the Gaussian structure of the specific processes being multiplied and a renormalisation procedure to make sense of the product of some of the terms in $B(f,g)$ . When $\alpha \in [\frac 12,\frac 34)$ , through these considerations, we will always be able to give meaning to $B(f,g)$ at each moment of time in all the needed cases. When $\alpha \in [\frac 34,1)$ , at times we must consider the time-integrated version of the drift term from the mild formulation (analogous to the second term from the right in equation (2.2)) and leverage time decorrelations of the Gaussian process to make sense of the nonlinear term in its integrated form $J(f,g)_t$ defined in equation (4.4).

3.5 Relation to trees and chaos expansions

We now briefly explore the relationship between tree representations of stochastic Gaussian objects from [Reference HairerHai13, Reference Mourrat, Weber and XuMWX15, Reference Gubinelli and PerkowskiGP17] and this work in a heuristic way. One way to view the trees in those works is to consider the expansion one obtains by formally substituting the integral representation of z given in equation (3.6) back into the first integral term on the right-hand side of equation (2.2). Repeated applications of this is one way to develop an expansion of the solution $u_t$ in terms of finite trees of z with a remainder. These tree representations of stochastic objects are key to the analysis in those works. We will later see that the drift terms of equation (3.9) can be decomposed into some of the same trees of z.

We push the idea of tree expansions further by grouping the trees formed by Z with different regularity and adding an extra stochastic forcing at each level. Here, looking back at equation (3.5), as we have subdivided our noise into n levels ( $Z^{(1)},\dots , Z^{(n)}$ ) and one remainder term ( $\widetilde {Z}^{(n)}$ ), we have a tree-like expansion mixing the Gaussian inputs of different levels. This work can be viewed as giving a more refined analysis of the stochastic objects in equation (2.1), since we finely decompose the noise into Gaussian processes of different regularity. We will see that our eventual assumptions on the Qs will imply that the regularity of the $Z^{(k)}$ increases with k.

Note that the drifts in equation (3.2) and equation (3.9) can be expanded fully in terms of the sum of trees of the Zs. Hence, we can understand the drifts in equation (3.2) and equation (3.9) as two different groupings of a subset of the tree objects from this expansion. We group them at each level in these two ways such that their regularity allows the use of the Cameron-Martin Theorem or the Time-Shifted Girsanov Method at that level. Clearly this grouping is not unique, but it makes analysis based on regularity more straightforward.

4.1 Function spaces and basic estimates

We shall denote by $\mathcal {C}^\gamma $ , $\gamma \in \mathbb {R}$ , the separable version of the Besov-Hölder space $B^\gamma _{\infty ,\infty }(\mathbb {T})$ of order $\gamma $ , namely the closure of periodic smooth functions with respect to the $B^\gamma _{\infty ,\infty }$ norm. See Appendix A for some details about Besov spaces. If $f \in \mathcal {C}^{\gamma }$ , we will say that f has (Hölder) regularity $\gamma $ . We will write $\mathcal {C}^{\gamma ^-}$ for the intersection of all of space $\mathcal {C}^{\beta }$ with $\beta < \gamma $ .Footnote ⁴

Given a Banach space $\mathbf {X}$ of functions $f(x)$ on $\mathbb {T}$ , we will write $C_T\mathbf {X}$ for the space of time-dependent functions $f(t,x)$ on $[0,T] \times \mathbb {T}$ such that for each $t \in [0,T]$ , $f(t,\;\cdot \;) \in \mathbf {X}$ and as $s \rightarrow t$ , we have that $\|f(s,\;\cdot \;)-f(t,\;\cdot \;)\|_{\mathbf {X}}\rightarrow 0$ . We will endow this space with the norm

$$ \begin{align*} \|f\|_{C_T\mathbf{X}} = \sup_{t \in [0,T]} \|f(t,\;\cdot\;)\|_{\mathbf{X}}. \end{align*} $$

Typical examples we will consider are $C_T\mathcal {C}^\beta $ and $C_TL^2$ . If $f \in C_T\mathcal {C}^{\gamma }$ , we will say that f has (Hölder) regularity $\gamma $ (in space). For convenience, we will write $C_T \mathcal {C}^{\gamma ^-}$ for the intersection of all the spaces $C_T \mathcal {C}^{\beta }$ with $\beta < \gamma $ .

As we are interested in solutions that might have a finite time of existence, we will introduce the one-point compactification of $\mathcal {C}^{\gamma }$ , . $\overline {\mathcal {C}}^{\gamma }$ is a topological space where the open neighbourhoods of are given by $\{ u \in \mathcal {C}^{\gamma }: \|u\|_{\mathcal {C}^\gamma }>R\}$ for $R>0$ . With a light abuse of notation, we will write $C_T \overline {\mathcal {C}}^{\gamma }$ to mean the space of all continuous functions on $[0,T]$ taking values in $\overline {\mathcal {C}}^{\gamma }$ . We do not place a norm on $C_T \overline {\mathcal {C}}^{\gamma }$ and view it only as a topological space. Observe that if $u \in C_T \overline {\mathcal {C}}^{\gamma }$ and $\tau \in (0,\infty ]$ such that for $t \in [0,T]$ , if $t \geq \tau $ and $u_t \in \mathcal {C}^{\gamma }$ if $t < \tau $ , then for all $t \in [0,T] \cap [0,\tau )$ , $u \in C_t \mathcal {C}^{\gamma }$ because u is continuous in $\overline {\mathcal {C}}^{\gamma }$ . We feel this justifies the notation $ C_T \overline {\mathcal {C}}^{\gamma }$ for continuous functions on $\mathcal {C}^{\gamma }$ even though the space is not endowed with the supremum norm. Additionally, because of our choice of open neighbourhoods of , if $\tau < \infty $ , then $\|u_t\|_{\mathcal {C}^\gamma } \rightarrow \infty $ as $t \rightarrow \tau $ . Again, we will write $C_T \overline {\mathcal {C}}^{\gamma ^-}$ for the intersection of all of the spaces $C_T \overline {\mathcal {C}}^{\beta }$ with $\beta < \gamma $ .

We collect a few useful properties of A and its semigroup in the following proposition. Here and throughout the text, we write $a \lesssim b$ to mean there exists a positive constant c so that $a \leq c b$ . When the constant c depends on some parameters, we will denote them by subscripts on $\lesssim $ . We will write $\gtrsim $ when the reverse inequality holds for some constant and $\eqsim $ when both $\lesssim $ and $\gtrsim $ hold (for possibly different constants).

Proposition 4.1. For $\gamma \in \mathbb {R}$ , $\partial _x:\mathcal {C}^\gamma \to \mathcal {C}^{\gamma -1}$ and $A:\mathcal {C}^\gamma \to \mathcal {C}^{\gamma -2}$ are bounded linear operators. Additionally, if $\delta \in \mathbb {R}$ with $\gamma \leq \delta $ , then

(4.1) $$ \begin{align} \|e^{-t A}f\|_{\mathcal{C}^\delta} \lesssim t^{\frac12(\gamma-\delta)}\|f\|_{\mathcal{C}^\gamma}\quad\text{and}\quad \|A^{\frac\delta2} e^{-t A}f\|_{L^2} \lesssim t^{\frac12(\gamma-\delta)}\|A^{\frac\gamma2} f\|_{L^2}. \end{align} $$

In particular, for any $\epsilon>0$ and $\delta ,t>0$ ,

(4.2) $$ \begin{align} \|A^{\frac\delta2}e^{-t A}f\|_{L^2} \lesssim \|A^{\frac\delta2}e^{-t A}f\|_{\mathcal{C}^\epsilon} \lesssim_\epsilon t^{\frac12(\gamma-\delta-\epsilon)}\|f\|_{\mathcal{C}^\gamma}. \end{align} $$

Using these estimates, one can obtain the regularity of the solution $z_t$ of equation (2.3).

Remark 4.2 (Regularity of stochastic convolution)

Using the results given in equation (4.1) and some classical embedding theorems, we have that if

(4.3) $$ \begin{align} z_t= \int_0^t e^{-(t-s) A} \Xi\,dW_s, \end{align} $$

with $\Xi \approx A^{\frac {\delta }2}$ (and hence a mild solution of $\mathscr {L} z_t = \Xi \,dW_t$ ), then $\|z_t\|_{\mathcal {C}^\gamma } < \infty $ uniformly on finite time intervals for all $\gamma <\frac 12-\delta $ . More compactly, $z \in C_t\mathcal {C}^{(\frac 12-\delta )^-}$ for any $t>0$ almost surely.

Given the structure of the nonlinearity B, we are particularly interested in the properties of the pointwise product of two functions. We now summarise the results in the classical setting and recall the results in the Gaussian setting.

With this fact about Gaussians in mind, we make the following definition to simplify discussions.

4.2 Regularity of the mild form of the nonlinearity

Looking back at equation (2.2), we see that the nonlinearity B integrated in time against the heat semigroup (namely, the first term on the right-hand side of this equation) will be a principal object of interest. We now pause to study the main properties of this object while postponing some more technical considerations to the Appendix.

In the sequel, it will be notationally convenient to define the bilinear operator $J(f,g)_t$ by

(4.4) $$ \begin{align} J(f, g)_t : = \int_0^t e^{-(t-s)A}B(f_s, g_s)\,ds, \end{align} $$

and $J(f)_t=J(f,f)_t$ .

Remark 4.5 (Canonical regularity of J)

If $f \in C_t\mathcal {C}^\gamma $ and $g \in C_t\mathcal {C}^\delta $ with $\gamma + \delta>0$ , then $B(f, g) \in C_t\mathcal {C}^{r -1}$ , where $r=\gamma \wedge \delta \wedge (\gamma +\delta )$ , which implies that

$$ \begin{align*} \| J(f,g)_t\|_{\mathcal{C}^\beta} \lesssim \int_0^t \|e^{-(t-s)A} B(f_s,g_s)\|_{\mathcal{C}^\beta}\, ds \lesssim \|f g \|_{C_t\mathcal{C}^{r}}\int_0^t \frac{1}{(t-s)^{\frac12(\beta-r+1)}}\, ds\,. \end{align*} $$

Here we have used the estimate from Proposition 4.1. Since this last integral is finite when $\frac 12(\beta -r+1)<1$ , we deduce that $\beta < r+1$ , implying that $J(f,g)_t \in \mathcal {C}^{(r+1)^-}$ , where $r=\gamma \wedge \delta \wedge (\gamma +\delta )$ .

As mentioned in Remark 4.3 (and proved in Appendix B), we can prove that product $fg$ is well-defined with canonical regularity in the specific examples needed in this work, when $f \in C_t\mathcal {C}^\gamma $ and $g \in C_t\mathcal {C}^\beta $ with $\gamma + \delta> -\frac 12$ . However, we will show in Appendix B that when $\gamma +\delta> -\frac 32$ , $J(f,g)$ is well-defined with $J(f,g) \in C_t\mathcal {C}^{(r+1)^-}$ , where $r=\gamma \wedge \delta \wedge (\gamma +\delta )$ , even though the product $fg$ might not be well-defined with its canonical regularity.

The regularising nature of J highlighted in Remark 4.7 is closely related to the use of fixed point methods to prove the existence and uniqueness of local in-time solutions with the needed regularity. This is explored further in Appendix C.

4.3 Regularity of solutions

We now turn to the regularity of the Burgers equation (2.1) and those in our decompositions in equation (3.2) and equation (3.9). Most of the equations are forced linear equations except for the remainder equations $R_t$ and $S_t$ and the original Burgers equation (2.1). While the following discussion will be illuminating in these later cases, it is most directly applicable in the setting of forced linear equations. A complete treatment in the nonlinear setting (namely $R_t$ , $S_t$ and equation (2.1)) involves a fixed point argument that we postpone to Appendix C. Nonetheless, the discussion in this section will still be illuminating to these cases while focusing on the forced linear equation setting.

We begin by studying a more general equation that can subsume most of the equations in our decompositions in equation (3.2) and equation (3.9). Since all of the forcing drift terms on the right-hand side of the equations are a finite sum of terms of the form $B(f,g)$ for some f and g, it is enough to consider the more general equation

(4.5) $$ \begin{align} \mathscr{L} v_t = B(f_t,g_t) dt + \Xi\,dW_t\, \end{align} $$

for some given $f \in C_t\mathcal {C}^\gamma $ and $g \in C_t\mathcal {C}^\delta $ and $\Xi \approx A^{\beta /2}$ for some $\beta , \gamma \in \mathbb {R}$ . All of the forced linear equations of interest are a finite sum of equations of this form.

The solution to equation (4.5) with initial condition $v_0$ is given by

$$ \begin{align*} v_t= e^{-t A}v_0 + J(f,g)_t + z_t, \end{align*} $$

where now $z_t$ is the stochastic convolution solving equation (4.3) and J is again defined by equation (4.4). We will assume that f and g are such that $J(f,g)_t$ has canonical regularity in the sense of Definition 4.6.

For any $t>0$ and any reasonable $v_0$ , $e^{-t A}v_0 \in \mathcal {C}^b$ for all $b \in \mathbb {R}$ . Hence, the first term will not be the term that fixes the regularity of the equation, and either J or z will determine the maximal regularity of the system.

By Remark 4.2, the stochastic convolution $z \in C_t\mathcal {C}^{(\frac 12-\beta )^-}$ . By Definition 4.6, we have that $J(f,g) \in C_t \mathcal {C}^{(r +1)^-}$ , where $r=\gamma \wedge \delta \wedge (\gamma +\delta )$ . Hence the regularity of the solution will be set by the stochastic convolution z to be $C_t\mathcal {C}^{(\frac 12-\beta )^-}$ if $r> \frac 12 - \beta $ . We will arrange our choice of parameters so that equations (3.2) and (3.9) will always satisfy this condition. Hence, the equations will always have their regularity set by the stochastic convolution term in the equation. With this motivation, we make the following definition.

The above considerations are also relevant to assessing the regularity of the remaining equations in (3.2) and (3.9) as well as the original Burgers equation. Observe that the solution to equation (2.1) can be written as

$$ \begin{align*} u_t= e^{-tA} u_0 + J(u)_t + z_t, \end{align*} $$

where $z_t$ solves equation (2.3). Since $Q \approx A^{\alpha /2}$ , we know from Remark 4.2 that $z \in C_t\mathcal {C}^{(\frac 12-\alpha )^-}$ . Since we are interested in $\alpha < 1$ , we have that $\frac 12-\alpha> -\frac 12$ . In light of Remark 4.7, we see that $u_t$ has the same regularity in space as $z_t$ ; then $J(u)_t$ will be more regular in space (assuming we can show that $J(u)_t$ has the canonical regularity dictated by u). Hence, it is expected that in our setting the regularity of equation (2.1) will be set by the regularity of the stochastic convolution term so $u \in C_t\mathcal {C}^{(\frac 12-\alpha )^-}$ . For more details, see the discussion in Appendix C.

5.1 The Cameron-Martin Theorem

The Cameron-Martin Theorem gives if and only if conditions describing when the $\mathrm {Law}(z_t)$ is equivalent to $\mathrm {Law}(v_t)$ , with $v_t=z_t+h_t$ , for a fixed time t and a deterministic shift $h_t$ . If $\Xi \approx A^{\beta /2}$ , then the covariance operator of the Gaussian random variable $z_t$ is (up to a compact operator) $A^{\beta -1}$ . Then Cameron-Martin Theorem requires that $\|A^{\frac {1-\beta }{2}} h_t\|_{L^2} < \infty $ (see [Reference Da PratoDP06, Theorem 2.8]). If $F_t$ is random but independent of the stochastic forcing $W_t$ , then we can still apply the Cameron-Martin Theorem by first conditioning on the trajectory of F. This produces the following sufficient condition for absolute continuity, which is a version of the classical Cameron-Martin Theorem adapted to our setting.

Remark 5.2. If $F=B(f,g)$ for some $f \in C_t\mathcal {C}^\gamma $ and $g\in C_t\mathcal {C}^\delta $ such that $J(f,g)_t$ has canonical regularity, then $h_t=J(f,g)_t \in \mathcal {C}^{(r+1)^-}$ , where $r=\gamma \wedge \delta \wedge (\gamma +\delta )$ . Hence, the condition in Theorem 5.1 becomes $r+\beta =\gamma \wedge \delta \wedge (\gamma +\delta )+\beta>0$ .

Notice that this remark extends to the setting where

(5.3) $$ \begin{align} F=\sum_{i=0}^m c_i B(f^{(i)},g^{(i)}) \end{align} $$

for some $ c_i \in \mathbb {R}$ , $f^{(i)} \in C_t\mathcal {C}^{\gamma _i}$ and $g^{(i)} \in C_t\mathcal {C}^{\delta _i}$ , where $r_i=\gamma _i\wedge \delta _i \wedge (\gamma _i+\delta _i)$ and the condition on the indexes becomes $r+\beta>0$ with $r=\min r_i$ .

5.2 The standard Girsanov Theorem

The Girsanov Theorem is essentially the specialisation of the Cameron-Martin Theorem to the path-space of a stochastic differential equation, while relaxing the assumptions to allow random shifts in the drift that are adapted to the Brownian motion forcing the SDE.

We again consider the setting of equation (5.1). Since we will be discussing path-space measures, we will write $v_{[0,t]}$ and $z_{[0,t]}$ for the random variable denoting the entire path of v and z respectively, on the time interval $[0,t]$ . We now give a version of the Girsanov Theorem adapted to our setting.

Theorem 5.5 (A version of Girsanov)

In the setting of equation (5.1) with $z_0=v_0$ , let $\mathcal {F}_t$ be a filtration to which W is an adapted Brownian motion, and let $\mathcal {G}_t$ be a filtration independent of $\mathcal {F}_t$ . Let $\tau $ be a stopping time adapted to $\mathcal {H}_t=\sigma (\mathcal {G}_t,\mathcal {F}_t)$ with $\mathbb {P}(\tau>0) > 0$ such that $v_t$ and $F_t$ are stochastic processes adapted to $\mathcal {H}_{t\wedge \tau }$ so that $v_t$ solves equation (5.1) for $t < \tau $ , and

(5.4) $$ \begin{align} \int_0^t \|A^{-\beta/2} F_s\|_{L^2}^2 \, ds < \infty, \end{align} $$

almost surely for all $t < \tau $ . Then $\mathrm {Law}( v_{[0,t]} \mid t < \tau , \mathcal {G}_t) \ll \mathrm {Law}( z_{[0,t]})$ almost surely.

In particular, it is sufficient that $F \in C_t\mathcal {C}^\sigma $ for $\sigma +\beta>0$ and any $t < \tau $ for equation (5.4) to hold almost surely.

The condition $r+\beta =\gamma \wedge \delta \wedge (\gamma +\delta )+\beta>1$ from Remark 5.6 should be contrasted with the condition $r+\beta =\gamma \wedge \delta \wedge (\gamma +\delta )+\beta>0$ from Remark 5.2. Relative to the Cameron-Martin Theorem 5.1, the basic Girsanov Theorem 5.5 does have the advantage that $F_t$ can be adapted to the forcing Brownian motion and not independent. Also, the results are not comparable, as Theorem 5.5 proves pathwise equivalence while Theorem 5.1 only proves equivalence at a fixed time t.

Proof of Theorem 5.5

We begin by defining

$$ \begin{align*} \tau_M=\tau \wedge \inf\Big\{ t>0 : \int_0^t \|A^{-\beta/2} F_s\|^2_{L^2} ds > M\Big\} \end{align*} $$

and

$$ \begin{align*} \mathscr{L} \, v_t^M = \mathbf{1}_{\{t < \tau_M\}}F_t \,dt+ \Xi\, dW_t. \end{align*} $$

Observe that $v_t^M$ is well-defined on $[0,t]$ for any $t> 0$ due to the stopping time and that $v_t= v_t^M$ on the event $\{t < \tau ^M\}$ . Then

$$ \begin{align*} \exp\Big( \int_0^t \|A^{-\beta/2}\mathbf{1}_{\{s < \tau_M\}}F_s\|^2_{L^2} ds\Big) < C(1+ e^M), \end{align*} $$

almost surely for some fixed constant C. Thus, the classical Kazamaki criterion (see for instance [Reference KrylovKry09]) ensures that the local-martingale in the Girsanov Theorem is an integrable martingale. Let $v_{[0,t]}^{M}$ and $z_{[0,t]}$ be the path-valued random variables over the time interval $[0,t]$ . We have that $\mathrm {Law}( v_{[0,t]}^{M} )$ is equivalent as a measure to $\mathrm {Law}( z_{[0,t]})$ . Since $\mathcal {G}_t$ is independent of the Brownian motion W, we have that $\mathrm {Law}( v_{[0,t]}^{M} \mid \mathcal {G}_t)$ is equivalent as a measure to $\mathrm {Law}(z_{[0,t]})$ almost surely.

We now show that we can remove the truncation level M. Now let E be a measurable subset of paths of length T such that $\mathbb {P}( z_{[0,T]} \in E ) =0$ . To prove the absolute continuity claim, we need to show that $\mathbb {P}\big (v_{[0,T]}\in E,\tau>T \mid \mathcal {G}_T\big )=0$ almost surely. If $\mathbb {P}(\tau>T \mid \mathcal {G}_T)=0$ almost surely, we are done. Hence we proceed assuming $\mathbb {P}(\tau>T \mid \mathcal {G}_T)>0$ almost surely.

Because, conditioned on $\mathcal {G}_T$ , the law of $ v^M_{[0,T]}$ is equivalent to the law of $z_{[0,T]}$ almost surely, we know that $\mathbb {P}( v^M_{[0,T]}\in E \mid \mathcal {G}_T ) =0$ for all $M>0$ almost surely. We also know from the construction of $ v^M$ that $\mathbb {P}( v^M_{[0,T]}\in E, \tau _M> T \mid \mathcal {G}_T ) =\mathbb {P}( v_{[0,T]}\in E, \tau _M > T \mid \mathcal {G}_T )$ . Now, since

$$ \begin{align*} \Big\{ v_{[0,T]}\in E,\tau>T\Big\} \subset\bigcup_{M>0}\Big\{ v_{[0,T]}\in E,\tau_M>T\Big\}, \end{align*} $$

we have that

$$ \begin{align*} \mathbb{P}\big(v_{[0,T]}\in E,\tau>T \mid \mathcal{G}_T\big) & \leq\sup_{M>0}\mathbb{P}\big(v_{[0,T]}\in E,\tau_M > T \mid \mathcal{G}_T\big)\\ & \leq\sup_{M>0}\mathbb{P}\big( v^M_{[0,T]} \in E,\tau_M > T \mid \mathcal{G}_T\big)\\ &\leq\sup_{M>0}\mathbb{P}\big( v^M_{[0,T]}\in E \mid \mathcal{G}_T\big) = 0, \end{align*} $$

as already noted, since $\mathbb {P}\big (z_{[0,T]}\in E \big )=0$ .

5.3 The Time-Shifted Girsanov Method

We now present the Time-Shifted Girsanov Method, which was developed in [Reference Mattingly and SuidanMS05, Reference Mattingly and SuidanMS08, Reference WatkinsWat10]. It will provide essentially the same regularity conditions in our setting as in Remark 5.2 while allowing adapted shifts as in the standard Girsanov Theorem. Interestingly, we will see that the classical Cameron-Martin Theorem still holds some advantages when dealing with extremely rough Gaussian objects.

Considering the mild-integral formulation of equation (5.1)

(5.5) $$ \begin{align} v_t - e^{-t A} v_0 = \int_0^t e^{-(t-s) A} F_s\, ds + \int_0^t e^{-(t-s) A} \Xi\, dW_s\,, \end{align} $$

we can understand the first term as the shift of the Gaussian measure, which is the second term. We will now recast the drift term in equation (5.5) to extend the applicability of the Girsanov Theorem.

We begin with the observation that for fixed $T>0$ ,

(5.6) $$ \begin{align} \begin{aligned} \int_0^T e^{-(T-s) A}F_sds &= \int_0^T e^{-\frac12(T-s) A}e^{-\frac12(T-s) A}F_s\,ds \\ &= 2\int_{\frac{T}2}^T e^{-(T-s) A}e^{-(T-s) A}F_{2s-T}\,ds= \int_{0}^T e^{-(T-s) A} \widetilde F_s \,ds, \end{aligned} \end{align} $$

where

(5.7) $$ \begin{align} \widetilde F_s=2\mathbf{1}_{[\frac{T}2,T]}(s) e^{-(T-s) A}F_{2s-T}\,. \end{align} $$

Since $2s-T \leq s$ for all $s \in [\frac {T}2,T]$ , $\widetilde F_s$ is adapted to the filtration of $\sigma $ -algebras generated by the forcing Wiener process W when $F_s$ is also adapted to the same filtration. Hence, we can define the auxiliary Itô stochastic differential equation

(5.8) $$ \begin{align} \mathscr{L} \,\widetilde v_t = \widetilde F_t\, dt + \Xi\, dW_t, \end{align} $$

which is driven by the same stochastic forcing as used to construct $v_t$ . Choosing the initial data to coincide with $v_0$ , the mild/integral formulation of this equation is

(5.9) $$ \begin{align} \widetilde v_t - e^{-t A}v_0 = \int_0^t e^{-(t-s)A} \widetilde F_s\, ds + \int_0^t e^{-(t-s) A} \Xi\, dW_s\,. \end{align} $$

By comparing equation (5.5) and equation (5.9), we see that $\widetilde v_T=v_T$ , while $\widetilde v_t$ need not equal $v_t$ for $t\neq T$ . Hence, if we use the standard Girsanov Theorem to show that the law of $\widetilde v_{[0,T]}$ on path-space is absolutely continuous with respect to the law of $z_{[0,T]}$ (the solution to equation (5.1)), then we can conclude that the law of $\widetilde v_T$ (at the specific time T) is absolutely continuous with respect to the law of $z_T$ (again at the specific time T). Finally, since $v_T= \widetilde v_T$ , we conclude that the law of $v_T$ is absolutely continuous with respect to the law of $z_T$ , both at the specific time T.

The power of this reformulation is seen when we write the condition needed to apply the Girsanov Theorem to remove the drift from equation (5.8). We now are required to have control over moments of

(5.10) $$ \begin{align} \int_0^T \| A^{-\beta/2}\widetilde F_s\|_{L^2}^2\, ds =\int_0^T \| A^{-\beta/2} e^{-(T-s)A} F_s\|_{L^2}^2\, ds \end{align} $$

to apply the standard Girsanov Theorem to transform the path-space law of $\widetilde v_{[0,T]}$ to that of $z_{[0,T]}$ . Comparing equation (5.4) with equation (5.10), we see that the additional semigroup $e^{-(T-s)A}$ in the integrand improves its regularity significantly.

Similarly, if we want to compare the distribution at time $T>0$ of two equations starting from different initial conditions $v_0,z_0 \in \mathcal {C}^b$ , for $\beta \in \mathbb {R}$ , then we can observe that

$$ \begin{align*} e^{-T A} v_0 = e^{-T A} z_0 + e^{-T A} (v_0-z_0) &=e^{-T A} z_0 + \int_{\frac{T}2}^T e^{-(T-s) A} \frac{2}{T}e^{-s A} (v_0 -z_0)\,ds\\ &= e^{-T A} z_0 + \int_{\frac{T}2}^T e^{-(T-s) A} \widetilde F_s^{(0)}\,ds, \end{align*} $$

where $\widetilde F_s^{(0)}= \mathbf {1}_{[\frac {T}2,T]}(s)\frac {2}{T}e^{-s A} (v_0 -z_0)$ . This is the observation at the core of the Bismut-Elworthy-Li formula [Reference BismutBis84, Reference Elworthy and LiEL94]. Observe that $\widetilde F^{(0)} \in C_T \mathcal {C}^b$ for any $b \in \mathbb {R}$ , regardless of the initial conditions, so $A^{-\frac {\beta }2} \widetilde F^{(0)} \in C_T L^2$ , and we will always be able to use the Girsanov Theorem to remove this term.

It will be convenient to consider a slightly generalised setting where $v_t$ , $\widetilde v_t$ and $\zeta _t$ , respectively, solve mild forms of the following equations,

(5.11) $$ \begin{align} \begin{aligned} \mathscr{L} \,v_t &= F_t\,dt +\,G_t\, dt+ \Xi\, dW_t, \\ \mathscr{L} \,\widetilde v_t &= (\widetilde F_t+\widetilde F_t^{(0)}) \,dt + G_t\, dt + \Xi\, dW_t,\\ \mathscr{L} \,\zeta_t &= G_t\,dt+ \Xi\, dW_t, \end{aligned} \end{align} $$

with initial conditions $v_0$ and $z_0$ , where $\zeta _0=\widetilde v_0 = z_0$ . Here $\widetilde F_t$ is defined as in equation (5.7), $\widetilde F_t^{(0)}$ as just above, and $G_t$ and $F_t$ are some stochastic processes.

Theorem 5.9 (Time-Shifted Girsanov Method)

In the setting of equation (5.11), let $\mathcal {F}_t$ be a filtration to which W is an adapted Brownian motion, let $\mathcal {G}_t$ be a filtration independent of $\mathcal {F}_t$ . Fix initial conditions $v_0$ and $z_0$ , and let $\tau $ be a stopping time adapted to $\mathcal {H}_t=\sigma (\mathcal {F}_t,\mathcal {G}_t)$ such that $\mathbb {P}(\tau>0) > 0$ . Let $G_t$ be stochastic processes adapted to $\mathcal {G}_t$ and defined for all $t \geq 0$ . Let $v_t$ and $F_t$ be stochastic process adapted to $\mathcal {H}_{t\wedge \tau }$ such that $v_t$ solves equation (5.11) for $t < \tau $ and

(5.12) $$ \begin{align} \int_0^t \|A^{-\frac{\beta}{2}} e^{-(t-s)A} F_s\|_{L^2}^2\, ds < \infty \end{align} $$

almost surely for all $t < \tau $ , and $v_t$ and $\zeta _t$ have initial conditions $v_0$ and $z_0$ , respectively. Then $\mathrm {Law}( v_{t} \mid t < \tau , \mathcal {G}_t ) \ll \mathrm {Law}( \zeta _{t} \mid \mathcal {G}_t)$ almost surely. Additionally, there exists a solution $\widetilde v_t$ , which solves equation (5.11) for $t < \tau $ and with $\mathrm {Law}(\widetilde v_{[0,t]} \mid t < \tau , \mathcal {G}_t ) \ll \mathrm {Law}( \zeta _{[0,t]} \mid \mathcal {G}_t)$ almost surely. In particular, it is sufficient that $F \in C_t\mathcal {C}^\sigma $ almost surely for $\sigma +\beta +1>0$ and for any $t < \tau $ to ensure that equation (5.12) holds.

Remark 5.10. In the setting of Theorem 5.9, we assume that there exist stochastic processes $f_t$ and $g_t$ so that $F_t=B(f_t,g_t)$ for all $t < \tau $ with $f \in C_t\mathcal {C}^\gamma $ , $g \in C_t\mathcal {C}^\delta $ and such that $F_t=B(f_t,g_t)$ has canonical regularity, namely $F \in C_t \mathcal {C}^{r-1}$ for $r=\gamma \wedge \delta \wedge (\gamma +\delta )$ and all $t < \tau $ . Then the regularity assumption of equation (5.4) is satisfied, provided that $r+\beta =\gamma \wedge \delta \wedge (\gamma +\delta )+\beta>0$ .

As in Remark 5.2, this remark extends to the setting where

$$ \begin{align*} F=\sum_{i=0}^m c_i B(f^{(i)},g^{(i)}) \end{align*} $$

for some $ c_i \in \mathbb {R}$ , $f^{(i)} \in C_t\mathcal {C}^{\gamma _i}$ and $g^{(i)} \in C_t\mathcal {C}^{\delta _i}$ , where $r_i=\gamma _i\wedge \delta _i \wedge (\gamma _i+\delta _i)$ and the condition on the indexes becomes $r+\beta>0$ with $r=\min r_i$ .

Proof of Theorem 5.9

Fixing a time T, we define

$$ \begin{align*} \int_0^T \|A^{-\frac{\beta}{2}} \widetilde F_s\|^2_{L^2}\, ds= \int_0^T \|A^{-\frac{\beta}{2}} e^{-(T-s)A} F_s\|^2_{L^2}\, ds <\infty, \end{align*} $$

almost surely on the event $\{T < \tau \}$ . Fix a positive integer M. The following stopping time

$$ \begin{align*} \tau_M=\tau \wedge \inf\Big\{ t>0 : \int_0^t \|A^{-\frac{\beta}{2}} \widetilde F_s\|^2_{L^2}\, ds > M\Big\} \end{align*} $$

is well-defined, and $\tau _M \rightarrow \tau $ monotonically as $M \rightarrow \infty $ . Let $\widetilde v_t$ be the solution to equation (5.8), and observe that it is well-defined on $[0,T]$ on the event $\{T < \tau \}$ with $v_T=\widetilde v_T$ on the same event. Now consider

$$ \begin{align*} \mathscr{L} \,\widetilde v_t^M = \mathbf{1}_{\{t < \tau_M\}}\widetilde F_t\, dt + \widetilde F_t^{(0)} dt +G_t\,dt+ \Xi\, dW_t, \end{align*} $$

with $\widetilde v_0=z_0$ . Clearly, $\widetilde v_t^M= \widetilde v_t$ for $t < \tau _M$ . Furthermore, because of the definition of $\tau _M$ and the fact that $\widetilde F^{(0)} \in C_T\mathcal {C}^b$ for any $b \in \mathbb {R}$ , the classical Girsanov Theorem implies that the law of the trajectories of $\widetilde v^M$ on $[0,T]$ , conditioned on $\mathcal {G}_T$ , are equivalent (i.e., mutually absolutely continuous) to the law of  $\zeta $ , conditioned on $\mathcal {G}_T$ , on $[0,T]$ . In the sequel, we will write $\zeta _{[0,T]}$ for the random variable on paths of lengths T induced by the law of $\zeta $ .

By the same argument as in Theorem 5.5, we remove the localisation by $\tau ^M$ to obtain $ \mathrm {Law}(\widetilde v_{[0,T]}\mid \tau>T) \ll \mathrm {Law}(\zeta _{[0,T]} \mid \mathcal {G}_T\big )$ almost surely.

To conclude the proof, we let D be any measurable subset such that $\mathbb {P}( \zeta _T \in D\mid \mathcal {G}_T)=0$ , where $z_T$ is the distribution of z at the fixed time T. Let $D_{[0,T]}$ be the subset of path-space of trajectories that are in D at time T. Then

(5.13) $$ \begin{align} & 0=\mathbb{P}\big( \zeta_T \in D \mid \mathcal{G}_T)\nonumber\\ &\qquad\qquad = \mathbb{P}( \zeta_{[0,T]} \in D_{[0,T]} \mid \mathcal{G}_T\big) = \mathbb{P}\big(\widetilde v_{[0,T]} \in D_{[0,T]} , \tau> T \mid \mathcal{G}_T\big) \nonumber\\ & \qquad\qquad\qquad\qquad\qquad\qquad= \mathbb{P}\big( \widetilde v_T\in D,\tau> T \mid \mathcal{G}_T\big) =\mathbb{P}\big( v_T\in D , \tau> T \mid \mathcal{G}_T\big), \end{align} $$

where the last equality follows from the fact that $ \widetilde v_T=v_T$ on the event $\{\tau> T\}$ . The chain of implications in equation (5.13) shows that the law of $v_T$ restricted to the event $\{\tau> T\}$ is absolutely continuous with respect to the law of $\zeta _T$ with both conditioned on $\mathcal {G}_T$ .

5.4 Range of applicability of methods

We now consider the regimes for which the Cameron-Martin Theorem and the Time-Shifted Girsanov Method can be applied directly to equation (2.1) to prove Theorem 2.1. We will proceed formally with the understanding that some neglected factors will lead to additional complications that will require more nuanced arguments.

For the moment, we assume that equation (2.1) has canonical regularity, namely the regularity dictated by the stochastic convolution term. Thus, $u \in C_t\mathcal {C}^{(\frac 12-\alpha )^-}$ , where recall that $\alpha $ is the exponent that sets the spatial regularity of the forcing.

When $\alpha <\frac 12$ , the solution $u_t$ to equation (2.1) has positive Hölder regularity with $u_t \in \mathcal {C}^{(\frac 12-\alpha )^-}$ . This implies that $B(u_t)$ has canonical regularity with $B(u_t) \in \mathcal {C}^{(-\frac 12-\alpha )^-}$ . Thus, the regularity condition to apply the Cameron-Martin Theorem or the Time-Shifted Girsanov Method to equation (2.2) becomes $\frac 12-\alpha + \alpha =\frac 12>0$ , which is always true. See Remark 5.2, Remark 5.10 and Remark 5.11.

When $\alpha \geq \frac 12$ , the solution $u_t$ to equation (2.1) has negative Hölder regularity since $u_t \in \mathcal {C}^{(\frac 12-\alpha )^-}$ still. If we proceed as if the relevant terms have canonical regularity ( $B(u_t)$ in the case of the Time-Shifted Girsanov Method and $J(u)_t$ in the case of the Cameron-Martin Theorem), then the regularity condition becomes $2(\frac 12-\alpha )+\alpha =1-\alpha>0$ , which restricts us to the setting of $\alpha <1$ . One cannot directly apply either the Time-Shifted Girsanov Method or the Cameron-Martin Theorem to equation (2.1) when $\alpha \geq \frac 12$ . We will see that we need the multilevel decomposition to incrementally improve the regularity of the solution to the point where we can apply the Time-Shifted Girsanov Method to the last level, namely $R^{(n)}$ or $S^{(n)}$ depending on the decomposition. Along the way, we typically use the Cameron-Martin Theorem to prove equivalence of the levels in the decomposition to the appropriate Gaussian processes. This is possible since the fed-forward structure of the decomposition means each level is conditionally independent from the previous. When $\alpha \in [\frac 34,1)$ , there is an added complication that the terms to be removed by the change of measure can be defined only when integrated against the heat semigroup. This necessitates the use of the Cameron-Martin Theorem rather than the Time-Shifted Girsanov Method.

5.5 Interpreting the Time-Shifted Girsanov Method

It is tempting to dismiss the manipulations in equation (5.6) as a trick of algebraic manipulation. We encourage you not to do so.

The standard Girsanov Theorem compares the two equations of the form in equation (5.1) and asks when we can shift the noise realisation to another ‘allowed’ noise realisation to absorb any differences in the drift terms, namely the $F_t$ in our setting. Here, ‘allowed’ means in a way that across all realisations, the resulting noise term’s distribution stays equivalent to the original noise term’s distribution. To keep the path measures equivalent on $[0,T]$ , one needs to do this instantaneously at every moment of time.

Since the equation for $z_t$ in equation (5.1) is a forced linear equation, the linear superposition principle (a.k.a. Duhamel’s principle or the variation of constants formula) applies. It states that we move an impulse injected into the system at time s to another time t by mapping it under the linear flow from the tangent space at time s to the tangent space at time t. Through this lens, we can interoperate equation (5.6) as a reordering of the impulses injected into the system by $F_s$ over the interval $s \in [0,T]$ . The impulse injected at time s is moved to time $t=\frac 12(T+s)$ via $e^{-(t-s)A}=e^{-\frac 12(T-s)A}$ . $\widetilde F_t$ is the resulting effective impulse at the time $t=\frac 12(T+s)$ . The Time-Shifted Girsanov Method compensates for the time-shifted impulse $\widetilde F_t$ using the forcing via a change of measure. The resulting $\widetilde F_t$ is more regular than $F_t$ for $t<T$ . The regularising effect of the semigroup $e^{tA}$ vanishes as we approach T. The requirement $\beta +\gamma +1>0$ ensures that the singularity at $t=T$ is sufficiently integrable to apply the classical Girsanov Theorem to the resulting process with its forcing impulses rearranged, which was denoted by $\widetilde v$ in the Time-Shifted Girsanov discussion in Section 5.3.

6.1 Regularity and existence times of solutions

We begin with a simple lemma that relates the maximal time of existence of $u_t$ with those of $X^{(0)}, X^{(1)}, \ldots $ , $X^{(n)}$ , $R^{(n)}$ satisfying equations (3.1) and (3.2) and $Y^{(0)}, Y^{(1)}, \ldots $ , $Y^{(n)}$ , $S^{(n)}$ satisfying equations (3.8) and (3.9).

Proof. The argument is the same in both cases. We detail the first case. $X^{(0)}_t$ , $X^{(1)}_t$ , $\ldots $ , $X^{(n)}_t$ exist for all time t because they are linear equations, and the drifts are well-defined for all time. If equation (3.1) holds, we have

$$ \begin{align*} u_t - \sum_{i=0}^n X^{(i)}_t \overset{\tiny{\mbox{dist}}}{=} R^{(n)}_t, \end{align*} $$

so the maximal time of existence for $R^{(n)}$ is almost surely the same as that of u. If equation (3.3) holds, then equation (3.2) combined with equation (3.3) implies equation (3.1).

6.2 Absolute continuity via decomposition

As already indicated, we will prove Theorem 2.1 using the decomposition in either equation (3.1) or equation (3.8). In the first case, we will prove equation (3.7), and in the second case equation (3.10). In both cases, Theorem 2.1 will follow from inductively applying the following lemma.

Proof of Lemma 6.3

We can assume $\mathrm {Law}(U' \mid \mathcal {G})(\omega ) \ll \mathrm {Law}(Z')$ for every $\omega $ . Let D be any measurable set with $\mathbb {P}(Z'+Z \in D)=0$ . Since Z is $\mathcal {G}$ -measurable and $Z'$ is independent of $\mathcal {G}$ , there exists a set E such that $\mathbb {P}(Z \in E) = 1$ and

$$ \begin{align*}0 = \mathbb{P}(Z' \in D - x \mid Z = x) = \mathbb{P}(Z' \in D - x)\end{align*} $$

for every $x \in E$ . Also, we have $\mathbb {P}(U \in E) = 1$ since $\mathrm {Law}(U) \ll \mathrm {Law}(Z)$ , and $\mathbb {P}(U' \in D - x \mid \mathcal {G}) = 0$ for every $x \in E$ since $\mathrm {Law}(U' \mid \mathcal {G}) \ll \mathrm {Law}(Z')$ . In particular, since U is $\mathcal {G}$ -measurable, the previous statement implies

$$ \begin{align*}\mathbb{P}(U + U' \in D) = \int_E \mathbb{P}(U' \in D - x \mid U = x) \,\mathbb{P}(U \in dx) = 0,\end{align*} $$

which completes the proof.

Corollary 6.4. Assume that, for some n, the system of equations (3.2) (respectively, (3.9)) is well-defined and satisfies the absolute continuity conditions given in equation (3.7) (respectively, equation (3.10)). Then in the first case,

$$ \begin{align*} \mathrm{Law}\left(R_t^{(n)} +\sum_{k=0}^n X_t^{(k)} \,\Big|\, \tau_\infty>t \right) \ll \mathrm{Law}\left(\widetilde{Z}_t^{(n)} +\sum_{k=0}^n Z_t^{(k)}\right) \end{align*} $$

holds, and in the second,

$$ \begin{align*} \mathrm{Law}\left(S_t^{(n)} +\sum_{k=0}^n Y_t^{(k)}\,\Big|\, \tau_\infty>t \right) \ll \mathrm{Law}\left(\widetilde{Z}_t^{(n)} +\sum_{k=0}^n Z_t^{(k)}\right) \end{align*} $$

holds.

The proof of the following corollary is completely analogous to that of Corollary 6.4.

Corollary 6.5. Assume that, for some n, the system of equations (3.2) and (3.9) is well-defined with equation (3.11) holding. Then

$$ \begin{align*} \mathrm{Law}\left(\sum_{k=0}^n X_t^{(k)}\right) \sim \mathrm{Law}\left(\sum_{k=0}^n Y_t^{(k)}\right). \end{align*} $$

6.3 Some informal computations

With the tools above, we can informally describe how to choose the number of levels and the $Q_i$ s in the decomposition given by equations (3.2) and (3.9). We focus on equation (3.2) because the equation structure of $X^{(i)}$ and the remainder $R^{(n)}$ are aligned, which makes discussions more intuitive. But the intuition is the same for equation (3.9), and later we will see that the result for equation (3.9) is actually more straightforward to prove rigorously. For simplicity, we do not distinguish between the Cameron-Martin Theorem and the Time-Shifted Girsanov Method, since they require the same condition on canonical regularity, as discussed in Remark 5.11.

Building on the preliminary discussion in Section 4, the first idea is that we assume every Gaussian term: that is, each of the $X^{(0,i)}$ , $B(X^{(0,i)})$ or $J(X^{(0,i)})$ in equation (3.2) can be well-defined with its canonical regularity. As a reminder, it means $X^{(i)}$ has the same Hölder regularity as $Z^{(i)}$ , and the B terms (and J terms) have canonical regularity following Remark 4.3 and Remark 4.5. The second idea is that we want $X^{(i)}$ to become smoother as i increases, so we can take $Q_i \approx A^{\alpha _i / 2}$ so that $Z^{(i)} \in C_T \mathcal {C}^{(\frac {1}{2} - \alpha _i)^-} $ , where $\alpha _i$ decreases as i increases. It is straightforward to show (later) that we can choose $Q_i$ , $\widetilde {Q}_n$ satisfying equation (3.3) with $\alpha _0 = \alpha $ .

When $\alpha < \frac {1}{2}$ , as discussed in Section 5.4, we can directly apply the Time-Shifted Girsanov Method to equation (2.2) and obtain $\mathrm {Law}(u_t) \ll \mathrm {Law}(z_t)$ . In fact, since the solutions can be seen to be almost surely global with finite control of some moments of the norm, one can show that $\mathrm {Law}(u_t) \sim \mathrm {Law}(z_t)$ .

When $\alpha \ge \frac {1}{2}$ , then $u_t \in \mathcal {C}^{(\frac {1}{2} - \alpha )^-}$ is a distribution with its canonical regularity, so we cannot make sense of $u_t^2$ classically. We first consider $u_t = X^{(0)}_t + X^{(1)}_t + R^{(1)}_t$ in equation (3.2) for $t < \tau _\infty $ . Clearly, $X^{(0)} = Z^{(0)}$ . To apply the Cameron-Martin Theorem or the Time-Shifted Girsanov Method on $X^{(1)}$ to show equation (3.7), the regularity condition is $2\alpha _0 - \alpha _1 < 1.$ On the other hand, note that the remainder $R^{(1)}$ is not a Gaussian object. We want $R^{(1)}_t$ to have positive regularity so that $B(R^{(1)}_t)$ is well-defined, so we can take $\widetilde {Q}_1 \approx A^{\beta _1}$ for some $\beta _1 < \frac {1}{2}.$ For convenience of computing regularity, we additionally impose $\alpha _1 < \frac {1}{2}$ so that $Z^{(1)}$ has positive regularity. Assume $R^{(1)}$ also has its canonical regularity as $\widetilde {Z}^{(1)}$ . Then $B(X^{(0,1)}_t, R^{(1)}_t)$ is well-defined classically if $1 - \beta _1 - \alpha _0> 0$ , and the roughest term in the drift is

$$\begin{align*}B(X^{(0,1)}) - B(X^{(0)}) = 2B(X^{(0)}, X^{(1)}) + B(X^{(1)}) \in C_T \mathcal{C}^{(-\frac12 - \alpha_0)^-}, \end{align*}$$

so the Time-Shifted Girsanov condition for $R^{(1)}$ is $\frac {1}{2} - \alpha _0 + \beta _1> 0.$ By collecting the above constraints on $\alpha _0$ , $\alpha _1$ and $\beta _1$ , we see that as long as $\alpha = \alpha _0 < \frac {3}{4}$ , we can find $\alpha _1$ and $\beta _1$ such that all constraints are satisfied and

$$ \begin{align*} \mathrm{Law}(u_t\mid t < \tau_\infty ) &= \mathrm{Law}(X^{(0)}_t + X^{(1)}_t + R^{(1)}_t\mid t < \tau_\infty)\\ & \ll \mathrm{Law}(Z^{(0)}_t + Z^{(1)}_t + \widetilde{Z}^{(1)}_t) = \mathrm{Law}(z_t) \end{align*} $$

follows from Corollary 6.4.

The previous informal computation is based on the decomposition equation (3.2) when $n = 1$ . Next, we consider the case $n = 2$ : that is, $u_t = X^{(0)}_t + X^{(1)}_t + X^{(2)}_t + R^{(2)}_t$ . Again, based on the same reasoning, we take $Q_i \approx A^{\alpha _i / 2}$ , $i = 0, 1$ and $\widetilde {Q}_2 \approx A^{\beta _2 / 2}$ , and we assume that $R^{(2)}$ has its canonical regularity as $\widetilde {Z}^{(2)}$ and $\alpha _2 < \frac {1}{2} \le \alpha _1 < \alpha _0$ for convenience. For the same reason as in the case $n = 1$ above, we need $\beta _2 < \frac {1}{2}$ , $2\alpha _0 - \alpha _1 < 1$ , $1 - \beta _2 - \alpha _0> 0$ and $\frac {1}{2} - \alpha _0 + \beta _2> 0.$ Similarly, to show equation (3.7) on $X^{(2)}$ , we need additionally $\alpha _0 + \alpha _1 - \alpha _2 < 1$ . However, one can check that the above constraints on $\alpha _i, \beta _2$ give no solutions if $\alpha _0 \ge \frac {3}{4}$ , and the bottleneck is the constraint $1 - \beta _2 - \alpha _0> 0$ for $B(X^{(0,2)}_t, R^{(2)}_t)$ to be classically well-defined. To resolve this issue, we employ the Da Prato-Debussche trick of interpreting $R^{(2)} = \eta ^{(2)} + \rho ^{(2)}$ , where $\eta ^{(2)}$ and $\rho ^{(2)}$ satisfy

$$\begin{align*}\mathscr{L} \eta^{(2)}_t = \widetilde{Q}_2\,dW^{(2)}_t, \end{align*}$$

$$ \begin{align*} \mathscr{L} \rho^{(2)}_t = B(X^{(0,2)}_t) - B(X^{(0,1)}_t) & + 2 B(X^{(0,2)}_t, \eta^{(2)}_t) \\+ 2B(X^{(0,2)}_t, \rho^{(2)}_t) &\quad + B(\rho^{(2)}_t) + 2B(\rho^{(2)}_t, \eta^{(2)}_t) + B(\eta^{(2)}_t), \end{align*} $$

with $\eta ^{(2)}_0 = 0$ and $\rho ^{(2)}_0 = u_0$ . Then we can interpret

$$\begin{align*}B(X^{(0,2)}_t, R^{(2)}_t) = B(X^{(0,2)}_t, \eta^{(2)}_t) + B(X^{(0,2)}_t, \rho^{(2)}_t), \end{align*}$$

where $B(X^{(0,2)}, \eta ^{(2)})$ is a well-defined Gaussian object, and one can check and will see that as long as $\alpha _0 < 1$ , $B(X^{(0,2)}, \rho ^{(2)})$ is classically well-defined. Now we do not need the constraint $1 - \beta _2 - \alpha _0> 0$ , and the remaining constraints can be satisfied as long as $\alpha = \alpha _0 < \frac {5}{6}.$

Following the heuristics above, we can increase the number n of levels of the decomposition given by equation (3.2) to obtain the main result up to $\alpha < 1$ . The informal computations above will be justified in a clean and rigorous way in the next section.

6.4 Basic assumptions on the factorisation of noise into levels

We now fix additional structure in the X and Y systems (equations (3.2) and (3.9), respectively) to allow us to better characterise the regularity of the different levels. We assume that there exists a sequence of real numbers

(6.1) $$ \begin{align} \beta_n < \alpha_n < \alpha_{n-1} < \ldots < \alpha_0 = \alpha, \end{align} $$

with

(6.2) $$ \begin{align} \alpha_n < \frac{1}{2} \le \alpha_{n-1} \end{align} $$

such that

(6.3) $$ \begin{align} Q_i \approx A^{\alpha_i/2}\quad\text{and}\quad \widetilde{Q}_n \approx A^{\beta_n/2}\,. \end{align} $$

We will see that the effect of the assumption in equation (6.1) is to make the levels in equations (3.2) and (3.9) have increasing spatial regularity as k increases. The conditions given by equations 6.1–6.3 will be our standing structural assumption on the noise.

The following lemma shows that one can choose the $\{\alpha _i : i = 0,\dots ,n\}$ and $\beta _n$ so that in addition to equations 6.1–6.3, the condition in equation (3.3) holds, which implies that the sum of the stochastic forcing in equation (3.2) or equation (3.9) has the same distribution as that of the Burgers equation in equation (2.1).

Proof of Lemma 6.9

We can take $Q_i = \frac {1}{\sqrt {n+2}}A^{\alpha _i/2}$ for $1 \le i \le n$ and $\widetilde {Q}_n = \frac {1}{\sqrt {n+2}}A^{\beta _n/2}$ . Note that the operator

$$ \begin{align*} QQ^* - \widetilde{Q}_n\widetilde{Q}_n^* - Q_nQ^*_n - \cdots - Q_2 Q_2^* - Q_1 Q_1^* \end{align*} $$

is symmetric and positive definite, so it is equal to $Q_0 Q_0^*$ for some operator $Q_0$ . Since $Q \approx A^{\alpha /2}$ and $\alpha = \alpha _0$ is the largest among $\alpha _i$ s and $\beta _n$ , we have $Q_0 \approx A^{\alpha _0/2}$ .

Remark 6.10 (Sums of $Z^{(i)}$ )

With the proofs of Lemma 6.9, we note that for $0 \le i \le n$ ,

$$\begin{align*}Z^{(0, i)}_t \overset{\tiny{\mbox{dist}}}{=} \int_0^t e^{-(t - s)A}Q^{(0,i)}dW_s \end{align*}$$

for some operator $Q^{(0,i)} \approx A^{\alpha _0/2}$ .

6.5 The $Y^{(i)}$ Equations

We will begin by establishing the needed structural and desired absolute continuity results for $Y^{(i)}$ . They will be leveraged to prove the results about the X system.

Proposition 6.11 (Canonical regularity of drifts)

Under the standing noise factorisation assumptions in equations 6.1–6.3, one has with probability one

(6.4) $$ \begin{align} \quad J(Z^{(0,i-1)}) - J(Z^{(0,i-2)}) \in C_T\mathcal{C}^{(2 - \alpha_0 - \alpha_{i-1})^-} \end{align} $$

for $1 \le i \le n$ . In particular, $J(Z^{(0,i-1)}) - J(Z^{(0,i-2)})$ has canonical regularity. In addition, the equations for $\{Y^{(i)}: i=0,\dots ,n\}$ are well-posed with global solutions.

Proof of Proposition 6.11

By the assumptions in equations 6.1–6.3 and Proposition B.8, $Z^{(i)} \in C_T\mathcal {C}^{(\frac {1}{2} - \alpha _i)^-}$ . The expression

(6.5) $$ \begin{align} J(Z^{(0,i-1)}) - J(Z^{(0,i-2)}) = J(Z^{(i-1)}) + 2\sum_{j=0}^{i-2} J(Z^{(j)}, Z^{(i-1)}) \end{align} $$

is well-defined and belongs to $C_T\mathcal {C}^{(2 - \alpha _0 - \alpha _{i-1})^-}$ almost surely, by Proposition B.12 and Proposition B.16, because $J(Z^{(0)}, Z^{(i-1)}) \in C_T\mathcal {C}^{(2 - \alpha _0 - \alpha _{i-1})^-}$ almost surely is the least regular term.

Proposition 6.12 (Constraints from canonical regularity of $Y^{(i)}$ )

Under the standing noise factorisation assumptions in equations 6.1–6.3, if in addition $\alpha _0 + \alpha _{i - 1} - \alpha _i < \frac {3}{2}$ for all $1 \le i \le n$ , then all the $Y^{(i)}$ equations have canonical regularity

$$ \begin{align*}Y^{(i)} \in C_T\mathcal{C}^{(\frac{1}{2} - \alpha_i)^-},\end{align*} $$

namely, that of the stochastic convolution in each equation.

Proof of Proposition 6.12

For any $1 \le i \le n$ and $t> 0$ , we only need to make sure that the drift

$$\begin{align*}J(Z^{(0,i-1)}) - J(Z^{(0,i-2)}) \in C_T \mathcal{C}^{(2-\alpha_0 - \alpha_{i-1})^-}\quad \text{a.s.} \end{align*}$$

is smoother than the stochastic convolution

$$\begin{align*}Z^{(i)} \in C_T\mathcal{C}^{(\frac{1}{2} - \alpha_i)^-} \quad \text{a.s.} \end{align*}$$

so that $Y^{(i)}$ has the same regularity as the stochastic convolution. This holds if $2 -\alpha _0 - \alpha _{i-1}> \frac {1}{2} - \alpha _i.$

We will apply the Cameron-Martin Theorem 5.1 in our setting to each level by conditioning on previous levels. As mentioned in Remark 5.11 and Section 5.4, the reason we cannot apply Time-Shifted Girsanov Method to some of the levels is that some of the terms involving $Z^{(i)}$ can only be defined when convolving with the heat kernel. For example, we cannot define $B(Z^{(0)}_t)$ but can only define $J(Z^{(0)})_t$ when $\alpha _0 \ge \frac {3}{4}$ (see Appendix B for more details). In this case, the Time-Shifted Girsanov Method, Theorem 5.9, cannot be applied.

Proposition 6.13 (Constraints from Cameron-Martin for $Y^{(i)}$ )

Under the standing noise factorisation assumptions in equations 6.1–6.3, if $\alpha _0 + \alpha _{i-1} - \alpha _i < 1$ for all $1 \le i \le n$ , then the regularity conditions given in Remark 5.2, needed to apply the Cameron-Martin Theorem 5.1, hold. More concretely, it implies that for $1 \le i \le n$ , for any $t> 0$ , it holds almost surely that

$$ \begin{align*}\mathrm{Law}(Y_t^{(i)}\mid \mathcal{F}_t^{(i-1)}) \sim \mathrm{Law}(Z_t^{(i)}),\end{align*} $$

where we recall $\mathcal {F}_t^{(i-1)} = \sigma ( W_s^{(j)}: j \leq i-1, s \leq t)$ .

Proof of Proposition 6.13

For each $1 \le i \le n$ , we have

$$\begin{align*}Y^{(i)}_t = J(Z^{(0,i-1)})_t - J(Z^{(0,i-2)})_t + Z^{(i)}_t. \end{align*}$$

Since $Q^{(i)} \approx A^{\alpha _i / 2}$ , by equation (6.4), the condition from Theorem 5.1 and Remark 5.2 for the equation of $Y^{(i)}$ is exactly $\alpha _0 + \alpha _{i-1} - \alpha _i < 1$ .

With all the constraints so far, we establish the relation between the range of $\alpha $ and the corresponding number n of levels needed in the decomposition (except for the remainder).

Proof of Proposition 6.15

First, we make sure the assumption in equation (6.2) is satisfied. Based on Proposition 6.12, Proposition 6.13 and Remark 6.14, we only need to make sure $\alpha _0 + \alpha _{i-1} - \alpha _i < 1$ for any $1 \le i \le n$ . In particular, we have $\alpha _0 + \alpha _{i-1} - 1 < \alpha _i$ , which implies

$$ \begin{align*} \alpha_1 &> 2\alpha_0 - 1 \\ \implies \alpha_2 &> \alpha_1 + \alpha_0 - 1 > 3\alpha_0 - 2 \\ \implies \alpha_3 &> \alpha_2 + \alpha_0 - 1 > 4\alpha_0 - 3 \\ \vdots \\ \implies \alpha_n &> \alpha_{n-1} + \alpha_0 - 1 > (n+1)\alpha_0 - n. \end{align*} $$

Since $\alpha _n < \frac {1}{2}$ , we deduce $\frac {1}{2} \le \alpha _0$ and $(n+1)\alpha _0 - n < \frac {1}{2}$ , so $\frac {1}{2} \le \alpha = \alpha _0 < \frac {2n+1}{2n+2}$ ; and starting from this constraint, we may find the possible values of $\alpha _1, \ldots , \alpha _n$ .

6.6 Analysis of remainder $S^{(n)}$ and associated constraints

Recall the remainder equation from equation (3.9):

(6.6) $$ \begin{align} \mathscr{L} S^{(n)}_t = \big(B(Y^{(0,n)}_t) - B(Z^{(0,n-1)}_t) + 2 B(Y^{(0,n)}_t, S^{(n)}_t) + B(S^{(n)}_t)\big)dt + \widetilde{Q}_nd\widetilde{W} ^{(n)}_t. \end{align} $$

Note that in this equation, the drift depends on the solution $S^{(n)}$ itself, so we cannot apply the Cameron-Martin Theorem 5.1 by conditioning on previous levels. However, unlike the previous level, the drift is regular enough that it can be defined without convolving with the heat kernel. This is reflected by the fact that $\alpha + \alpha _n < \frac {3}{2}$ and $\alpha + \beta _n < \frac {3}{2}.$ We are in a good position to use the Time-Shifted Girsanov Method, Theorem 5.9.

We first study the well-posedness of $S^{(n)}$ and canonical regularity of the terms. We start with the term $B(Y^{(0,n)}_t) - B(Z^{(0,n-1)}_t)$ .

Proposition 6.17. Under the standing noise factorisation assumptions in equations 6.1–6.3, if the $Y^{(i)}$ equations are well-posed, with all their terms possessing canonical regularity (as guaranteed, for example, by Proposition 6.11 and Proposition 6.12), then if additionally $\alpha _0 < 1$ , with probability one for any $t> 0$ , we have

(6.7) $$ \begin{align} B(Y^{(0,n)}) - B(Z^{(0,n-1)}) \in C_T\mathcal{C}^{(-\frac{1}{2}-\alpha_0)^-}. \end{align} $$

That is to say, these terms are well-defined with their canonical regularity.

Proof of Proposition 6.17

Note that

$$ \begin{align*}Y^{(0,n)} = J(Z^{(0, n-1)}) + Z^{(0,n)},\end{align*} $$

which yields

$$ \begin{align*} B(Y^{(0,n)}) - B(Z^{(0,n-1)}) & = B\big(J(Z^{(0, n-1)}) + Z^{(0,n)}\big) - B(Z^{(0,n-1)}) \\ & = B\big(J(Z^{(0, n-1)})\big) + 2 B\big(J(Z^{(0, n-1)}), Z^{(0,n)}\big)\\ &\qquad+ B( Z^{(0,n)}) - B(Z^{(0,n-1)}); \end{align*} $$

and to leverage independence among the $Z^{(i)}$ , we note that

$$\begin{align*}B(J(Z^{(0, n-1)}), Z^{(0,n)}) = B(J(Z^{(0, n-1)}), Z^{(0,n-1)}) + B(J(Z^{(0, n-1)}), Z^{(n)}), \end{align*}$$

$$\begin{align*}B(Z^{(0,n)}) - B(Z^{(0,n-1)}) = B(Z^{(n)}) + 2 B(Z^{(0,n-1)}, Z^{(n)}). \end{align*}$$

By $\alpha _0 < 1, \alpha _n < \frac {1}{2}$ , Remark 6.10, Section B.3–B.7,

$$ \begin{align*} B(J(Z^{(0, n-1)})) & \in C_T\mathcal{C}^{(1 - 2\alpha_0)^-}, \\ B(J(Z^{(0, n-1)}), Z^{(0,n)}) & \in C_T\mathcal{C}^{(-\frac{1}{2} - \alpha_0)^-}, \\ B(Z^{(0,n)}) - B(Z^{(0,n-1)}) & \in C_T\mathcal{C}^{(-\frac{1}{2} - \alpha_0)^-}, \end{align*} $$

almost surely. Therefore, since $\alpha _0 < 1$ ,

$$\begin{align*}B(Y^{(0,n)}) - B(Z^{(0,n-1)}) \in C_T\mathcal{C}^{(-\frac{1}{2} - \alpha_0)^-} \end{align*}$$

almost surely.

Unlike the system in equation (3.9) of $Y^{(i)}$ , where all terms are Gaussian objects, some product terms in equation (6.6) may not be a priori well-defined. Assume for the moment that $S^{(n)}$ is well-defined with its canonical regularity. From the assumptions of Lemma 6.17, we have $\beta _n < \alpha _n < \frac {1}{2}$ , so $S^{(n)}_t$ is function-valued almost surely, which makes $B(S^{(n)}_t)$ well-defined and belong to $\mathcal {C}^{(-\frac {1}{2} - \beta _n)^-}$ almost surely. The only term we need to define appropriately is $B(Y^{(0,n)}_t, S^{(n)}_t)$ .

As motivated in Section 6.3, we interpret the term $B(Y^{(0,n)}_t, S^{(n)}_t)$ as

$$\begin{align*}B(Y^{(0,n)}_t, \eta^{(n)}_t) + B(Y^{(0,n)}_t, \rho^{(n)}_t), \end{align*}$$

where $\eta ^{(n)}$ and $\rho ^{(n)}$ solve

$$ \begin{align*} \mathscr{L} \eta^{(n)}_t = \widetilde{Q}_n d\widetilde{W} ^{(n)}_t, \end{align*} $$

(6.8) $$ \begin{align} & \mathscr{L} \rho^{(n)}_t = B(Y^{(0,n)}_t) - B(Z^{(0,n-1)}_t) + 2 B(Y^{(0,n)}_t, \eta^{(n)}_t) \nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad+ 2B(Y^{(0,n)}_t, \rho^{(n)}_t) + B(\rho^{(n)}_t) + 2B(\rho^{(n)}_t, \eta^{(n)}_t) + B(\eta^{(n)}_t), \end{align} $$

with $\eta ^{(n)}_0 = 0$ and $\rho ^{(n)}_0 = u_0$ . Note that the stochastic forcing term $\eta ^{(n)}$ is just $\widetilde {Z}^{(n)}$ but with zero initial condition. In this case, $B(Y^{(0,n)}, \eta ^{(n)})$ can be defined with Sections B.3–B.7 since $\eta ^{(n)}$ is an Ornstein-Uhlenbeck process with zero initial condition. On the other hand, as we remove the stochastic forcing term, $\rho ^{(n)}$ has better regularity so that $B(Y^{(0,n)}, \rho ^{(n)})$ can be classically defined with the appropriate choice of parameters.

Lemma 6.18. Assume the standing noise factorisation assumptions in equations 6.1–6.3 hold. If the $Y^{(i)}$ equations are well-posed, with all their terms possessing canonical regularity, then with probability one,

(6.9) $$ \begin{align} \eta^{(n)} \in C_T\mathcal{C}^{(\frac{1}{2}-\beta_n)^-}, \quad B(Y^{(0,n)}, \eta^{(n)}) \in C_T\mathcal{C}^{(-\frac{1}{2} - \alpha_0)^-}. \end{align} $$

If $\alpha _0 < 1$ , then $\rho ^{(n)}$ and $B(Y^{(0,n)}, \rho ^{(n)})$ are well-defined locally in time, and for any $T < \tau _\infty $ , the maximal existence time,

(6.10) $$ \begin{align} \rho^{(n)} \in C_T \mathcal{C}^{(\frac{3}{2}-\alpha_0)^-}, \quad B(Y^{(0,n)}, \rho^{(n)}) \in C_T\mathcal{C}^{(-\frac{1}{2} - \alpha_0)^-}. \end{align} $$

In particular, $ S^{(n)} = \eta ^{(n)} + \rho ^{(n)} $ is well-defined locally in time, and for $T < \tau _\infty $ ,

$$\begin{align*}B(Y^{(0,n)}, S^{(n)}) = B(Y^{(0,n)}, \eta^{(n)}) + B(Y^{(0,n)}, \rho^{(n)}) \in C_T\mathcal{C}^{(-\frac{1}{2} - \alpha_0)^-}. \end{align*}$$

Proof of Lemma 6.18

The first statement is proved as before with Remark 6.10, Proposition B.8, and Section B.6 and B.7.

We turn to the second statement. Note that we can rewrite equation (6.8) as

$$\begin{align*}\mathscr{L} \rho^{(n)}_t = B(\rho^{(n)}_t) + 2B(\rho^{(n)}_t, \eta_t^{(n)}) + F^{(n)}_t, \end{align*}$$

where $F^{(n)}$ is given by

$$\begin{align*}F^{(n)}_t := B(Y^{(0,n)}_t) - B(Z^{(0,n-1)}_t) + 2 B(Y^{(0,n)}_t, \eta^{(n)}_t) + B(\eta^{(n)}_t). \end{align*}$$

By equation (6.9), we know $\eta ^{(n)} \in C_T \mathcal {C}^{(\frac {1}{2} - \beta _n)^-}$ . By Sections B.6 and B.7, since $\alpha _0 + \beta _n < \frac {3}{2}$ , it is clear that

$$\begin{align*}B(Y^{(0,n)}, \eta^{(n)}) \in C_T \mathcal{C}^{(-\frac{1}{2} - \alpha_0)^-}. \end{align*}$$

Hence, by equation (6.7), we have $F^{(n)} \in C_T \mathcal {C}^{(-\frac {1}{2} - \alpha _0)^-}.$ By a standard fixed-point argument in Appendix C, we obtain that equation (6.8) is well-posed with local in time solutions such that $\rho ^{(n)} \in C_T \mathcal {C}^{(\frac {3}{2}-\alpha _0)^-}.$ In particular, since $\alpha _0 < 1$ , $B(Y^{(0,n)}, \rho ^{(n)}) \in C_T\mathcal {C}^{(-\frac {1}{2} - \alpha _0)^-} $ is well-defined classically.

Now we are in a good place to figure out the needed constraints for $S^{(n)}$ to have its canonical regularity and for the application of the Time-Shifted Girsanov Method.

Proposition 6.19 (Constraint from canonical regularity for $S^{(n)}$ )

Under the standing noise factorisation assumptions in equations 6.1–6.3, if the $Y^{(i)}$ equations are well-posed with all their terms possessing canonical regularity, then if in addition $\alpha _0 < 1$ and $\alpha _0 - \beta _n < 1$ , then $S^{(n)}$ has canonical regularity

$$\begin{align*}S^{(n)} \in C_T\mathcal{C}^{(\frac{1}{2}-\beta_n)^-} \end{align*}$$

for any $T < \tau _\infty $ , namely that of the stochastic convolution in the equation. Setting for $t \geq \tau _\infty $ , we have that $ S^{(n)} \in C_T\overline {\mathcal {C}}^{(\frac {1}{2}-\beta _n)^-}$ for any $T>0$ (see Section 4.1 for the definition of $C_T\overline { \mathcal {C}}^\delta $ ).

Proposition 6.20 (Constraint from Time-Shifted Girsanov for $S^{(n)}$ )

Under the standing noise factorisation assumptions in equations 6.1–6.3, if the $Y^{(i)}$ equations are well-posed, with all their terms possessing canonical regularity, and if in addition $\alpha _0 < 1$ and $\alpha _0 - \beta _n < \frac {1}{2}$ , then the regularity conditions needed to apply Time-Shifted Girsanov Method to $S^{(n)}$ hold. More concretely, it implies that for any $t> 0$ , it holds almost surely that

$$ \begin{align*}\mathrm{Law}(S_t^{(n)}\mid t < \tau_\infty,\mathcal{F}_t^{(n)}) \ll \mathrm{Law}(\widetilde{Z}_t^{(n)}),\end{align*} $$

where we recall $\mathcal {F}_t^{(n)} = \sigma ( W_s^{(j)}: j \leq n, s \leq t)$ .

In particular, as long as $\alpha _0 < 1$ , $\beta _n$ (and $\alpha _n$ ) can be taken close enough to $\frac {1}{2}$ to satisfy the condition $\alpha _0 - \beta _n < \frac {1}{2}$ .

Finally, by collecting all the results above, we prove our main absolute continuity of the law of $u_t$ with respect to the law of $z_t$ defined in equation (2.3), for $\alpha < 1$ .

Corollary 6.23 (The overall result on the Y system)

Fix an n and an $\alpha $ so that $\frac {1}{2} \le \alpha < \frac {2n+1}{2n+2}$ . Then there exists a sequence of real numbers $\beta _n < \alpha _n < \ldots < \alpha _0 = \alpha $ such that the standing noise factorisation assumptions on the $\{\alpha _j: j=0,\dots ,n\}$ in equations 6.1–6.3 hold as well as the hypothesis of Proposition 6.11, Proposition 6.12, Proposition 6.13, Proposition 6.17, Lemma 6.18, Proposition 6.19 and Proposition 6.20. More concretely, it implies that for any $t> 0$ , it holds almost surely that

$$ \begin{align*} \mathrm{Law}(u_t \mid \tau_\infty>t) &= \mathrm{Law}\left(S_t^{(n)} +\sum_{k=0}^n Y_t^{(k)}\,\Big|\, \tau_\infty>t\right) \\ &\ll \mathrm{Law}\left(\widetilde{Z}_t^{(n)} +\sum_{k=0}^n Z_t^{(k)}\right) = \mathrm{Law}(z_t), \end{align*} $$

where, as a reminder, $z_t$ is the linear part of $u_t$ with initial condition $z_0$ as defined in equation (2.3).