2.1 Setting

In this subsection, we introduce a stochastic process that will play a main role in this paper. From now on we denote by $(w_{t})_{t\geqslant 0}$ the $d$-dimensional fBm with Hurst parameter $H$. Throughout this paper we assume $1/4<H\leqslant 1/2$. It is a unique $d$-dimensional, mean-zero, continuous Gaussian process with covariance $\mathbf{E}[w_{s}^{i}w_{t}^{j}]=\unicode[STIX]{x1D6FF}_{ij}R(s,t)$ ($s,t\geqslant 0$), where

$$\begin{eqnarray}R(s,t)={\textstyle \frac{1}{2}}\{|s|^{2H}+|t|^{2H}-|t-s|^{2H}\}.\end{eqnarray}$$

Note that, for any $c>0$, $(w_{ct})_{t\geqslant 0}$ and $(c^{H}w_{t})_{t\geqslant 0}$ have the same law. This property is called self-similarity or scale invariance. When $H=1/2$, it is the usual Brownian motion. In what follows the time interval will always be $[0,1]$. It is well known that $(w_{t})_{0\leqslant t\leqslant 1}$ admits a canonical rough path lift $\mathbf{w}=(\mathbf{w}^{1},\ldots ,\mathbf{w}^{\lfloor 1/H\rfloor })$, which is called fractional Brownian rough path [Reference Coutin and QianCQ02].

Let $V_{i}:\mathbf{R}^{n}\rightarrow \mathbf{R}^{n}$ be $C_{b}^{\infty }$, that is, $V_{i}$ is a bounded smooth function with bounded derivatives of all order ($0\leqslant i\leqslant d$). We shall identify $V_{i}$ with its corresponding vector field and denote the vector field by the same symbol. We consider the following RDE:

(2.1)$$\begin{eqnarray}dy_{t}=\mathop{\sum }_{i=1}^{d}V_{i}(y_{t})\,dw_{t}^{i}+V_{0}(y_{t})\,dt\quad \text{with }y_{0}=a,\end{eqnarray}$$

where $a\in \mathbf{R}^{n}$ is a deterministic initial point. This RDE is driven by the Young pairing $(\mathbf{w},\boldsymbol{\unicode[STIX]{x1D706}})$, where $\unicode[STIX]{x1D706}_{t}=t$. The unique solution is denoted by $\mathbf{y}=(\mathbf{y}^{1},\ldots ,\mathbf{y}^{\lfloor 1/H\rfloor })$ and we set $y_{t}:=a+\mathbf{y}_{0,t}^{1}$ as usual. We will sometimes write $y_{t}=y_{t}(a)=y_{t}(a,\mathbf{w})$ etc. to make explicit the dependence on $a$ and $\mathbf{w}$. We often use a matrix notation for (2.1), that is, we set $b=V_{0}$ and $\unicode[STIX]{x1D70E}=[V_{1},\ldots ,V_{d}]$, which is $n\times d$ matrix-valued, and write (2.1) as

$$\begin{eqnarray}dy_{t}=\unicode[STIX]{x1D70E}(y_{t})\,dw_{t}+b(y_{t})\,dt\quad \text{with }y_{0}=a.\end{eqnarray}$$

2.2 Assumptions

In this subsection, we introduce assumptions of the main theorem. First, we introduce the Hörmander condition:

Definition 2.1. (Hörmander condition) Set

$$\begin{eqnarray}{\mathcal{V}}_{m}=\left\{\begin{array}{@{}ll@{}}\{V_{i}\mid 1\leqslant i\leqslant d\},\quad & m=0,\\ \{[V_{i},U]\mid U\in {\mathcal{V}}_{m-1},0\leqslant i\leqslant d\},\quad & m\geqslant 1,\end{array}\right.\qquad {\mathcal{V}}=\mathop{\bigcup }_{m=0}^{\infty }{\mathcal{V}}_{m}.\end{eqnarray}$$

For $x\in \mathbf{R}^{n}$, ${\mathcal{V}}_{m}(x)$ (resp. ${\mathcal{V}}(x)$) stands for the subset of $\mathbf{R}^{n}$ obtained by plugging $x$ into the vector field of ${\mathcal{V}}_{m}$ (resp. ${\mathcal{V}}$). We say that the vector fields $V_{0},V_{1},\ldots ,V_{d}$ satisfy the Hörmander condition at $x\in \mathbf{R}^{n}$ if ${\mathcal{V}}(x)$ linearly spans $\mathbf{R}^{n}$.

We assume the following:

(A1)$V_{0},V_{1},\ldots ,V_{d}$ satisfy the Hörmander condition at the initial point $a\in \mathbf{R}^{n}$.

It is known that, under (A1), the law of the solution $y_{t}$ has a density $p_{t}(a,a^{\prime })$ with respect to the Lebesgue measure on $\mathbf{R}^{n}$ for any $t>0$ (see [Reference Hairer and PillaiHP13, Reference Cass, Hairer, Litterer and TindelCHLT15]). Hence, for any Borel subset $A\subset \mathbf{R}^{n}$, $\mathbf{P}(y_{t}(a)\in A)=\int _{A}p_{t}(a,a^{\prime })\,da^{\prime }$.

Let $\mathfrak{H}=\mathfrak{H}^{H}$ be the Cameron–Martin space of fBm on the time interval $[0,1]$. For every $H\in (1/4,1/2]$, there exists $q\in [1,2)$ such that every $\unicode[STIX]{x1D6FE}\in \mathfrak{H}$ is continuous and of finite$q$-variation. For $\unicode[STIX]{x1D6FE}\in \mathfrak{H}$, we denote by $\unicode[STIX]{x1D719}_{t}^{0}=\unicode[STIX]{x1D719}_{t}^{0}(\unicode[STIX]{x1D6FE})$ the solution of the following Young ordinary differential equation (ODE):

(2.2)$$\begin{eqnarray}d\unicode[STIX]{x1D719}_{t}^{0}=\mathop{\sum }_{i=1}^{d}V_{i}(\unicode[STIX]{x1D719}_{t}^{0})\,d\unicode[STIX]{x1D6FE}_{t}^{i}\quad \text{with }\unicode[STIX]{x1D719}_{0}^{0}=a.\end{eqnarray}$$

Set, for $a^{\prime }\neq a$, $K_{a}^{a^{\prime }}=\{\unicode[STIX]{x1D6FE}\in \mathfrak{H}\mid \unicode[STIX]{x1D719}_{1}^{0}(\unicode[STIX]{x1D6FE})=a^{\prime }\}$. We only consider the case where $K_{a}^{a^{\prime }}$ is not empty. From goodness of the rate function in Schilder-type large deviation for fractional Brownian rough path, it follows that $\inf \{\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}\mid \unicode[STIX]{x1D6FE}\in K_{a}^{a^{\prime }}\}=\min \{\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}\mid \unicode[STIX]{x1D6FE}\in K_{a}^{a^{\prime }}\}$. Now we introduce the following assumption:

(A2)$\bar{\unicode[STIX]{x1D6FE}}\in K_{a}^{a^{\prime }}$ which minimizes $\mathfrak{H}$-norm exists uniquely.

In what follows, $\bar{\unicode[STIX]{x1D6FE}}$ denotes the minimizer in (A2). Note that the map $\unicode[STIX]{x1D6FE}\in \mathfrak{H}{\hookrightarrow}C^{q\text{-}\text{var}}([0,1];\mathbf{R}^{d})\mapsto \unicode[STIX]{x1D719}_{1}^{0}(\unicode[STIX]{x1D6FE})\in \mathbf{R}^{n}$ is Fréchet differentiable; $D\unicode[STIX]{x1D719}_{1}^{0}(\unicode[STIX]{x1D6FE})$ stands for the Fréchet derivative and is expressed by

(2.3)$$\begin{eqnarray}\langle D[\unicode[STIX]{x1D719}_{1}^{0}(\unicode[STIX]{x1D6FE})]^{k},h\rangle _{\mathfrak{ H}}=\mathop{\sum }_{i=1}^{d}\int _{0}^{1}[J_{1}(\unicode[STIX]{x1D6FE})K_{s}(\unicode[STIX]{x1D6FE})\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719}_{s}^{0}(\unicode[STIX]{x1D6FE}))]_{i}^{k}\,dh_{s}^{i},\end{eqnarray}$$

where $[\bullet ]^{k}$ and $[\bullet ]_{i}^{k}$ are the $k$th component of the vector and the $(k,i)$-component of the matrix, respectively, and $J(\unicode[STIX]{x1D6FE})$ and $K(\unicode[STIX]{x1D6FE})$ are the Jacobi process of $\unicode[STIX]{x1D719}^{0}(\unicode[STIX]{x1D6FE})$ and its inverse, respectively. We define the deterministic Malliavin covariance matrix $Q(\unicode[STIX]{x1D6FE})=(Q(\unicode[STIX]{x1D6FE})_{kl})_{1\leqslant k,l\leqslant n}$ by

(2.4)$$\begin{eqnarray}Q(\unicode[STIX]{x1D6FE})_{kl}=\langle D[\unicode[STIX]{x1D719}_{1}^{0}(\unicode[STIX]{x1D6FE})]^{k},D[\unicode[STIX]{x1D719}_{1}^{0}(\unicode[STIX]{x1D6FE})]^{l}\rangle _{\mathfrak{ H}^{\ast }}.\end{eqnarray}$$

We assume that $Q$ is nondegenerate at the minimizer $\bar{\unicode[STIX]{x1D6FE}}$:

(A3) There is a positive constant $c$ such that $Q(\bar{\unicode[STIX]{x1D6FE}})\geqslant cI$, where $I$ stands for the identity matrix.

Finally, we assume that $\Vert \bullet \Vert _{\mathfrak{H}}^{2}/2$ is not so degenerate at $\bar{\unicode[STIX]{x1D6FE}}$ in the following sense:

(A4) At $\bar{\unicode[STIX]{x1D6FE}}$, the Hessian of the functional $K_{a}^{a^{\prime }}\ni \unicode[STIX]{x1D6FE}\mapsto \Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}^{2}/2$ is strictly positive in the quadratic form sense. More precisely, if $(-\unicode[STIX]{x1D716}_{0},\unicode[STIX]{x1D716}_{0})\ni u\mapsto f(u)\in K_{a}^{a^{\prime }}$ is a smooth curve in $K_{a}^{a^{\prime }}$ such that $f(0)=\bar{\unicode[STIX]{x1D6FE}}$ and $f^{\prime }(0)\neq 0$, then $(d/du)^{2}|_{u=0}\Vert f(u)\Vert _{\mathfrak{H}}^{2}/2>0$.

2.3 Index sets

In this subsection we introduce several index sets for the exponent of the small time parameter $t>0$, which will be used in the asymptotic expansion. Unfortunately, these index sets are not subsets of $\mathbf{N}=\{0,1,2,\ldots \}$ and are rather complicated in general. (However, they are discrete subsets of $(\mathbf{Z}+H^{-1}\mathbf{Z})\cap [0,\infty )$ with the minimum $0$.)

Set

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{1}=\bigg\{n_{1}+\frac{n_{2}}{H}\bigg|n_{1},n_{2}\in \mathbf{N}\bigg\}.\end{eqnarray}$$

We denote by $0=\unicode[STIX]{x1D705}_{0}<\unicode[STIX]{x1D705}_{1}<\unicode[STIX]{x1D705}_{2}<\cdots \,$ all the elements of $\unicode[STIX]{x1D6EC}_{1}$ in increasing order. Several smallest elements are explicitly given as follows: for $1/3<H<1/2$,

$$\begin{eqnarray}\unicode[STIX]{x1D705}_{1}=1,\qquad \unicode[STIX]{x1D705}_{2}=2,\qquad \unicode[STIX]{x1D705}_{3}=\frac{1}{H},\qquad \unicode[STIX]{x1D705}_{4}=3,\qquad \unicode[STIX]{x1D705}_{5}=1+\frac{1}{H},\qquad \unicode[STIX]{x1D705}_{6}=4,\ldots\end{eqnarray}$$

and, for $1/4<H<1/3$,

$$\begin{eqnarray}\unicode[STIX]{x1D705}_{1}=1,\qquad \unicode[STIX]{x1D705}_{2}=2,\qquad \unicode[STIX]{x1D705}_{3}=3,\qquad \unicode[STIX]{x1D705}_{4}=\frac{1}{H},\qquad \unicode[STIX]{x1D705}_{5}=4,\qquad \unicode[STIX]{x1D705}_{6}=1+\frac{1}{H},\ldots .\end{eqnarray}$$

We also set

$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6EC}_{2}=\{\unicode[STIX]{x1D705}-1\mid \unicode[STIX]{x1D705}\in \unicode[STIX]{x1D6EC}_{1}\setminus \{\unicode[STIX]{x1D705}_{0}\}\}=\bigg\{0,1,\frac{1}{H}-1,2,\frac{1}{H},3,\ldots \bigg\}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D6EC}_{2}^{\prime }=\{\unicode[STIX]{x1D705}-2\mid \unicode[STIX]{x1D705}\in \unicode[STIX]{x1D6EC}_{1}\setminus \{\unicode[STIX]{x1D705}_{0},\unicode[STIX]{x1D705}_{1}\}\}=\bigg\{0,\frac{1}{H}-2,1,\frac{1}{H}-1,2,\ldots \bigg\}. & \displaystyle \nonumber\end{eqnarray}$$

Note that in the above explicit expression of these two index sets the elements are not sorted in increasing order when $1/4<H\leqslant 1/3$. Next we set

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}_{3}=\{a_{1}+a_{2}+\cdots +a_{m}\mid m\in \mathbf{N}_{+}\text{ and }a_{1},\ldots ,a_{m}\in \unicode[STIX]{x1D6EC}_{2}\}, & & \displaystyle \nonumber\\ \displaystyle \unicode[STIX]{x1D6EC}_{3}^{\prime }=\{a_{1}+a_{2}+\cdots +a_{m}\mid m\in \mathbf{N}_{+}\text{ and }a_{1},\ldots ,a_{m}\in \unicode[STIX]{x1D6EC}_{2}^{\prime }\}, & & \displaystyle \nonumber\end{eqnarray}$$

where $\mathbf{N}_{+}=\{1,2,\ldots \!\,\}$. In the sequel, $0=\unicode[STIX]{x1D708}_{0}<\unicode[STIX]{x1D708}_{1}<\unicode[STIX]{x1D708}_{2}<\cdots \,$ and $0=\unicode[STIX]{x1D70C}_{0}<\unicode[STIX]{x1D70C}_{1}<\unicode[STIX]{x1D70C}_{2}<\cdots \,$ stand for all the elements of $\unicode[STIX]{x1D6EC}_{3}$ and $\unicode[STIX]{x1D6EC}_{3}^{\prime }$ in increasing order, respectively. Finally,

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{4}=\unicode[STIX]{x1D6EC}_{3}+\unicode[STIX]{x1D6EC}_{3}^{\prime }=\{\unicode[STIX]{x1D708}+\unicode[STIX]{x1D70C}\mid \unicode[STIX]{x1D708}\in \unicode[STIX]{x1D6EC}_{3},\unicode[STIX]{x1D70C}\in \unicode[STIX]{x1D6EC}_{3}^{\prime }\}.\end{eqnarray}$$

We denote by $0=\unicode[STIX]{x1D706}_{0}<\unicode[STIX]{x1D706}_{1}<\unicode[STIX]{x1D706}_{2}<\cdots \,$ all the elements of $\unicode[STIX]{x1D6EC}_{4}$ in increasing order.

We remark that when $H=1/2$ and $H=1/3$, all these index sets $\unicode[STIX]{x1D6EC}_{i},\unicode[STIX]{x1D6EC}_{j}^{\prime }$ above are just $\mathbf{N}$.

2.4 Statement of the main result

Now we state our main theorem. This is basically analogous to many preceding works on the standard Brownian motion such as [Reference Ben ArousBA88, Reference WatanabeWat87]. However, when $H\neq 1/2,1/3$ and the drift term exists, there are some differences. First, the exponents of $t$ are not (a constant multiple of) natural numbers. Second, cancelation of “odd terms” (see e.g., [Reference WatanabeWat87, pp. 20 and 34]) does not occur in general. (These phenomena were already observed in [Reference InahamaIna16a] and [Reference InahamaIna16b] when $H\in (1/3,1)$.) Our proof uses the Watanabe distribution theory. Therefore, the asymptotic expansion is actually obtained at the level of Watanabe distributions.

Theorem 2.3. Assume $a\neq a^{\prime }$ and $\text{(A1)}$–$\text{(A4)}$. Then, we have the following asymptotic expansion as $t{\searrow}0$:

$$\begin{eqnarray}p_{t}(a,a^{\prime })\sim \exp \bigg(-\frac{\Vert \bar{\unicode[STIX]{x1D6FE}}\Vert _{\mathfrak{H}}^{2}}{2t^{2H}}\bigg)\frac{1}{t^{nH}}\{\unicode[STIX]{x1D6FC}_{0}+\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}_{1}}t^{\unicode[STIX]{x1D706}_{1}H}+\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}_{2}}t^{\unicode[STIX]{x1D706}_{2}H}+\,\cdots \}\end{eqnarray}$$

for a certain positive constant $\unicode[STIX]{x1D6FC}_{0}$ and certain real constants $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}_{j}}\;(j=1,2,\ldots )$. Here, $0=\unicode[STIX]{x1D706}_{0}<\unicode[STIX]{x1D706}_{1}<\unicode[STIX]{x1D706}_{2}<\cdots \,$ are all the elements of $\unicode[STIX]{x1D6EC}_{4}$ in increasing order.

The above result improves that in [Reference InahamaIna16b]. The reason is as follows. First, we work under Hörmander condition instead of the ellipticity assumption. Second, the case $1/4<H\leqslant 1/3$ is also treated. Hence, we must calculate the third level rough paths.

Remark 2.4. If RDE (2.1) has no drift term, that is, $V_{0}\equiv 0$, then we can replace $\unicode[STIX]{x1D6EC}_{4}$ by $2\mathbf{N}$ in Theorem 2.3, namely,

(2.5)$$\begin{eqnarray}p_{t}(a,a^{\prime })\sim \exp \bigg(-\frac{\Vert \bar{\unicode[STIX]{x1D6FE}}\Vert _{\mathfrak{H}}^{2}}{2t^{2H}}\bigg)\frac{1}{t^{nH}}\{\unicode[STIX]{x1D6FC}_{0}+\unicode[STIX]{x1D6FC}_{2}t^{2H}+\unicode[STIX]{x1D6FC}_{4}t^{4H}+\,\cdots \}\end{eqnarray}$$

as $t{\searrow}0$. The reason is as follows. First, if $V_{0}\equiv 0$, the index sets $\unicode[STIX]{x1D6EC}_{i}$, $\unicode[STIX]{x1D6EC}_{j}^{\prime }$$(1\leqslant i\leqslant 4,2\leqslant j\leqslant 3)$ in the sequel can be replaced by $\mathbf{N}$. Second, the odd-numbered terms in the asymptotics of the generalized Wiener functional are also odd as generalized Wiener functionals. Hence, their generalized expectations all vanish.

When $H=1/2$, (2.5) holds for the same reason. Thus, we reprove the main result of Ben Arous [Reference Ben ArousBA88] via rough path theory.

3.1 Rough path analysis

In this paper we basically work in Lyons’ original framework of rough path theory. (The only exception is § 5.2, where the controlled path theory is used.) We borrow most of notations and terminologies from [Reference Lyons and QianLQ02, Reference Lyons, Caruana and LévyLCL07, Reference Friz and VictoirFV10]. Let $p\in [2,4)$ be the roughness constant and let $q\in [1,2)$ be such that $1/p+1/q>1$. Set $\unicode[STIX]{x1D6FC}=1/p$ and let $m\in \mathbf{N}_{+}$ satisfy $\unicode[STIX]{x1D6FC}-1/m>0$. We denote by $G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})$, $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC}}^{\text{H}}(\mathbf{R}^{d})$, and $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},m}^{\text{B}}(\mathbf{R}^{d})$ the geometric rough path spaces with $p$-variation, $\unicode[STIX]{x1D6FC}$-Hölder and $(\unicode[STIX]{x1D6FC},m)$-Besov topologies, respectively. In this paper, $\mathbf{x}=(\mathbf{x}^{1},\ldots ,\mathbf{x}^{\lfloor p\rfloor })$ stands for a generic element in these rough path spaces. Recall that the $p$-variation, $\unicode[STIX]{x1D6FC}$-Hölder and $(\unicode[STIX]{x1D6FC},m)$-Besov topologies are induced by

$$\begin{eqnarray}\displaystyle \Vert \mathbf{x}^{i}\Vert _{p/i\text{-}\text{var}} & = & \displaystyle \sup _{0=t_{0}<\cdots <t_{K}=1}\bigg(\mathop{\sum }_{k=1}^{K}|\mathbf{x}_{t_{k-1},t_{k}}^{i}|^{p/i}\bigg)^{i/p},\nonumber\\ \displaystyle \Vert \mathbf{x}^{i}\Vert _{i\unicode[STIX]{x1D6FC}\text{-}\text{Hol}} & = & \displaystyle \sup _{0\leqslant s<t\leqslant 1}\frac{|\mathbf{x}_{s,t}^{i}|}{|t-s|^{i\unicode[STIX]{x1D6FC}}},\nonumber\\ \displaystyle \Vert \mathbf{x}^{i}\Vert _{(i\unicode[STIX]{x1D6FC},m/i)\text{-}\text{Bes}} & = & \displaystyle \bigg(\iint _{0\leqslant s<t\leqslant 1}\frac{|\mathbf{x}_{s,t}^{i}|^{m/i}}{|t-s|^{1+i\unicode[STIX]{x1D6FC}\cdot m/i}}\,dsdt\bigg)^{i/m}\quad (i=1,\ldots ,\lfloor p\rfloor ).\nonumber\end{eqnarray}$$

Next we introduce homogeneous norms to $G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})$, $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC}}^{\text{H}}(\mathbf{R}^{d})$, and $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},m}^{\text{B}}(\mathbf{R}^{d})$ which are consistent with their topology. More explicitly, they are given by

$$\begin{eqnarray}\displaystyle & \displaystyle \Vert \mathbf{x}\Vert _{p\text{-}\text{var}}=\mathop{\sum }_{i=1}^{\lfloor p\rfloor }\Vert \mathbf{x}^{i}\Vert _{p/i\text{-}\text{var}}^{1/i},\qquad \Vert \mathbf{x}\Vert _{\unicode[STIX]{x1D6FC}\text{-}\text{Hol}}=\mathop{\sum }_{i=1}^{\lfloor 1/\unicode[STIX]{x1D6FC}\rfloor }\Vert \mathbf{x}^{i}\Vert _{i\unicode[STIX]{x1D6FC}\text{-}\text{Hol}}^{1/i}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \Vert \mathbf{x}\Vert _{(\unicode[STIX]{x1D6FC},m)\text{-}\text{Bes}}=\mathop{\sum }_{i=1}^{\lfloor 1/\unicode[STIX]{x1D6FC}\rfloor }\Vert \mathbf{x}^{i}\Vert _{(i\unicode[STIX]{x1D6FC},m/i)\text{-}\text{Bes}}^{1/i}. & \displaystyle \nonumber\end{eqnarray}$$

We denote by $\Vert \mathbf{x}^{i}\Vert _{p/i\text{-}\text{var};[s,t]}$, $\Vert \mathbf{x}\Vert _{p\text{-}\text{var};[s,t]}$, etc.  the norms restricted on the subinterval $[s,t]$.

For a rough path $\mathbf{x}$, we write $x_{t}=\mathbf{x}_{0,t}^{1}$ as usual. For $x\in C_{0}^{r\text{-}\text{var}}([0,1];\mathbf{R}^{d})$ with $r\in [1,2)$, we denote the natural lift of $x$ (i.e., the smooth rough path lying above $x$) by the corresponding boldface letter $\mathbf{x}$. Note that, for $x\in C_{0}^{r\text{-}\text{var}}([0,1];\mathbf{R}^{d})$ and $y\in C_{0}^{r\text{-}\text{var}}([0,1];\mathbf{R}^{e})$, $(\mathbf{x},\mathbf{y})\in G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d+e})$ stands for the natural lift of $(x,y)\in C_{0}^{r\text{-}\text{var}}([0,1];\mathbf{R}^{d+e})$, not for the pair $(\mathbf{x},\mathbf{y})\in G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})\times G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{e})$. In a similar way, for $\mathbf{x}\in G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})$ and $h\in C_{0}^{q\text{-}\text{var}}(\mathbf{R}^{e})$ with $1/p+1/q>1$, $(\mathbf{x},\mathbf{h})\in G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d+e})$ stands for the Young pairing. For $\mathbf{x}\in G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})$ and $h\in C_{0}^{q\text{-}\text{var}}(\mathbf{R}^{d})$ with $1/p+1/q>1$, $\mathbf{x}+\mathbf{h}=\unicode[STIX]{x1D70F}_{h}(\mathbf{x})\in G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})$ stands for the Young translation. (See [Reference Friz and VictoirFV10, Section 9.4].) We denote by $c\mathbf{x}$ the dilation of a rough path $\mathbf{x}$ by $c\in \mathbf{R}$. These notations may be somewhat misleading. But, they make many operations intuitively clear and easy to understand when we treat rough paths over a direct sum of many vector spaces.

Note that the spaces $G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})$, $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC}}^{\text{H}}(\mathbf{R}^{d})$, and $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},m}^{\text{B}}(\mathbf{R}^{d})$ enjoy the next continuous embeddings. We have $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},m}^{\text{B}}(\mathbf{R}^{d}){\hookrightarrow}G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})$ and

(3.1)$$\begin{eqnarray}\Vert \mathbf{x}\Vert _{p\text{-}\text{var};[s,t]}\leqslant c\Vert \mathbf{x}\Vert _{(\unicode[STIX]{x1D6FC},m)\text{-}\text{Bes}}(t-s)^{\unicode[STIX]{x1D6FC}-(1/m)}\quad (\mathbf{x}\in G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},m}^{\text{B}}(\mathbf{R}^{d}))\end{eqnarray}$$

from [Reference Friz and VictoirFV10, Corollary A.3]. (The continuity of this embedding is not explicitly written in [Reference Friz and VictoirFV10], but can be shown by standard and slightly lengthy argument.) For $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FD}\leqslant 1/\lfloor 1/\unicode[STIX]{x1D6FC}\rfloor$, a direct computation implies Besov–Hölder embedding $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FD}}^{\text{H}}(\mathbf{R}^{d}){\hookrightarrow}G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},m}^{\text{B}}(\mathbf{R}^{d})$.

For a control function $\unicode[STIX]{x1D714}$ in the sense of [Reference Lyons and QianLQ02, p. 16], we write $\bar{\unicode[STIX]{x1D714}}:=\unicode[STIX]{x1D714}(0,1)$. For any $\mathbf{x}\in G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})$,

(3.2)$$\begin{eqnarray}\unicode[STIX]{x1D714}_{\mathbf{x}}(s,t):=\mathop{\sum }_{i=1}^{\lfloor p\rfloor }\Vert \mathbf{x}^{i}\Vert _{p/i\text{-}\text{var};[s,t]}^{p/i}\quad (0\leqslant s\leqslant t\leqslant 1)\end{eqnarray}$$

defines a control function. (This control function is equivalent to the one defined by Carnot–Caratheodory metric.) Similarly, we set $\unicode[STIX]{x1D714}_{h}(s,t):=\Vert h\Vert _{q\text{-}\text{var};[s,t]}^{q}$ for $h\in C_{0}^{q\text{-}\text{var}}([0,1];\mathbf{R}^{e})$.

For $\unicode[STIX]{x1D6FF}>0$ and $\mathbf{x}\in G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})$, set $\unicode[STIX]{x1D70F}_{0}(\unicode[STIX]{x1D6FF})=0$ and

$$\begin{eqnarray}\unicode[STIX]{x1D70F}_{i+1}(\unicode[STIX]{x1D6FF})=\inf \{t\in (\unicode[STIX]{x1D70F}_{i}(\unicode[STIX]{x1D6FF}),1]\mid \unicode[STIX]{x1D714}_{\mathbf{x}}(\unicode[STIX]{x1D70F}_{i}(\unicode[STIX]{x1D6FF}),t)\geqslant \unicode[STIX]{x1D6FF}\}\wedge 1\quad (i=1,2,\ldots \,).\end{eqnarray}$$

Define

(3.3)$$\begin{eqnarray}N_{\unicode[STIX]{x1D6FF}}(\mathbf{x})=\sup \{i\in \mathbf{N}\mid \unicode[STIX]{x1D70F}_{i}(\unicode[STIX]{x1D6FF})<1\}.\end{eqnarray}$$

Superadditivity of $\unicode[STIX]{x1D714}_{\mathbf{x}}$ yields $\unicode[STIX]{x1D6FF}N_{\unicode[STIX]{x1D6FF}}(\mathbf{x})\leqslant \overline{\unicode[STIX]{x1D714}_{\mathbf{x}}}$. This quantity (3.3) was first studied by Cass, Litterer, and Lyons [Reference Cass, Litterer and LyonsCLL13].

Let $\mathbf{x}\in G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d+1})$. We consider an RDE

(3.4)$$\begin{eqnarray}dy_{t}=\mathop{\sum }_{i=0}^{d}V_{i}(y_{t})\,dx_{t}^{i}\quad \text{with }y_{0}=a.\end{eqnarray}$$

It is well known that the solution map $G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d+1})\ni \mathbf{x}\mapsto \mathbf{y}=(\mathbf{y}^{1},\ldots ,\mathbf{y}^{\lfloor p\rfloor })\in G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{n})$ is continuous. Set $\unicode[STIX]{x1D6F7}(\mathbf{x})=\mathbf{y}$ and $y_{t}=a+\mathbf{y}_{0,t}^{1}$. The Jacobi process $J=((J_{t}^{kl}=\frac{dy_{t}^{k}}{da^{l}})_{t\in [0,1]})_{1\leqslant k,l\leqslant n}$ and its inverse $K=(K^{kl})_{1\leqslant k,l\leqslant n}$ exist and satisfy

(3.5)$$\begin{eqnarray}\displaystyle dJ_{t} & = & \displaystyle +\mathop{\sum }_{i=0}^{d}\unicode[STIX]{x1D6FB}V_{i}(y_{t})J_{t}\,dx_{t}^{i}\quad \text{with }J_{0}=I,\end{eqnarray}$$

(3.6)$$\begin{eqnarray}\displaystyle dK_{t} & = & \displaystyle -\mathop{\sum }_{i=0}^{d}K_{t}\unicode[STIX]{x1D6FB}V_{i}(y_{t})\,dx_{t}^{i}\quad \text{with }K_{0}=I.\end{eqnarray}$$

Here, $I$ stands for the identity matrix with size $n$. It is well known that the system of RDEs (3.4)–(3.6) is globally well-posed. In particular, the map $\mathbf{x}\mapsto (\mathbf{y},\mathbf{J},\mathbf{K})$ is continuous.

Here, we make a remark on continuity of the solution map. Let $\unicode[STIX]{x1D706}$ be a one-dimensional path defined by $\unicode[STIX]{x1D706}_{t}=t$. The solution map $\mathbf{x}\mapsto (\mathbf{y},\mathbf{J},\mathbf{K})$ is continuous on $G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d+1})$ as stated above. In addition, $G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})\times \mathbf{R}\langle \unicode[STIX]{x1D706}\rangle$ is embedded in $G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d+1})$ continuously. Hence, the composition $(\mathbf{x},c\unicode[STIX]{x1D706})\mapsto (\mathbf{x},c\boldsymbol{\unicode[STIX]{x1D706}})\mapsto (\mathbf{y},\mathbf{J},\mathbf{K})$ of the two maps is continuous on $G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})\times \mathbf{R}\langle \unicode[STIX]{x1D706}\rangle$. This continuity plays an important role in §§ 6.1 and 5.3, in which we will use results on large deviation for fractional Brownian rough path.

3.2 RDEs driven by fBm

In this paper, we consider the case $x=(w,\unicode[STIX]{x1D706})$, where $w$ is an fBm with the Hurst parameter $1/4<H\leqslant 1/2$ and $\unicode[STIX]{x1D706}$ is a one-dimensional path defined by $\unicode[STIX]{x1D706}_{t}=t$. Let $p$ be larger than but sufficiently close to $1/H$. Since $w$ admits a canonical lift $\mathbf{w}\in G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})$ (see [Reference Coutin and QianCQ02]), we can deal with (2.1) in the framework of rough path analysis and obtain $\mathbf{y}=\unicode[STIX]{x1D6F7}(\mathbf{w},\boldsymbol{\unicode[STIX]{x1D706}})$. Next we introduce a scaled (and shifted) RDE of (2.1). For every $0<\unicode[STIX]{x1D716}<1$ and $\unicode[STIX]{x1D6FE}\in \mathfrak{H}$, we consider a scaled RDE

(3.7)$$\begin{eqnarray}dy_{t}^{\unicode[STIX]{x1D716}}=\mathop{\sum }_{i=1}^{d}V_{i}(y_{t}^{\unicode[STIX]{x1D716}})\,d(\unicode[STIX]{x1D716}w)_{t}^{i}+V_{0}(y_{t}^{\unicode[STIX]{x1D716}})\,d(\unicode[STIX]{x1D716}^{1/H}t)\quad \text{with }y_{0}=a,\end{eqnarray}$$

and a scaled–shifted RDE

(3.8)$$\begin{eqnarray}d{\tilde{y}}_{t}^{\unicode[STIX]{x1D716}}=\mathop{\sum }_{i=1}^{d}V_{i}({\tilde{y}}_{t}^{\unicode[STIX]{x1D716}})\,d(\unicode[STIX]{x1D716}w+\unicode[STIX]{x1D6FE})_{t}^{i}+V_{0}({\tilde{y}}_{t}^{\unicode[STIX]{x1D716}})\,d(\unicode[STIX]{x1D716}^{1/H}t)\quad \text{with }y_{0}=a.\end{eqnarray}$$

The solutions are respectively given by

$$\begin{eqnarray}\mathbf{y}^{\unicode[STIX]{x1D716}}=\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D716}\mathbf{w},\unicode[STIX]{x1D716}^{1/H}\boldsymbol{\unicode[STIX]{x1D706}}),\qquad \tilde{\mathbf{y}}^{\unicode[STIX]{x1D716}}=\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D716}\mathbf{w}+\boldsymbol{\unicode[STIX]{x1D6FE}},\unicode[STIX]{x1D716}^{1/H}\boldsymbol{\unicode[STIX]{x1D706}}).\end{eqnarray}$$

It is known that the processes $(y_{t}^{\unicode[STIX]{x1D716}})$ and $(y_{\unicode[STIX]{x1D716}^{1/H}t})$ have the same law. We denote the Jacobi processes and its inverses of $y^{\unicode[STIX]{x1D716}}$ and ${\tilde{y}}^{\unicode[STIX]{x1D716}}$ by $J^{\unicode[STIX]{x1D716}}$, $K^{\unicode[STIX]{x1D716}}$, $\tilde{J}^{\unicode[STIX]{x1D716}}$ and $\tilde{K}^{\unicode[STIX]{x1D716}}$. (In the next subsection, a brief summary of the Cameron–Martin space $\mathfrak{H}$ and the Young translation by $\unicode[STIX]{x1D6FE}\in \mathfrak{H}$ will be given.)

We remark that everything in this subsection still holds if $p$-variation topology is replaced by $1/p$-Hölder topology. See [Reference Friz and VictoirFV10, Reference Friz and VictoirFV06].

3.3 The Cameron–Martin space and its isometric space

Next we introduce the Cameron–Martin space $\mathfrak{H}$ associated to a $d$-dimensional fBm on the time interval $[0,1]$. Let $\mathfrak{E}_{1}$ be the linear span of $\{R(t,\bullet )\}_{t\in [0,1]}$ and define the inner product

$$\begin{eqnarray}\mathop{\langle \mathop{\sum }_{k=1}^{m}a_{k}R(s_{k},\bullet ),\mathop{\sum }_{l=1}^{n}b_{l}R(t_{l},\bullet )\rangle }\nolimits_{\mathfrak{E}_{1}}=\mathop{\sum }_{k=1}^{m}\mathop{\sum }_{l=1}^{n}a_{k}b_{l}R(s_{k},t_{l}).\end{eqnarray}$$

The Cameron–Martin space $\mathfrak{H}$ associated to a $d$-dimensional fBm is given by the completion of $\mathfrak{E}=\mathfrak{E}_{1}^{d}$ with respect to the norm $\Vert \bullet \Vert _{\mathfrak{H}}=\langle \bullet ,\bullet \rangle _{\mathfrak{H}}$, where $\langle f,g\rangle _{\mathfrak{H}}=\sum _{i=1}^{d}\langle f^{i},g^{i}\rangle _{\mathfrak{E}_{1}}$ for $f=(f^{1},\ldots ,f^{d})^{\top }$, $g=(g^{1},\ldots ,g^{d})^{\top }\in \mathfrak{E}$. Note that $\mathfrak{H}$ is unitarily isometric to the first Wiener chaos ${\mathcal{C}}_{1}$, which is the Hilbert space defined by the $\Vert \bullet \Vert _{L^{2}(\unicode[STIX]{x1D6FA};\mathbf{R})}$-completion of the linear span of $(w_{t})_{t\in [0,1]}$. From [Reference Friz and VictoirFV06, Corollary 1], we see the continuous embedding $\mathfrak{H}{\hookrightarrow}C_{0}^{q\text{-}\text{var}}([0,1];\mathbf{R}^{d})$ for $(H+1/2)^{-1}<q<2$. According to [Reference Friz, Gess, Gulisashvili and RiedelFGGR16], the embedding holds for $q=(H+1/2)^{-1}$. (Note that if $p>1/H$ is sufficiently close to $1/H$, then $1/p+1/q>1$ holds since $H>1/4$. Therefore, the Young translation by $\unicode[STIX]{x1D6FE}\in \mathfrak{H}$ is well-defined on $G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})$.)

Next we introduce another Hilbert space $\tilde{\mathfrak{H}}$, which is isometric to $\mathfrak{H}$. Let $\tilde{\mathfrak{E}}_{1}$ be the linear span of $\{{\mathtt{1}_{[0,t]}\}}_{t\in [0,1]}$ and define an inner product by

$$\begin{eqnarray}\mathop{\langle \mathop{\sum }_{k=1}^{m}a_{k}\mathtt{1}_{[0,s_{k}]},\mathop{\sum }_{l=1}^{n}b_{l}\mathtt{1}_{[0,t_{l}]}\rangle }\nolimits_{\tilde{\mathfrak{E}}_{1}}=\mathop{\sum }_{k=1}^{m}\mathop{\sum }_{l=1}^{n}a_{k}b_{l}R(s_{k},t_{l}).\end{eqnarray}$$

We define $\tilde{\mathfrak{H}}$ by the completion of $\tilde{\mathfrak{E}}=\tilde{\mathfrak{E}}_{1}^{d}$ with respect to the norm $\Vert \bullet \Vert _{\tilde{\mathfrak{H}}}=\langle \bullet ,\bullet \rangle _{\tilde{\mathfrak{H}}}$, where $\langle f,g\rangle _{\tilde{\mathfrak{H}}}=\sum _{i=1}^{d}\langle f^{i},g^{i}\rangle _{\tilde{\mathfrak{E}}_{1}}$ for $f=(f^{1},\ldots ,f^{d})^{\top }$, $g=(g^{1},\ldots ,g^{d})^{\top }\in \tilde{\mathfrak{E}}$. Note that $\tilde{\mathfrak{H}}=I_{1-}^{1/2-H}(L^{2}([0,1];\mathbf{R}^{d}))$ from [Reference Decreusefond and ÜstünelDÜ99] and [Reference NualartNua06, p. 284]. Here, $I_{1-}^{1/2-H}$ denotes the fractional integral of order $1/2-H$. Hence, the inclusions $C^{(H-\unicode[STIX]{x1D6FF})\text{-}\text{Hol}}([0,1];\mathbf{R}^{d})\subset \tilde{\mathfrak{H}}\subset L^{2}([0,1];\mathbf{R}^{d})$ hold for $0<\unicode[STIX]{x1D6FF}<2(H-1/4)$. The linear map ${\mathcal{R}}=({\mathcal{R}}^{1},\ldots ,{\mathcal{R}}^{d})^{\top }:\tilde{{\mathcal{E}}}\rightarrow \mathfrak{H}$ defined by ${\mathcal{R}}^{i}\mathtt{1}_{[0,t]}=R(t,\bullet )$ extends to a unitary isometry from $\tilde{\mathfrak{H}}$ to $\mathfrak{H}$.

Remark 3.1. Under the condition $1/4<H\leqslant 1/2$, we can choose two numbers $1/H<p<1+\lfloor 1/H\rfloor$ and $(H+1/2)^{-1}<q<2$ with $1/p+1/q>1$. For every $A=(A_{1},\ldots ,A_{d})\in C^{p\text{-}\text{var}}([0,1];\mathbf{R}^{d})$, we can define $\unicode[STIX]{x1D719}_{A}\in \mathfrak{H}^{\ast }$ by

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{A}(h)=\mathop{\sum }_{i=1}^{d}\int _{0}^{1}A_{i}(s)\,dh_{s}^{i},\end{eqnarray}$$

where the right-hand side is the Young integral since $h\in \mathfrak{H}\subset C_{0}^{q\text{-}\text{var}}([0,1];\mathbf{R}^{d})$. By using the isometry between $\mathfrak{H}^{\ast }$ and ${\mathcal{C}}_{1}$, the definition of $\tilde{\mathfrak{H}}$ and the inequality $\Vert \bullet \Vert _{L^{2}([0,1];\mathbf{R}^{d})}\leqslant c\Vert \bullet \Vert _{\tilde{\mathfrak{H}}}$ for some positive constant $c$, we have

$$\begin{eqnarray}\Vert \unicode[STIX]{x1D719}_{A}\Vert _{\mathfrak{H}^{\ast }}^{2}=\mathbf{E}\Bigg[\Bigg(\mathop{\sum }_{i=1}^{d}\int _{0}^{1}A_{i}(s)\,dw_{s}^{i}\Bigg)^{2}\Bigg]=\Vert A\Vert _{\tilde{\mathfrak{ H}}}^{2}\geqslant c^{-2}\Vert A\Vert _{L^{2}([0,1];\mathbf{R}^{d})}^{2}.\end{eqnarray}$$

3.4 Large deviations

Let $(1+\lfloor 1/H\rfloor )^{-1}<\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FD}<H$ and $m\in \mathbf{N}_{+}$ satisfy $\unicode[STIX]{x1D6FC}-1/m>0$. Recall that fBm $(w_{t})$ admits a natural lift almost surely via the dyadic piecewise linear approximation and the lift $\mathbf{w}$ is a random variable taking values in $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FD}}^{\text{H}}(\mathbf{R}^{d})$. For $0<\unicode[STIX]{x1D716}<1$, the law of $\unicode[STIX]{x1D716}\mathbf{w}$ on this space is denoted by $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}=\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{H}$. Note that the lift of Cameron–Martin space $\mathfrak{H}$ is contained in $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FD}}^{\text{H}}(\mathbf{R}^{d})$. Moreover, as $\unicode[STIX]{x1D716}{\searrow}0$, Schilder-type large deviations hold for $\{{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}\}}_{\unicode[STIX]{x1D716}>0}$. (See [Reference Friz and VictoirFV07] and [Reference Friz and VictoirFV10, Sections 13.6 and 15.7].) Because of the Besov–Hölder embedding mentioned above, these properties also hold in $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},m}^{\text{B}}(\mathbf{R}^{d})$. As usual, the good rate function ${\mathcal{I}}$ is given as follows: ${\mathcal{I}}(\mathbf{x})=\Vert h\Vert _{\mathfrak{H}}^{2}/2$ if $\mathbf{x}$ is the lift of some $h\in \mathfrak{H}$ and ${\mathcal{I}}(\mathbf{x})=\infty$ if otherwise.

Next, set $\hat{\unicode[STIX]{x1D708}}_{\unicode[STIX]{x1D716}}=\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}\otimes \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D716}^{1/H}\unicode[STIX]{x1D706}}$, where $\unicode[STIX]{x1D706}$ is a one-dimensional path defined by $\unicode[STIX]{x1D706}_{t}=t$ and $\otimes$ stands for the product of probability measures. This measure is supported on $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},m}^{\text{B}}(\mathbf{R}^{d})\times \mathbf{R}\langle \unicode[STIX]{x1D706}\rangle$. The Young pairing map $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},m}^{\text{B}}(\mathbf{R}^{d})\times \mathbf{R}\langle \unicode[STIX]{x1D706}\rangle \rightarrow G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},m}^{\text{B}}(\mathbf{R}^{d+1})$ is continuous. The law of this map under $\hat{\unicode[STIX]{x1D708}}_{\unicode[STIX]{x1D716}}$ is the law of $(\unicode[STIX]{x1D716}\mathbf{w},\unicode[STIX]{x1D716}^{1/H}\boldsymbol{\unicode[STIX]{x1D706}})$, the Young pairing of $\unicode[STIX]{x1D716}\mathbf{w}$ and $\unicode[STIX]{x1D716}^{1/H}\unicode[STIX]{x1D706}$. Define $\hat{{\mathcal{I}}}(\mathbf{x},l)=\Vert h\Vert _{\mathfrak{H}}^{2}/2$ if $\mathbf{x}$ is the lift of some $h\in \mathfrak{H}$ and $l_{t}\equiv 0$ and define $\hat{{\mathcal{I}}}(\mathbf{x},l)=\infty$ if otherwise. Here, $l$ is a one-dimensional path. We can easily show that ${\{\hat{\unicode[STIX]{x1D708}}_{\unicode[STIX]{x1D716}}\}}_{\unicode[STIX]{x1D716}>0}$ also satisfies a large deviation principle as $\unicode[STIX]{x1D716}{\searrow}0$ with a good rate function $\hat{{\mathcal{I}}}$. We will use this in Lemma 6.1 to show that we may localize on a neighborhood of the minimizer $\bar{\unicode[STIX]{x1D6FE}}$ in order to obtain the asymptotic expansion. (The existence of $\bar{\unicode[STIX]{x1D6FE}}$ is assumed in (A2).)

3.5 The Young translation

In this subsection, we show that the Young translation is well-defined and continuous. In this subsection, we fix $1/4<H\leqslant 1/2$. Let $(1+\lfloor 1/H\rfloor )^{-1}<\unicode[STIX]{x1D6FC}<H$ and choose $m\in \mathbf{N}_{+}$ such that $H-\unicode[STIX]{x1D6FC}>2/m$. Set $p=1/\unicode[STIX]{x1D6FC}$ and $q=(H+1/2-1/m)^{-1}$. Then we have $1/p+1/q>1$ and $1/q-1/2-\unicode[STIX]{x1D6FC}>1/m$. Since $(H+1/2)^{-1}\leqslant q<2$, [Reference Friz and VictoirFV06, Corollary 1] implies

$$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D6FE}\Vert _{q\text{-}\text{var};[s,t]}\leqslant c\Vert \unicode[STIX]{x1D6FE}\Vert _{W^{1/q,2}}(t-s)^{(1/q-1/2)}\leqslant c\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}(t-s)^{(1/q-1/2)}\quad (\unicode[STIX]{x1D6FE}\in \mathfrak{H}). & & \displaystyle \nonumber\end{eqnarray}$$

Throughout this subsection, $c$ denotes a positive constant and may change from line to line.

We define the Young translation $\unicode[STIX]{x1D70F}=(\unicode[STIX]{x1D70F}^{1},\ldots ,\unicode[STIX]{x1D70F}^{\lfloor 1/\unicode[STIX]{x1D6FC}\rfloor })$ from $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},12m}^{\text{B}}(\mathbf{R}^{d})\times \mathfrak{H}$ to $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},12m}^{\text{B}}(\mathbf{R}^{d})$; for $\mathbf{x}\in G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},12m}^{\text{B}}(\mathbf{R}^{d})$ and $\unicode[STIX]{x1D6FE}\in \mathfrak{H}$, define

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FE}}(\mathbf{x})_{s,t}^{1} & = & \displaystyle \mathbf{x}_{s,t}^{1}+\boldsymbol{\unicode[STIX]{x1D6FE}}_{s,t}^{1},\nonumber\\ \displaystyle \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FE}}(\mathbf{x})_{s,t}^{2} & = & \displaystyle \mathbf{x}_{s,t}^{2}+A_{s,t}^{1}+A_{s,t}^{2}+\boldsymbol{\unicode[STIX]{x1D6FE}}_{s,t}^{2},\nonumber\\ \displaystyle \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FE}}(\mathbf{x})_{s,t}^{3} & = & \displaystyle \mathbf{x}_{s,t}^{3}+B_{s,t}^{1}+B_{s,t}^{2}+B_{s,t}^{3}+C_{s,t}^{1}+C_{s,t}^{2}+C_{s,t}^{3}+\boldsymbol{\unicode[STIX]{x1D6FE}}_{s,t}^{3},\nonumber\end{eqnarray}$$

where

$$\begin{eqnarray}\displaystyle \begin{array}{@{}ll@{}}\displaystyle \boldsymbol{\unicode[STIX]{x1D6FE}}_{s,t}^{1}=\unicode[STIX]{x1D6FE}_{t}-\unicode[STIX]{x1D6FE}_{s},\quad & \displaystyle \boldsymbol{\unicode[STIX]{x1D6FE}}_{s,t}^{i}=\int _{s}^{t}\boldsymbol{\unicode[STIX]{x1D6FE}}_{s,u}^{i-1}\otimes d\unicode[STIX]{x1D6FE}_{u}\quad (i=2,3),\\ \displaystyle A_{s,t}^{1}=\int _{s}^{t}\mathbf{x}_{s,u}^{1}\otimes d\unicode[STIX]{x1D6FE}_{u},\quad & \displaystyle A_{s,t}^{2}=\int _{s}^{t}\boldsymbol{\unicode[STIX]{x1D6FE}}_{s,u}^{1}\otimes dx_{u},\\ \displaystyle B_{s,t}^{1}=\int _{s}^{t}\mathbf{x}_{s,u}^{2}\otimes d\unicode[STIX]{x1D6FE}_{u},\quad & \displaystyle B_{s,t}^{2}=\int _{s}^{t}\int _{s}^{v}\mathbf{x}_{s,u}^{1}\otimes d\unicode[STIX]{x1D6FE}_{u}\otimes dx_{v},\\ \displaystyle B_{s,t}^{3}=-\int _{s}^{t}\boldsymbol{\unicode[STIX]{x1D6FE}}_{s,u}^{1}\otimes d\mathbf{x}_{u,t}^{2},\quad & \displaystyle C_{s,t}^{1}=-\int _{s}^{t}\mathbf{x}_{s,u}^{1}\otimes d\boldsymbol{\unicode[STIX]{x1D6FE}}_{u,t}^{2},\\ \displaystyle C_{s,t}^{2}=\int _{s}^{t}\int _{s}^{v}\boldsymbol{\unicode[STIX]{x1D6FE}}_{s,u}^{1}\otimes dx_{u}\otimes d\unicode[STIX]{x1D6FE}_{v},\quad & \displaystyle C_{s,t}^{3}=\int _{s}^{t}\boldsymbol{\unicode[STIX]{x1D6FE}}_{s,u}^{2}\otimes dx_{u}.\end{array} & & \displaystyle \nonumber\end{eqnarray}$$

We should understand the integrals in the Young sense since $1/p+1/q>1$. At first glance, $B^{3}$ and $C^{1}$ look strange. However, if $x$ and $\unicode[STIX]{x1D6FE}$ are smooth and $\mathbf{x}$ is the natural lift of $x$, we have

$$\begin{eqnarray}B_{s,t}^{3}=\int _{s}^{t}\boldsymbol{\unicode[STIX]{x1D6FE}}_{s,u}^{1}\otimes dx_{u}\otimes (x_{t}-x_{u})=\int _{s}^{t}\int _{s}^{v}\boldsymbol{\unicode[STIX]{x1D6FE}}_{s,u}^{1}\otimes dx_{u}\otimes dx_{v}.\end{eqnarray}$$

In the first equality, we used $\frac{d\mathbf{x}_{u,t}^{2}}{du}=-\frac{dx_{u}}{du}\otimes (x_{t}-x_{u})$. Since a similar equality for $C_{s,t}^{3}$ holds, we see $B^{3}$ and $C^{1}$ are defined appropriately. Then we see the next proposition.

Proof. Let $0\leqslant s\leqslant u<v\leqslant t\leqslant 1$. Recall that $\mathbf{x}_{s,\cdot }^{i}$ and $\mathbf{x}_{\cdot ,t}^{i}$ have finite $p$-variation and that $\Vert \mathbf{x}_{s,\cdot }^{i}\Vert _{p\text{-}\text{var};[u,v]}$ and $\Vert \mathbf{x}_{\cdot ,t}^{i}\Vert _{p\text{-}\text{var};[u,v]}$ are bounded above by $c\Vert \mathbf{x}\Vert _{p\text{-}\text{var};[s,t]}^{i-1}\Vert \mathbf{x}\Vert _{p\text{-}\text{var};[u,v]}$ for $i=1,2,3$.

We show well-definedness of the map $\unicode[STIX]{x1D70F}$. Since $1/q+1/q>1$, we can define $\boldsymbol{\unicode[STIX]{x1D6FE}}^{2}$ in the Young sense and see that it is of finite $q$-variation and satisfies

$$\begin{eqnarray}\displaystyle \Vert \boldsymbol{\unicode[STIX]{x1D6FE}}_{s,\cdot }^{2}\Vert _{q\text{-}\text{var};[u,v]},\,\,\Vert \boldsymbol{\unicode[STIX]{x1D6FE}}_{\cdot ,t}^{2}\Vert _{q\text{-}\text{var};[u,v]}\leqslant c\Vert \unicode[STIX]{x1D6FE}\Vert _{q\text{-}\text{var};[s,t]}\Vert \unicode[STIX]{x1D6FE}\Vert _{q\text{-}\text{var};[u,v]}. & & \displaystyle \nonumber\end{eqnarray}$$

In this estimate, we used [Reference Friz and VictoirFV10, Theorem 6.8]. By the same reason, $A^{1}$ and $A^{2}$ are well-defined in the Young sense and they are of finite $q$- and $p$-variation, respectively; more precisely, it holds that

$$\begin{eqnarray}\displaystyle \Vert A_{s,\cdot }^{1}\Vert _{q\text{-}\text{var};[u,v]} & {\leqslant} & \displaystyle c\Vert x\Vert _{p\text{-}\text{var};[s,t]}\Vert \unicode[STIX]{x1D6FE}\Vert _{q\text{-}\text{var};[u,v]},\nonumber\\ \displaystyle \Vert A_{s,\cdot }^{2}\Vert _{p\text{-}\text{var};[u,v]} & {\leqslant} & \displaystyle c\Vert \unicode[STIX]{x1D6FE}\Vert _{q\text{-}\text{var};[s,t]}\Vert x\Vert _{p\text{-}\text{var};[u,v]}.\nonumber\end{eqnarray}$$

Note that $B_{s,t}^{2}=\int _{s}^{t}A_{s,r}^{1}\otimes dx_{r}$ and $C_{s,t}^{2}=\int _{s}^{t}A_{s,r}^{2}\otimes d\unicode[STIX]{x1D6FE}_{r}$ can be defined and

$$\begin{eqnarray}\displaystyle \Vert B_{s,\cdot }^{2}\Vert _{p\text{-}\text{var};[u,v]} & {\leqslant} & \displaystyle c\Vert A_{s,\cdot }^{1}\Vert _{q\text{-}\text{var};[s,t]}\Vert x\Vert _{p\text{-}\text{var};[u,v]},\nonumber\\ \displaystyle \Vert C_{s,\cdot }^{2}\Vert _{q\text{-}\text{var};[u,v]} & {\leqslant} & \displaystyle c\Vert A_{s,\cdot }^{2}\Vert _{p\text{-}\text{var};[s,t]}\Vert \unicode[STIX]{x1D6FE}\Vert _{q\text{-}\text{var};[u,v]}.\nonumber\end{eqnarray}$$

By the same reason, we see that $B_{s,t}^{1}$, $B_{s,t}^{3}$, $C_{s,t}^{1}$, and $C_{s,t}^{3}$ are well-defined and

$$\begin{eqnarray}\displaystyle \Vert \boldsymbol{\unicode[STIX]{x1D6FE}}_{s,\cdot }^{3}\Vert _{q\text{-}\text{var};[u,v]} & {\leqslant} & \displaystyle c\Vert \boldsymbol{\unicode[STIX]{x1D6FE}}_{s,\cdot }^{2}\Vert _{q\text{-}\text{var};[s,t]}\Vert \unicode[STIX]{x1D6FE}\Vert _{q\text{-}\text{var};[u,v]},\nonumber\\ \displaystyle \Vert B_{s,\cdot }^{1}\Vert _{q\text{-}\text{var};[u,v]} & {\leqslant} & \displaystyle c\Vert \mathbf{x}_{s,\cdot }^{2}\Vert _{p\text{-}\text{var};[s,t]}\Vert \unicode[STIX]{x1D6FE}\Vert _{q\text{-}\text{var};[u,v]},\nonumber\\ \displaystyle \Vert B_{s,\cdot }^{3}\Vert _{p\text{-}\text{var};[u,v]} & {\leqslant} & \displaystyle c\Vert \unicode[STIX]{x1D6FE}\Vert _{q\text{-}\text{var};[s,t]}\Vert \mathbf{x}_{\cdot ,t}^{2}\Vert _{p\text{-}\text{var};[u,v]},\nonumber\\ \displaystyle \Vert C_{s,\cdot }^{1}\Vert _{q\text{-}\text{var};[u,v]} & {\leqslant} & \displaystyle c\Vert x\Vert _{p\text{-}\text{var};[s,t]}\Vert \boldsymbol{\unicode[STIX]{x1D6FE}}_{\cdot ,t}^{2}\Vert _{q\text{-}\text{var};[u,v]},\nonumber\\ \displaystyle \Vert C_{s,\cdot }^{3}\Vert _{p\text{-}\text{var};[u,v]} & {\leqslant} & \displaystyle c\Vert \boldsymbol{\unicode[STIX]{x1D6FE}}_{s,\cdot }^{2}\Vert _{q\text{-}\text{var};[s,t]}\Vert x\Vert _{p\text{-}\text{var};[u,v]}.\nonumber\end{eqnarray}$$

By recalling upper bounds of $\Vert \unicode[STIX]{x1D6FE}\Vert _{q\text{-}\text{var};[u,v]}$, $\Vert \boldsymbol{\unicode[STIX]{x1D6FE}}_{s,\cdot }^{2}\Vert _{q\text{-}\text{var};[u,v]}$, $\Vert \boldsymbol{\unicode[STIX]{x1D6FE}}_{\cdot ,t}^{2}\Vert _{q\text{-}\text{var};[u,v]}$, $\Vert x\Vert _{q\text{-}\text{var};[u,v]}$, $\Vert \mathbf{x}_{s,\cdot }^{2}\Vert _{q\text{-}\text{var};[u,v]}$, and $\Vert \mathbf{x}_{\cdot ,t}^{2}\Vert _{q\text{-}\text{var};[u,v]}$, we have

$$\begin{eqnarray}\displaystyle |\boldsymbol{\unicode[STIX]{x1D6FE}}_{s,t}^{2}| & {\leqslant} & \displaystyle c\Vert \unicode[STIX]{x1D6FE}\Vert _{q\text{-}\text{var};[s,t]}^{2}\leqslant c\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}^{2}(t-s)^{2(1/q-1/2)},\nonumber\\ \displaystyle |A_{s,t}^{1}|,|A_{s,t}^{2}| & {\leqslant} & \displaystyle c\Vert \mathbf{x}\Vert _{p\text{-}\text{var};[s,t]}\Vert \unicode[STIX]{x1D6FE}\Vert _{q\text{-}\text{var};[s,t]}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle c\Vert \mathbf{x}\Vert _{(\unicode[STIX]{x1D6FC},12m)\text{-}\text{Bes}}\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}\cdot (t-s)^{(\unicode[STIX]{x1D6FC}-1/12m)+(1/q-1/2)}.\nonumber\end{eqnarray}$$

Here, we used (3.1). Moreover, we obtain

$$\begin{eqnarray}\displaystyle |B_{s,t}^{1}|,|B_{s,t}^{2}|,|B_{s,t}^{3}| & {\leqslant} & \displaystyle c\Vert \mathbf{x}\Vert _{p\text{-}\text{var};[s,t]}^{2}\Vert \unicode[STIX]{x1D6FE}\Vert _{q\text{-}\text{var};[s,t]}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle c\Vert \mathbf{x}\Vert _{(\unicode[STIX]{x1D6FC},12m)\text{-}\text{Bes}}^{2}\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}\cdot (t-s)^{2(\unicode[STIX]{x1D6FC}-1/12m)+(1/q-1/2)},\nonumber\\ \displaystyle |C_{s,t}^{1}|,|C_{s,t}^{2}|,|C_{s,t}^{3}| & {\leqslant} & \displaystyle c\Vert \mathbf{x}\Vert _{p\text{-}\text{var};[s,t]}\Vert \unicode[STIX]{x1D6FE}\Vert _{q\text{-}\text{var};[s,t]}^{2}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle c\Vert \mathbf{x}\Vert _{(\unicode[STIX]{x1D6FC},12m)\text{-}\text{Bes}}\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}^{2}\cdot (t-s)^{(\unicode[STIX]{x1D6FC}-1/12m)+2(1/q-1/2)}.\nonumber\end{eqnarray}$$

Note

$$\begin{eqnarray}\displaystyle & \displaystyle \bigg(\unicode[STIX]{x1D6FC}-\frac{1}{12m}\bigg)+\bigg(\frac{1}{q}-\frac{1}{2}\bigg)=2\unicode[STIX]{x1D6FC}+\bigg(\frac{1}{q}-\frac{1}{2}-\unicode[STIX]{x1D6FC}-\frac{1}{12m}\bigg)>2\unicode[STIX]{x1D6FC}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle 2\bigg(\unicode[STIX]{x1D6FC}-\frac{1}{12m}\bigg)+\bigg(\frac{1}{q}-\frac{1}{2}\bigg)=3\unicode[STIX]{x1D6FC}+\bigg(\frac{1}{q}-\frac{1}{2}-\unicode[STIX]{x1D6FC}-\frac{2}{12m}\bigg)>3\unicode[STIX]{x1D6FC}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \bigg(\unicode[STIX]{x1D6FC}-\frac{1}{12m}\bigg)+2\bigg(\frac{1}{q}-\frac{1}{2}\bigg)=3\unicode[STIX]{x1D6FC}+2\bigg(\frac{1}{q}-\frac{1}{2}-\unicode[STIX]{x1D6FC}-\frac{1}{2\cdot 12m}\bigg)>3\unicode[STIX]{x1D6FC}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \frac{1}{q}-\frac{1}{2}=\unicode[STIX]{x1D6FC}+\bigg(\frac{1}{q}-\frac{1}{2}-\unicode[STIX]{x1D6FC}\bigg)>\unicode[STIX]{x1D6FC}. & \displaystyle \nonumber\end{eqnarray}$$

Hence, $A^{i}$ and $\boldsymbol{\unicode[STIX]{x1D6FE}}^{2}$ are of finite $2(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF})$-Hölder norm for some $\unicode[STIX]{x1D6FF}>0$ and therefore they are of finite $(2\unicode[STIX]{x1D6FC},6m)$-Besov norm. We also see that $B^{i}$, $C^{i}$ and $\boldsymbol{\unicode[STIX]{x1D6FE}}^{3}$ are of finite $(3\unicode[STIX]{x1D6FC},4m)$-Besov norm. Hence, $\unicode[STIX]{x1D70F}$ is well-defined.

The continuity follows from the continuous embedding $C_{0}^{q\text{-}\text{var}}([0,1];\mathbf{R}^{d})$ and the continuity of the Young integration.◻

5.1 Notation and results of this section

We recall Watanabe’s theory of generalized Wiener functionals (Watanabe distributions) in Malliavin calculus. We borrow the notations used in this paper from Ikeda and Watanabe [Reference Ikeda and WatanabeIW89, V.8-V.10]. We also refer to Nualart [Reference NualartNua06], Shigekawa [Reference ShigekawaShi04], Matsumoto and Taniguchi [Reference Matsumoto and TaniguchiMT17] and Hu [Reference HuHu17].

We introduce a measure. Let $\mathbf{P}=\mathbf{P}^{H}$ be the law of the fBm with Hurst parameter $H$. This is a probability measure on $\unicode[STIX]{x1D6FA}=\overline{\mathfrak{H}}$, which is the closure of the Cameron–Martin space $\mathfrak{H}=\mathfrak{H}^{H}$ in $C_{0}^{p\text{-}\text{var}}([0,1];\mathbf{R}^{d})$. Then, the triple $(\unicode[STIX]{x1D6FA},\mathfrak{H},\mathbf{P})$ is an abstract Wiener space. We denote by $\mathbf{D}_{r,s}(K)$ the $K$-valued Gaussian–Sobolev space, where $1<r<\infty$, $s\in \mathbf{R}$ and $K$ is a real separable Hilbert space. Note that $r$ and $s$ stand for the integrability index and the differentiability index, respectively. As usual, we set the spaces $\mathbf{D}_{\infty }(K)=\bigcap _{s=1}^{\infty }\bigcap _{1<r<\infty }\mathbf{D}_{r,s}(K)$ and $\tilde{\mathbf{D}}_{\infty }(K)=\bigcap _{s=1}^{\infty }\bigcup _{1<r<\infty }\mathbf{D}_{r,s}(K)$ of test functions and the spaces $\mathbf{D}_{-\infty }(K)=\bigcup _{s=1}^{\infty }\bigcup _{1<r<\infty }\mathbf{D}_{r,-s}(K)$ and $\tilde{\mathbf{D}}_{-\infty }(K)=\bigcup _{s=1}^{\infty }\bigcap _{1<r<\infty }\mathbf{D}_{n,-s}(K)$ of Watanabe distributions. For $F\in \mathbf{D}_{r,s}(K)$, $D^{k}F$ denotes the $k$th derivative of $F$ for $k\in \mathbf{N}_{+}$.

We set $V_{0}^{\unicode[STIX]{x1D716}}=\unicode[STIX]{x1D716}^{1/H}V_{0}$, $V_{i}^{\unicode[STIX]{x1D716}}=\unicode[STIX]{x1D716}V_{i}$ ($1\leqslant i\leqslant d$), and $\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D716}}=[V_{1}^{\unicode[STIX]{x1D716}},\ldots ,V_{d}^{\unicode[STIX]{x1D716}}]$, which is viewed as an $n\times d$ matrix. It holds that $\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D716}}=\unicode[STIX]{x1D716}\unicode[STIX]{x1D70E}$. Set $\tilde{A}^{\unicode[STIX]{x1D716}}(s)=\tilde{J}_{1}^{\unicode[STIX]{x1D716}}\tilde{K}_{s}^{\unicode[STIX]{x1D716}}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D716}}({\tilde{y}}_{s}^{\unicode[STIX]{x1D716}})$, whose size is again $n\times d$. Let $\tilde{A}^{\unicode[STIX]{x1D716},k}(s)$ and $\tilde{A}_{i}^{\unicode[STIX]{x1D716},k}(s)$ denote the $k$th row and the $(k,i)$-component of $\tilde{A}^{\unicode[STIX]{x1D716}}(s)$, respectively.

Under this setting, we see that ${\tilde{y}}_{1}^{\unicode[STIX]{x1D716}}$ is differentiable in the sense of Malliavin and the derivatives admit good estimate as follows:

Proposition 5.1. We have ${\tilde{y}}_{1}^{\unicode[STIX]{x1D716}}\in \mathbf{D}_{\infty }(\mathbf{R}^{n})$ and

(5.1)$$\begin{eqnarray}\langle D{\tilde{y}}_{1}^{\unicode[STIX]{x1D716},k},h\rangle _{\mathfrak{ H}}=\unicode[STIX]{x1D719}_{\tilde{A}^{\unicode[STIX]{x1D716},k}}(h)=\mathop{\sum }_{i=1}^{d}\int _{0}^{1}\tilde{A}_{i}^{\unicode[STIX]{x1D716},k}(s)\,dh_{s}^{i}\quad (k=1,\ldots ,n).\end{eqnarray}$$

In addition, for any $m=0,1,2,\ldots$ and $1<r<\infty$, there exists a positive constant $c=c_{m,r}$ such that

$$\begin{eqnarray}\mathbf{E}[\Vert D^{m}{\tilde{y}}_{1}^{\unicode[STIX]{x1D716},k}\Vert _{\mathfrak{ H}^{\otimes m}}^{r}]^{1/r}\leqslant c\unicode[STIX]{x1D716}^{m}\quad (k=1,\ldots ,n).\end{eqnarray}$$

Proposition 5.2. We have the following asymptotic expansion as $\unicode[STIX]{x1D716}{\searrow}0$:

$$\begin{eqnarray}{\tilde{y}}_{1}^{\unicode[STIX]{x1D716}}\sim \unicode[STIX]{x1D719}_{1}^{0}+\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D705}_{1}}\unicode[STIX]{x1D719}_{1}^{\unicode[STIX]{x1D705}_{1}}+\cdots +\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D705}_{k}}\unicode[STIX]{x1D719}_{1}^{\unicode[STIX]{x1D705}_{k}}+\cdots \quad \text{in }\mathbf{D}_{\infty }(\mathbf{R}^{n}).\end{eqnarray}$$

This means that for each $k$, (i) $\unicode[STIX]{x1D719}_{1}^{\unicode[STIX]{x1D705}_{k}}\in \mathbf{D}_{\infty }(\mathbf{R}^{n})$ and (ii) $\mathbf{D}_{r,s}$-norm of $r_{\unicode[STIX]{x1D716},1}^{\unicode[STIX]{x1D705}_{k+1}}$ is $O(\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D705}_{k+1}})$ for any $1<r<\infty$ and $s\geqslant 0$. Here, $r_{\unicode[STIX]{x1D716},1}^{\unicode[STIX]{x1D705}_{k+1}}$ is defined by (4.6).

The Malliavin covariance matrices $Q=(Q_{kl})_{1\leqslant l,k\leqslant n}$ of $y_{1}$, $Q^{\unicode[STIX]{x1D716}}=(Q_{kl}^{\unicode[STIX]{x1D716}})_{1\leqslant l,k\leqslant n}$ of $y_{1}^{\unicode[STIX]{x1D716}}$, and $\tilde{Q}^{\unicode[STIX]{x1D716}}=(\tilde{Q}_{kl}^{\unicode[STIX]{x1D716}})_{1\leqslant l,k\leqslant n}$ of ${\tilde{y}}_{1}^{\unicode[STIX]{x1D716}}$ are defined by

(5.2)$$\begin{eqnarray}Q_{kl}=\langle Dy_{1}^{k},Dy_{1}^{l}\rangle _{\mathfrak{ H}},\qquad Q_{kl}^{\unicode[STIX]{x1D716}}=\langle Dy_{1}^{\unicode[STIX]{x1D716},k},Dy_{1}^{\unicode[STIX]{x1D716},l}\rangle _{\mathfrak{ H}},\qquad \tilde{Q}_{kl}^{\unicode[STIX]{x1D716}}=\langle D{\tilde{y}}_{1}^{\unicode[STIX]{x1D716},k},D{\tilde{y}}_{1}^{\unicode[STIX]{x1D716},l}\rangle _{\mathfrak{ H}},\end{eqnarray}$$

respectively. In this paper we do not express these covariance matrices as two-parameter Young integrals (cf. [Reference Cass, Hairer, Litterer and TindelCHLT15, (6.1)]). In this notation, we will state the following two propositions.

The first one is Kusuoka–Stroock type estimate for the solution of the scaled RDE. Although this is not surprising, there seems to be no literature that actually proves it.

Proposition 5.3. Suppose that (A1) holds. Then, there are positive constants $c$ and $\unicode[STIX]{x1D707}$ such that for every $0<\unicode[STIX]{x1D716}<1$ and $1<r<\infty$, we have

$$\begin{eqnarray}\mathbf{E}[|\!\det Q^{\unicode[STIX]{x1D716}}|^{-r}]^{1/r}\leqslant c\unicode[STIX]{x1D716}^{-\unicode[STIX]{x1D707}}.\end{eqnarray}$$

Here, $c=c(r)$ can be chosen independent of $\unicode[STIX]{x1D716}$, while $\unicode[STIX]{x1D707}$ is independent of both $r$ and $\unicode[STIX]{x1D716}$.

The second one is about the uniform non-degeneracy in the sense of Malliavin calculus of the solution of the scaled–shifted RDE under (A3). The Schilder-type large deviations for fractional Brownian rough path are used in the proof.

Proposition 5.4. Suppose that (A1) holds. Let $\unicode[STIX]{x1D6FE}$ in the definition of ${\tilde{y}}^{\unicode[STIX]{x1D716}}$ in (3.8) satisfy $Q(\unicode[STIX]{x1D6FE})\geqslant cI$ for some $c>0$. Then, for every $1<r<\infty$, we have

$$\begin{eqnarray}\sup _{0<\unicode[STIX]{x1D716}<1}\mathbf{E}[|\!\det \unicode[STIX]{x1D716}^{-2}\tilde{Q}^{\unicode[STIX]{x1D716}}|^{-r}]^{1/r}<\infty .\end{eqnarray}$$

Let $a^{\prime }\in \mathbf{R}^{n}$. Then, we have

$$\begin{eqnarray}\unicode[STIX]{x1D716}^{-2}\tilde{Q}_{kl}^{\unicode[STIX]{x1D716}}=\langle D\bigg(\frac{{\tilde{y}}_{1}^{\unicode[STIX]{x1D716},k}-(a^{\prime })^{k}}{\unicode[STIX]{x1D716}}\bigg),D\bigg(\frac{{\tilde{y}}_{1}^{\unicode[STIX]{x1D716},l}-(a^{\prime })^{l}}{\unicode[STIX]{x1D716}}\bigg)\rangle _{\mathfrak{H}}.\end{eqnarray}$$

It follows from Proposition 5.4 and this identity that $({\tilde{y}}_{1}^{\unicode[STIX]{x1D716}}-a^{\prime })/\unicode[STIX]{x1D716}$ with $\unicode[STIX]{x1D6FE}=\bar{\unicode[STIX]{x1D6FE}}$ is uniformly nondegenerate in the sense of Malliavin calculus under (A1), (A2) and (A3).

Propositions 5.3 and 5.4 will be proved in §§ 5.2 and 5.3, respectively.

5.2 Covariance matrix of the solution of scaled RDE driven by fBm

In this subsection, we show Proposition 5.3. Set

$$\begin{eqnarray}C^{\unicode[STIX]{x1D716}}=\int _{0}^{1}K_{s}^{\unicode[STIX]{x1D716}}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D716}}(y_{s}^{\unicode[STIX]{x1D716}})\{K_{s}^{\unicode[STIX]{x1D716}}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D716}}(y_{s}^{\unicode[STIX]{x1D716}})\}^{\top }\,ds.\end{eqnarray}$$

Then we see Proposition 5.3 from the following three lemmas.

Lemma 5.5. For every $0<\unicode[STIX]{x1D716}<1$, we have

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\min }(Q^{\unicode[STIX]{x1D716}})\geqslant c\unicode[STIX]{x1D706}_{\min }(C^{\unicode[STIX]{x1D716}})\unicode[STIX]{x1D706}_{\min }(J_{1}^{\unicode[STIX]{x1D716}}(J_{1}^{\unicode[STIX]{x1D716}})^{\top }).\end{eqnarray}$$

Here, $c$ is a positive constant independent of $\unicode[STIX]{x1D716}$.

Lemma 5.6. Suppose that (A1) holds. Then, there are positive constants $c$ and $\unicode[STIX]{x1D707}$ such that for every $0<\unicode[STIX]{x1D716}<1$ and $1<r<\infty$, we have

$$\begin{eqnarray}\mathbf{E}[\unicode[STIX]{x1D706}_{\min }(C^{\unicode[STIX]{x1D716}})^{-r}]^{1/r}\leqslant c\unicode[STIX]{x1D716}^{-\unicode[STIX]{x1D707}}.\end{eqnarray}$$

Here, $c=c(r)$ can be chosen independent of $\unicode[STIX]{x1D716}$, while $\unicode[STIX]{x1D707}$ is independent of both $r$ and $\unicode[STIX]{x1D716}$.

Lemma 5.7. [Reference Cass, Litterer and LyonsCLL13]

Let $p>1/H$. For every $1<r<\infty$, we have

$$\begin{eqnarray}\sup _{0<\unicode[STIX]{x1D716}<1}\mathbf{E}[\Vert J^{\unicode[STIX]{x1D716}}\Vert _{C^{p\text{-}\text{var}}([0,1];\mathbf{R}^{n^{2}})}^{r}]^{1/r}<\infty ,\qquad \sup _{0<\unicode[STIX]{x1D716}<1}\mathbf{E}[\Vert K^{\unicode[STIX]{x1D716}}\Vert _{C^{p\text{-}\text{var}}([0,1];\mathbf{R}^{n^{2}})}^{r}]^{1/r}<\infty .\end{eqnarray}$$

Now we show Proposition 5.3 by using lemmas above.

Proof of Proposition 5.3.

From Lemma 5.5, we see

$$\begin{eqnarray}\det Q^{\unicode[STIX]{x1D716}}\geqslant \unicode[STIX]{x1D706}_{\min }(Q^{\unicode[STIX]{x1D716}})^{n}\geqslant c^{n}\unicode[STIX]{x1D706}_{\min }(C^{\unicode[STIX]{x1D716}})^{n}\unicode[STIX]{x1D706}_{\min }(J_{1}^{\unicode[STIX]{x1D716}}(J_{1}^{\unicode[STIX]{x1D716}})^{\top })^{n}.\end{eqnarray}$$

Hence,

$$\begin{eqnarray}\displaystyle \mathbf{E}[|\!\det Q^{\unicode[STIX]{x1D716}}|^{-r}]^{1/r} & {\leqslant} & \displaystyle c^{-n}\mathbf{E}[\unicode[STIX]{x1D706}_{\min }(C^{\unicode[STIX]{x1D716}})^{-nr}\unicode[STIX]{x1D706}_{\min }(J_{1}^{\unicode[STIX]{x1D716}}(J_{1}^{\unicode[STIX]{x1D716}})^{\top })^{-nr}]^{1/r}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle c^{-n}\mathbf{E}[\unicode[STIX]{x1D706}_{\min }(C^{\unicode[STIX]{x1D716}})^{-2nr}]^{1/(2r)}\mathbf{E}[\unicode[STIX]{x1D706}_{\min }(J_{1}^{\unicode[STIX]{x1D716}}(J_{1}^{\unicode[STIX]{x1D716}})^{\top })^{-2nr}]^{1/(2r)}.\nonumber\end{eqnarray}$$

Noting $\unicode[STIX]{x1D706}_{\min }(J_{1}^{\unicode[STIX]{x1D716}}(J_{1}^{\unicode[STIX]{x1D716}})^{\top })^{-1}=\unicode[STIX]{x1D706}_{\max }(K_{1}^{\unicode[STIX]{x1D716}}(K_{1}^{\unicode[STIX]{x1D716}})^{\top })$ and applying Lemmas 5.6 and 5.7, we see the assertion.

Next we show Lemma 5.5.

Proof of Lemma 5.5.

Set $A^{\unicode[STIX]{x1D716}}(s)=\mathtt{1}_{[0,1]}(s)J_{1}^{\unicode[STIX]{x1D716}}K_{s}^{\unicode[STIX]{x1D716}}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D716}}(y_{s}^{\unicode[STIX]{x1D716}})$ and let $A^{\unicode[STIX]{x1D716},k}(s)$ denote the $k$th row of $A^{\unicode[STIX]{x1D716}}(s)$. Then, for every $v=(v_{1},\ldots ,v_{n})^{\top }\in \mathbf{R}^{n}$, we have $\langle v,Q^{\unicode[STIX]{x1D716}}v\rangle _{\mathbf{R}^{n}}=\Vert \!\sum _{k=1}^{n}v_{k}Dy_{1}^{\unicode[STIX]{x1D716},k}\Vert _{\mathfrak{H}}^{2}$. It follows from the Riesz representation theorem and Remark 3.1 that

$$\begin{eqnarray}\biggl\|\mathop{\sum }_{k=1}^{n}v_{k}Dy_{1}^{\unicode[STIX]{x1D716},k}\biggr\|_{\mathfrak{ H}}^{2}=\biggl\|\mathop{\sum }_{k=1}^{n}v_{k}Dy_{1}^{\unicode[STIX]{x1D716},k}\biggr\|_{\mathfrak{ H}^{\ast }}^{2}\geqslant c\biggl\|\mathop{\sum }_{k=1}^{n}v_{k}A^{\unicode[STIX]{x1D716},k}\biggr\|_{L^{2}([0,1];\mathbf{R}^{d})}^{2}.\end{eqnarray}$$

Here, $c$ is a positive constant independent of $v$ and $\unicode[STIX]{x1D716}$. Noting $|\sum _{k=1}^{n}v_{k}A^{\unicode[STIX]{x1D716},k}(s)|_{\mathbf{R}^{d}}^{2}=\langle v,A^{\unicode[STIX]{x1D716}}(s)A^{\unicode[STIX]{x1D716}}(s)^{\top }v\rangle _{\mathbf{R}^{n}}$, we have

$$\begin{eqnarray}\biggl\|\mathop{\sum }_{k=1}^{n}v_{k}A^{\unicode[STIX]{x1D716},k}\biggr\|_{L^{2}([0,1];\mathbf{R}^{d})}^{2}=\int _{0}^{1}\langle v,A^{\unicode[STIX]{x1D716}}(s)A^{\unicode[STIX]{x1D716}}(s)^{\top }v\rangle _{\boldsymbol{ R}^{n}}\,ds=\langle v,J_{1}^{\unicode[STIX]{x1D716}}C_{\unicode[STIX]{x1D716}}(J_{1}^{\unicode[STIX]{x1D716}})^{\top }v\rangle _{\boldsymbol{ R}^{n}}.\end{eqnarray}$$

These imply

$$\begin{eqnarray}\langle v,Q^{\unicode[STIX]{x1D716}}v\rangle _{\mathbf{R}^{n}}\geqslant c\langle (J_{1}^{\unicode[STIX]{x1D716}})^{\top }v,C^{\unicode[STIX]{x1D716}}(J_{1}^{\unicode[STIX]{x1D716}})^{\top }v\rangle _{\boldsymbol{ R}^{n}}.\end{eqnarray}$$

Hence,

$$\begin{eqnarray}\langle v,Q^{\unicode[STIX]{x1D716}}v\rangle _{\mathbf{R}^{n}}\geqslant c\unicode[STIX]{x1D706}_{\min }(C^{\unicode[STIX]{x1D716}})\langle (J_{1}^{\unicode[STIX]{x1D716}})^{\top }v,(J_{1}^{\unicode[STIX]{x1D716}})^{\top }v\rangle _{\boldsymbol{ R}^{n}}\geqslant c\unicode[STIX]{x1D706}_{\min }(C^{\unicode[STIX]{x1D716}})\unicode[STIX]{x1D706}_{ min}(J_{1}^{\unicode[STIX]{x1D716}}(J_{1}^{\unicode[STIX]{x1D716}})^{\top })|v|^{2}.\end{eqnarray}$$

The proof has been completed.

In the rest of this subsection we show Lemma 5.6, following [Reference Cass, Hairer, Litterer and TindelCHLT15] closely. To end this, we regard (3.7) as an RDE driven by $\mathbf{w}$ with the coefficients $V_{0}^{\unicode[STIX]{x1D716}}$, $V_{1}^{\unicode[STIX]{x1D716}},\ldots ,V_{d}^{\unicode[STIX]{x1D716}}$. (Except in the rest of this subsection, we regard (3.7) as an RDE driven by $\unicode[STIX]{x1D716}\mathbf{w}$ with the coefficients $V_{0},V_{1},\ldots ,V_{d}$.) Keeping this in mind, we introduce a quantity which plays an important role in a Norris-type lemma. Note that the quantity is defined in the framework of the controlled path theory. Let $(1+\lfloor 1/H\rfloor )^{-1}<\unicode[STIX]{x1D6FC}<H$. Fix $a\in \mathbf{R}^{n}$ and $0<\unicode[STIX]{x1D703}<1$. For every $0<\unicode[STIX]{x1D716}<1$, define

$$\begin{eqnarray}{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)=1+L_{\unicode[STIX]{x1D703}}(w)^{-1}+|a|+\Vert (y^{\unicode[STIX]{x1D716}},J^{\unicode[STIX]{x1D716}},K^{\unicode[STIX]{x1D716}})\Vert _{Q_{\mathbf{w}}^{\unicode[STIX]{x1D6FC}}}+{\mathcal{N}}_{\mathbf{w},\unicode[STIX]{x1D6FC}}.\end{eqnarray}$$

Here, ${\mathcal{N}}_{\mathbf{w},\unicode[STIX]{x1D6FC}}=\sum _{i=1}^{\lfloor 1/H\rfloor }\Vert \mathbf{w}^{i}\Vert _{i\unicode[STIX]{x1D6FC}\text{-}\text{Hol}}$ and $Q_{\mathbf{w}}^{\unicode[STIX]{x1D6FC}}$ stands for the Banach space of controlled paths with respect to $\mathbf{w}$. We refer to [Reference Hairer and PillaiHP13, Definition 3] and [Reference Cass, Hairer, Litterer and TindelCHLT15, Definition 5.2] for $L_{\unicode[STIX]{x1D703}}(w)$, which is called the modulus of $\unicode[STIX]{x1D703}$-Hölder roughness of $w$, and to [Reference Cass, Hairer, Litterer and TindelCHLT15, Definition 5.1] for $\Vert (y^{\unicode[STIX]{x1D716}},J^{\unicode[STIX]{x1D716}},K^{\unicode[STIX]{x1D716}})\Vert _{Q_{\mathbf{w}}^{\unicode[STIX]{x1D6FC}}}$.

Although we do not discuss the details of ${\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)$ for concise, we note that expectations of $r$th power of ${\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)$ are bounded in $\unicode[STIX]{x1D716}$ (Lemma 5.8) and it gives a good estimate of $\langle v,C^{\unicode[STIX]{x1D716}}v\rangle _{\mathbf{R}^{n}}$ (Lemma 5.10). Combining these two lemmas, we can prove Lemma 5.6.

Let us start to prove Lemma 5.6 with the next lemma.

Lemma 5.8. For every $1<r<\infty$, we have

$$\begin{eqnarray}\sup _{0<\unicode[STIX]{x1D716}<1}\mathbf{E}[{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{r}]<\infty .\end{eqnarray}$$

Before stating the next lemma, we make a remark.

Remark 5.9. For every $v=(v_{1},\ldots ,v_{n})^{\top }\in \mathbf{R}^{n}$, we have

(5.3)$$\begin{eqnarray}\langle v,C^{\unicode[STIX]{x1D716}}v\rangle _{\mathbf{R}^{n}}=\mathop{\sum }_{i=1}^{d}\int _{0}^{1}\langle v,K_{s}^{\unicode[STIX]{x1D716}}V_{i}^{\unicode[STIX]{x1D716}}(y_{s}^{\unicode[STIX]{x1D716}})\rangle _{\boldsymbol{ R}^{n}}^{2}\,ds,\end{eqnarray}$$

which follows from

$$\begin{eqnarray}\langle v,C^{\unicode[STIX]{x1D716}}v\rangle _{\mathbf{R}^{n}}=\int _{0}^{1}\langle v,K_{s}^{\unicode[STIX]{x1D716}}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D716}}(y_{s}^{\unicode[STIX]{x1D716}})\{K_{s}^{\unicode[STIX]{x1D716}}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D716}}(y_{s}^{\unicode[STIX]{x1D716}})\}^{\top }v\rangle _{\boldsymbol{ R}^{n}}\,ds\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle \langle v,K_{s}^{\unicode[STIX]{x1D716}}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D716}}(y_{s}^{\unicode[STIX]{x1D716}})\{K_{s}^{\unicode[STIX]{x1D716}}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D716}}(y_{s}^{\unicode[STIX]{x1D716}})\}^{\top }v\rangle _{\mathbf{R}^{n}} & = & \displaystyle \langle \{K_{s}^{\unicode[STIX]{x1D716}}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D716}}(y_{s}^{\unicode[STIX]{x1D716}})\}^{\top }v,\{K_{s}^{\unicode[STIX]{x1D716}}\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D716}}(y_{s}^{\unicode[STIX]{x1D716}})\}^{\top }v\rangle _{\mathbf{R}^{d}}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{i=1}^{d}\langle v,K_{s}^{\unicode[STIX]{x1D716}}V_{i}^{\unicode[STIX]{x1D716}}(y_{s}^{\unicode[STIX]{x1D716}})\rangle _{\mathbf{R}^{n}}^{2}.\nonumber\end{eqnarray}$$

We denote by ${\mathcal{V}}_{m}^{\unicode[STIX]{x1D716}}$ and ${\mathcal{V}}^{\unicode[STIX]{x1D716}}$ sets of vectors which are defined by replacing $V_{i}$ by $V_{i}^{\unicode[STIX]{x1D716}}$ in Definition 2.1. Then, we see the relationship of ${\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)$ and $\langle v,C^{\unicode[STIX]{x1D716}}v\rangle _{\mathbf{R}^{n}}$ from the following lemma.

Lemma 5.10. Let $m\in \mathbf{N}$. For every $0<\unicode[STIX]{x1D716}<1$, $W\in {\mathcal{V}}_{m}^{\unicode[STIX]{x1D716}}$, $v\in \mathbf{R}^{n}$ with $|v|=1$ and $0\leqslant s\leqslant 1$, we have

$$\begin{eqnarray}|\langle v,K_{s}^{\unicode[STIX]{x1D716}}W(y_{s}^{\unicode[STIX]{x1D716}})\rangle _{\boldsymbol{ R}^{n}}|\leqslant c_{m}{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{\unicode[STIX]{x1D707}(m)}\langle v,C^{\unicode[STIX]{x1D716}}v\rangle _{\boldsymbol{ R}^{n}}^{\unicode[STIX]{x1D70B}(m)},\end{eqnarray}$$

where $c_{m}$, $\unicode[STIX]{x1D707}(m)$ and $\unicode[STIX]{x1D70B}(m)$ are certain positive constants independent of $\unicode[STIX]{x1D716}$, $W$, $v$ and $s$.

Proof. The proof is done by induction on $m$. Let $m=0$. Then, $W=V_{i}^{\unicode[STIX]{x1D716}}$ for some $1\leqslant i\leqslant d$. Since $f_{i}^{\unicode[STIX]{x1D716}}=\langle v,K^{\unicode[STIX]{x1D716}}V_{i}^{\unicode[STIX]{x1D716}}(y^{\unicode[STIX]{x1D716}})\rangle _{\mathbf{R}^{n}}$ is $\unicode[STIX]{x1D6FC}$-Hölder continuous, we can use [Reference Hairer and PillaiHP11, Lemma A.3] to obtain

$$\begin{eqnarray}\Vert f_{i}^{\unicode[STIX]{x1D716}}\Vert _{\infty }\leqslant 2\Vert f_{i}^{\unicode[STIX]{x1D716}}\Vert _{\unicode[STIX]{x1D6FC}\text{-}\text{Hol}}^{1/(2\unicode[STIX]{x1D6FC}+1)}\Vert f_{i}^{\unicode[STIX]{x1D716}}\Vert _{L^{2}([0,1];\mathbf{R})}^{2\unicode[STIX]{x1D6FC}/(2\unicode[STIX]{x1D6FC}+1)}.\end{eqnarray}$$

Since

$$\begin{eqnarray}\Vert f_{i}^{\unicode[STIX]{x1D716}}\Vert _{\unicode[STIX]{x1D6FC}\text{-}\text{Hol}}\leqslant c\{1+\Vert K^{\unicode[STIX]{x1D716}}\Vert _{\unicode[STIX]{x1D6FC}\text{-}\text{Hol}}\}\{|a|+\Vert y^{\unicode[STIX]{x1D716}}\Vert _{\unicode[STIX]{x1D6FC}\text{-}\text{Hol}}\}\leqslant c{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{2}\end{eqnarray}$$

holds and $\Vert f_{i}^{\unicode[STIX]{x1D716}}\Vert _{L^{2}([0,1];\mathbf{R})}\leqslant \langle v,C^{\unicode[STIX]{x1D716}}v\rangle _{\mathbf{R}^{n}}^{1/2}$ follows from (5.3), we have

$$\begin{eqnarray}\Vert f_{i}^{\unicode[STIX]{x1D716}}\Vert _{\infty }\leqslant 2c^{1/(2\unicode[STIX]{x1D6FC}+1)}{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{2/(2\unicode[STIX]{x1D6FC}+1)}\langle v,C^{\unicode[STIX]{x1D716}}v\rangle _{\boldsymbol{ R}^{n}}^{\unicode[STIX]{x1D6FC}/(2\unicode[STIX]{x1D6FC}+1)}.\end{eqnarray}$$

This is the conclusion for $m=0$.

Assuming the conclusion to hold for $m-1$, we will prove it for $m$. Note that for every $W\in {\mathcal{V}}_{m}^{\unicode[STIX]{x1D716}}$, there exists $U\in {\mathcal{V}}_{m-1}^{\unicode[STIX]{x1D716}}$ and $0\leqslant i\leqslant d$ such that $W=[V_{i}^{\unicode[STIX]{x1D716}},U]$. We have

$$\begin{eqnarray}\displaystyle & & \displaystyle \langle v,K_{t}^{\unicode[STIX]{x1D716}}U(y_{t}^{\unicode[STIX]{x1D716}})\rangle _{\mathbf{R}^{n}}-\langle v,U(a)\rangle _{\mathbf{R}^{n}}\nonumber\\ \displaystyle & & \displaystyle \qquad =\mathop{\sum }_{i=1}^{d}\int _{0}^{t}\langle v,K_{s}^{\unicode[STIX]{x1D716}}[V_{i}^{\unicode[STIX]{x1D716}},U](y_{s}^{\unicode[STIX]{x1D716}})\rangle _{\mathbf{R}^{n}}\,dw_{s}^{i}+\int _{0}^{t}\langle v,K_{s}^{\unicode[STIX]{x1D716}}[V_{0},U](y_{s}^{\unicode[STIX]{x1D716}})\rangle _{\mathbf{R}^{n}}\,ds.\nonumber\end{eqnarray}$$

From a Norris-type lemma ([Reference Cass, Hairer, Litterer and TindelCHLT15, Theorem 5.6], [Reference Hairer and PillaiHP13, Theorem 3.1]), there exist positive constants $Q$ and $R$ such that

$$\begin{eqnarray}\displaystyle & & \displaystyle \Vert \langle v,K_{\bullet }^{\unicode[STIX]{x1D716}}W(y_{\bullet }^{\unicode[STIX]{x1D716}})\rangle _{\mathbf{R}^{n}}\Vert _{\infty }\nonumber\\ \displaystyle & & \displaystyle \qquad \leqslant M{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{Q}\Vert \langle v,K_{\bullet }^{\unicode[STIX]{x1D716}}U(y_{\bullet }^{\unicode[STIX]{x1D716}})\rangle _{\mathbf{R}^{n}}-\langle v,U(a)\rangle _{\mathbf{R}^{n}}\Vert _{\infty }^{R}\nonumber\\ \displaystyle & & \displaystyle \qquad \leqslant M{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{Q}(2c_{m-1}{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{\unicode[STIX]{x1D707}(m-1)}\langle v,C^{\unicode[STIX]{x1D716}}v\rangle _{\mathbf{R}^{n}}^{\unicode[STIX]{x1D70B}(m-1)})^{R}\nonumber\\ \displaystyle & & \displaystyle \qquad =M(2c_{m-1})^{R}{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{Q+\unicode[STIX]{x1D707}(m-1)R}\langle v,C^{\unicode[STIX]{x1D716}}v\rangle _{\mathbf{R}^{n}}^{\unicode[STIX]{x1D70B}(m-1)R}\nonumber\end{eqnarray}$$

for some $M$ depending only on $d$ and $n$. This is the conclusion for $m$. The proof is finished.◻

We are now in a position to show Lemma 5.6.

Proof of Lemma 5.6.

Let $\unicode[STIX]{x1D708}$ be a positive constant specified later. We will show that, for every $1<r<\infty$, there exist positive constants $c_{r,1}$ and $c_{r,2}$ such that

(5.4)$$\begin{eqnarray}\displaystyle & \displaystyle \mathbf{P}(|C^{\unicode[STIX]{x1D716}}|>1/\unicode[STIX]{x1D709})\leqslant c_{r,1}\unicode[STIX]{x1D709}^{r}, & \displaystyle\end{eqnarray}$$

(5.5)$$\begin{eqnarray}\displaystyle & \displaystyle \sup _{|v|=1}\mathbf{P}(\langle v,C^{\unicode[STIX]{x1D716}}v\rangle _{\mathbf{R}^{n}}<\unicode[STIX]{x1D709})\leqslant c_{r,2}(\unicode[STIX]{x1D716}^{-\unicode[STIX]{x1D708}})^{r}\unicode[STIX]{x1D709}^{r} & \displaystyle\end{eqnarray}$$

for any $0<\unicode[STIX]{x1D716}<1$ and $0<\unicode[STIX]{x1D709}<1$. Due to Lemma 8.1, these two estimates are sufficient for Lemma 5.6. Indeed the assertion holds with $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D708}+1$.

Because (5.4) follows from Lemma 5.7, we show (5.5) in the rest of the proof. Since the vector fields $V_{0},V_{1},\ldots ,V_{d}$ satisfy the Hörmander condition at $a$, we can choose $W_{1},\ldots ,W_{n}\in {\mathcal{V}}$ so that $W_{1}(a),\ldots ,W_{n}(a)$ linearly spans $\mathbf{R}^{n}$. Then, for every $1\leqslant k\leqslant n$, there exists a non-negative integer $i_{k}\in \mathbf{N}$ such that $W_{k}\in {\mathcal{V}}_{i_{k}}$. From the definition of ${\mathcal{V}}_{i_{k}}^{\unicode[STIX]{x1D716}}$, we can choose a positive constant $\unicode[STIX]{x1D70C}_{k}$ such that $W_{k}^{\unicode[STIX]{x1D716}}=\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D70C}_{k}}W_{k}\in {\mathcal{V}}_{i_{k}}^{\unicode[STIX]{x1D716}}$. Set $\unicode[STIX]{x1D70C}=\max \{\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{n}\}$. We define

$$\begin{eqnarray}\unicode[STIX]{x1D719}(u)=\max _{1\leqslant k\leqslant n}|\langle u,W_{k}(a)\rangle _{\mathbf{R}^{n}}|,\qquad \unicode[STIX]{x1D719}^{\unicode[STIX]{x1D716}}(u)=\max _{1\leqslant k\leqslant n}|\langle u,W_{k}^{\unicode[STIX]{x1D716}}(a)\rangle _{\boldsymbol{ R}^{n}}|\end{eqnarray}$$

for every $u\in \mathbf{R}^{n}$ with $|u|=1$. Then, $\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D716}}(u)\geqslant \unicode[STIX]{x1D716}^{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D719}(u)$ for any $|u|=1$. Since $W_{1}^{\unicode[STIX]{x1D716}}(a),\ldots ,W_{n}^{\unicode[STIX]{x1D716}}(a)$ spans linearly $\mathbf{R}^{n}$ and $\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D716}}$ is continuous, $\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D716}}$ attains the positive minimum. Let $c_{m}$, $\unicode[STIX]{x1D707}(m)$ and $\unicode[STIX]{x1D70B}(m)$ be the same symbols as Lemma 5.10 and set $c=\max \{c_{i_{1}},\ldots ,c_{i_{n}}\}$, $\unicode[STIX]{x1D707}=\max \{\unicode[STIX]{x1D707}_{i_{1}},\ldots ,\unicode[STIX]{x1D707}_{i_{n}}\}$, and $\unicode[STIX]{x1D70B}=\min \{\unicode[STIX]{x1D70B}_{i_{1}},\ldots ,\unicode[STIX]{x1D70B}_{i_{n}}\}$. Set $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70B}$.

We choose $v\in \mathbf{R}^{n}$ with $|v|=1$ and $\unicode[STIX]{x1D716}>0$ arbitrarily. For $v$ and $\unicode[STIX]{x1D716}$, there exists $1\leqslant k_{0}\equiv k_{0}(v,\unicode[STIX]{x1D716})\leqslant n$ such that $\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D716}}(v)=|\langle v,W_{k_{0}}^{\unicode[STIX]{x1D716}}(a)\rangle _{\boldsymbol{ R}^{n}}|$. From Lemma 5.10 and the above, we have

$$\begin{eqnarray}\mathbf{P}(\langle v,C^{\unicode[STIX]{x1D716}}v\rangle _{\mathbf{R}^{n}}<\unicode[STIX]{x1D709})\leqslant \mathbf{P}(|\langle v,W_{k_{0}}^{\unicode[STIX]{x1D716}}(a)\rangle _{\mathbf{R}^{n}}|<c_{i_{k_{0}}}{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{\unicode[STIX]{x1D707}(i_{k_{0}})}\unicode[STIX]{x1D709}^{\unicode[STIX]{x1D70B}(i_{k_{0}})})\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle & & \displaystyle \{|\langle v,W_{k_{0}}^{\unicode[STIX]{x1D716}}(a)\rangle _{\boldsymbol{ R}^{n}}|<c_{i_{k_{0}}}{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{\unicode[STIX]{x1D707}(i_{k_{0}})}\unicode[STIX]{x1D709}^{\unicode[STIX]{x1D70B}(i_{k_{0}})}\}\nonumber\\ \displaystyle & & \displaystyle \qquad \subset \{\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D719}(v)<c{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D709}^{\unicode[STIX]{x1D70B}}\}\nonumber\\ \displaystyle & & \displaystyle \qquad =\{\unicode[STIX]{x1D719}(v)<c{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D716}^{-\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D709}^{\unicode[STIX]{x1D70B}}\}\nonumber\\ \displaystyle & & \displaystyle \qquad \subset \{\min _{|v|=1}\unicode[STIX]{x1D719}(v)<c{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D716}^{-\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D709}^{\unicode[STIX]{x1D70B}}\}.\nonumber\end{eqnarray}$$

From the Markov inequality, we see

$$\begin{eqnarray}\mathbf{P}(\langle v,C^{\unicode[STIX]{x1D716}}v\rangle _{\mathbf{R}^{n}}<\unicode[STIX]{x1D709})\leqslant \frac{1}{\{\min _{|v|=1}\unicode[STIX]{x1D719}(v)\}^{r/\unicode[STIX]{x1D70B}}}c^{p/\unicode[STIX]{x1D70B}}\mathbf{E}[{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{\unicode[STIX]{x1D707}r/\unicode[STIX]{x1D70B}}](\unicode[STIX]{x1D716}^{-\unicode[STIX]{x1D708}})^{r}\unicode[STIX]{x1D709}^{r}.\end{eqnarray}$$

Noting $\mathbf{E}[{\mathcal{L}}_{w}^{\unicode[STIX]{x1D716}}(a,\unicode[STIX]{x1D703},1)^{\unicode[STIX]{x1D707}r/\unicode[STIX]{x1D70B}}]$ are bounded from above in $\unicode[STIX]{x1D716}$, we obtain (5.5).

5.3 Covariance matrix of the solution of scaled–shifted RDE driven by fBm

In this subsection, we will show Proposition 5.4. Let $1/H<p<\lfloor 1/H\rfloor +1$ and $(H+1/2)^{-1}<q<2$ satisfy $1/p+1/q>1$. We start our discussion with making a remark on continuity of $Q$ on $G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})\times \mathbf{R}\langle \unicode[STIX]{x1D706}\rangle$, where $\unicode[STIX]{x1D706}$ is a one-dimensional path defined by $\unicode[STIX]{x1D706}_{t}=t$. Recall that the solution maps $(\mathbf{x},c\unicode[STIX]{x1D706})\mapsto y,J,K$ are continuous on $G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})\times \mathbf{R}\langle \unicode[STIX]{x1D706}\rangle$, where $y$, $J$ and $K$ are solutions to (3.4), (3.5) and (3.6) driven by $(x,c\unicode[STIX]{x1D706})$, respectively. Furthermore, $Dy_{1}$ is continuous in $y$, $J$ and $K$ since the right-hand side of (5.1) is Young integration and Young integration is continuous in both integrands and integrators. Note that the embedding $\mathfrak{H}\subset C_{0}^{q\text{-}\text{var}}([0,1];\mathbf{R}^{d})$ with $1/p+1/q>1$ holds. Hence, we see that $Q$ is continuous on $G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})\times \mathbf{R}\langle \unicode[STIX]{x1D706}\rangle$. We are now in a position to prove Proposition 5.4.

Proof of Proposition 5.4.

From the continuity of $Q$ at $(\boldsymbol{\unicode[STIX]{x1D6FE}},0)\in G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})\times \mathbf{R}\langle \unicode[STIX]{x1D706}\rangle$ and the assumption on $\unicode[STIX]{x1D6FE}$, there exists an open set $O\subset G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d})\times \mathbf{R}\langle \unicode[STIX]{x1D706}\rangle$ such that

(5.6)$$\begin{eqnarray}Q(\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FE}}(\mathbf{x}),\mathbf{k})\geqslant c_{1}I\end{eqnarray}$$

for any $(\mathbf{x},k)\in O$. Note that $O$ contains $(\mathbf{0},0)$. (Here, $(\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FE}}(\mathbf{x}),\mathbf{k})$ is the Young pairing and $(\mathbf{x},k)$ is a pair. In this proof, both of them will appear.)

We decompose $\mathbf{E}[|\!\det \tilde{Q}^{\unicode[STIX]{x1D716}}|^{-r}]$ into expectations on $U_{\unicode[STIX]{x1D716}}=\{w\in \unicode[STIX]{x1D6FA}\mid (\unicode[STIX]{x1D716}\mathbf{w},\unicode[STIX]{x1D716}^{1/H}\unicode[STIX]{x1D706})\in O\}$ and $U_{\unicode[STIX]{x1D716}}^{\complement }$. First, we study the expectation on $U_{\unicode[STIX]{x1D716}}$. Since $\tilde{Q}^{\unicode[STIX]{x1D716}}=\unicode[STIX]{x1D716}^{2}Q(\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x1D716}\mathbf{w}),\unicode[STIX]{x1D716}^{1/H}\boldsymbol{\unicode[STIX]{x1D706}})$ and (5.6), we see

$$\begin{eqnarray}\det \tilde{Q}^{\unicode[STIX]{x1D716}}=\unicode[STIX]{x1D716}^{2n}\det Q(\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x1D716}\mathbf{w}),\unicode[STIX]{x1D716}^{1/H}\boldsymbol{\unicode[STIX]{x1D706}})\geqslant \unicode[STIX]{x1D716}^{2n}c_{1}^{n},\end{eqnarray}$$

which implies

(5.7)$$\begin{eqnarray}\mathbf{E}[|\!\det \tilde{Q}^{\unicode[STIX]{x1D716}}|^{-r};U_{\unicode[STIX]{x1D716}}]\leqslant (c_{1}\unicode[STIX]{x1D716}^{2})^{-nr}.\end{eqnarray}$$

Next we consider the expectation on $U_{\unicode[STIX]{x1D716}}^{\complement }$. From the Hölder inequality, we have

$$\begin{eqnarray}\mathbf{E}[|\!\det \tilde{Q}^{\unicode[STIX]{x1D716}}|^{-r};U_{\unicode[STIX]{x1D716}}^{\complement }]\leqslant \mathbf{E}[|\!\det \tilde{Q}^{\unicode[STIX]{x1D716}}|^{-2r}]^{1/2}\mathbf{P}(U_{\unicode[STIX]{x1D716}}^{\complement })^{1/2}.\end{eqnarray}$$

Since the rate function of the Schilder-type large deviation principle is good, we see that there exists a positive constant $c_{2}$ such that

$$\begin{eqnarray}\mathbf{P}(U_{\unicode[STIX]{x1D716}}^{\complement })=\mathbf{P}((\unicode[STIX]{x1D716}\mathbf{w},\unicode[STIX]{x1D716}^{1/H}\unicode[STIX]{x1D706})\in O^{\complement })=\hat{\unicode[STIX]{x1D708}}_{\unicode[STIX]{x1D716}}(O^{\complement })\leqslant \exp \bigg(-\frac{c_{2}}{2\unicode[STIX]{x1D716}^{2}}\bigg)\end{eqnarray}$$

for small $\unicode[STIX]{x1D716}>0$. Here, we choose $1<q<\infty$ so that $(q-1)\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}^{2}<c_{2}$ and set $1<q^{\prime }<\infty$ as the Hölder conjugate of $q$. The Girsanov theorem and Proposition 5.3 imply

$$\begin{eqnarray}\displaystyle \mathbf{E}[|\!\det \tilde{Q}^{\unicode[STIX]{x1D716}}|^{-2r}]^{1/2} & = & \displaystyle \mathbf{E}\biggl[|\!\det Q^{\unicode[STIX]{x1D716}}|^{-2r}\exp \biggl(\langle w,\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D716}}\rangle -\frac{\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}^{2}}{2\unicode[STIX]{x1D716}^{2}}\biggr)\biggr]^{1/2}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \mathbf{E}[|\!\det Q^{\unicode[STIX]{x1D716}}|^{-2rq^{\prime }}]^{1/2q^{\prime }}\mathbf{E}\bigg[\exp \biggl(q\bigg\{\langle w,\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D716}}\rangle -\frac{\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}^{2}}{2\unicode[STIX]{x1D716}^{2}}\bigg\}\biggr)\bigg]^{1/2q}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle (c_{3}\unicode[STIX]{x1D716}^{-\unicode[STIX]{x1D707}})^{r}\exp \bigg(\frac{(q-1)\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}^{2}}{4\unicode[STIX]{x1D716}^{2}}\bigg).\nonumber\end{eqnarray}$$

From the above, we see

(5.8)$$\begin{eqnarray}\mathbf{E}[|\!\det \tilde{Q}^{\unicode[STIX]{x1D716}}|^{-r};U_{\unicode[STIX]{x1D716}}^{\complement }]\leqslant c_{3}^{r}\unicode[STIX]{x1D716}^{-r\unicode[STIX]{x1D707}}\exp \bigg(\frac{(q-1)\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}^{2}}{4\unicode[STIX]{x1D716}^{2}}\bigg)\exp \bigg(-\frac{c_{2}}{4\unicode[STIX]{x1D716}^{2}}\bigg).\end{eqnarray}$$

The estimates (5.7) and (5.8) imply

$$\begin{eqnarray}\displaystyle \mathbf{E}[|\!\det \unicode[STIX]{x1D716}^{-2}\tilde{Q}^{\unicode[STIX]{x1D716}}|^{-r}] & = & \displaystyle \unicode[STIX]{x1D716}^{2nr}\mathbf{E}[|\!\det \tilde{Q}^{\unicode[STIX]{x1D716}}|^{-r}]\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \unicode[STIX]{x1D716}^{2nr}\bigg\{(c_{1}\unicode[STIX]{x1D716}^{2})^{-nr}+c_{3}^{r}\unicode[STIX]{x1D716}^{-r\unicode[STIX]{x1D707}}\exp \bigg(-\frac{c_{2}-(q-1)\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}^{2}}{4\unicode[STIX]{x1D716}^{2}}\bigg)\bigg\}\nonumber\\ \displaystyle & = & \displaystyle c_{1}^{-nr}+c_{3}^{r}\unicode[STIX]{x1D716}^{(2n-\unicode[STIX]{x1D707})r}\exp \bigg(-\frac{c_{2}-(q-1)\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}^{2}}{4\unicode[STIX]{x1D716}^{2}}\bigg).\nonumber\end{eqnarray}$$

The right-hand side is bounded as $\unicode[STIX]{x1D716}{\searrow}0$. The proof has finished.◻

6.1 Localization around energy minimizing path

Let us introduce a cut-off function for the localization. Let $\bar{\unicode[STIX]{x1D6FE}}\in \mathfrak{H}$ be as in (A2) and $\unicode[STIX]{x1D716}>0$. Since $\Vert \mathbf{w}^{i}\Vert _{(i\unicode[STIX]{x1D6FC},12m/i)\text{-}\text{Bes}}^{12m/i}$ is an element of an inhomogeneous Wiener chaos of order $12m$, so is its Cameron–Martin shift $\Vert \unicode[STIX]{x1D70F}_{-\bar{\unicode[STIX]{x1D6FE}}}(\unicode[STIX]{x1D716}\mathbf{w})^{i}\Vert _{(i\unicode[STIX]{x1D6FC},12m/i)\text{-}\text{Bes}}^{12m/i}$. Here, $\unicode[STIX]{x1D70F}_{-\bar{\unicode[STIX]{x1D6FE}}}$ is the Young translation by $-\bar{\unicode[STIX]{x1D6FE}}$. It is a continuous map from $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},12m}^{\text{B}}(\mathbf{R}^{d})$ to itself. So, this Wiener functional is defined for almost all $w\in \unicode[STIX]{x1D6FA}$. For any $r\in (1,\infty )$, $L^{r}$-norm of this Wiener functional is bounded in $\unicode[STIX]{x1D716}$. Hence, so is its $\mathbf{D}_{r,k}$-norm for any $r,k$. Due to this fact, the localization is allowed even in the framework of Watanabe distribution theory. This is the reason why we use this Besov-type norm on the geometric rough path space. Let $\unicode[STIX]{x1D713}:\mathbf{R}\rightarrow [0,1]$ be a smooth function such that $\unicode[STIX]{x1D713}(u)=1$ if $|u|\leqslant 1/2$ and $\unicode[STIX]{x1D713}(u)=0$ if $|u|\geqslant 1$. For each $\unicode[STIX]{x1D702}>0$ and $\unicode[STIX]{x1D716}>0$, we set

$$\begin{eqnarray}\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D702}}(\unicode[STIX]{x1D716},w)=\mathop{\prod }_{i=1}^{\lfloor 1/H\rfloor }\unicode[STIX]{x1D713}\bigg(\frac{\Vert \unicode[STIX]{x1D70F}_{-\bar{\unicode[STIX]{x1D6FE}}}(\unicode[STIX]{x1D716}\mathbf{w})^{i}\Vert _{(i\unicode[STIX]{x1D6FC},12m/i)\text{-}\text{Bes}}^{12m/i}}{\unicode[STIX]{x1D702}^{12m}}\bigg).\end{eqnarray}$$

The following lemma states that only rough paths sufficiently close to the lift of the minimizer $\bar{\unicode[STIX]{x1D6FE}}$ contribute to the asymptotics. The keys of the proof are (i) the Schilder-type large deviations for fractional Brownian rough path and (ii) the Kusuoka–Stroock type estimate of the Malliavin covariance of $y_{1}^{\unicode[STIX]{x1D716}}$ (Proposition 5.3).

Lemma 6.1. Suppose that $\text{(A1)}$ and $\text{(A2)}$ hold. Then, for any $\unicode[STIX]{x1D702}>0$, there exists $c=c_{\unicode[STIX]{x1D702}}>0$ such that

$$\begin{eqnarray}0\leqslant \mathbf{E}[(1-\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D702}}(\unicode[STIX]{x1D716},w))\cdot \unicode[STIX]{x1D6FF}_{a^{\prime }}(y_{1}^{\unicode[STIX]{x1D716}})]=O\bigg(\exp \bigg\{-\frac{\Vert \bar{\unicode[STIX]{x1D6FE}}\Vert _{\mathfrak{H}}^{2}+c}{2\unicode[STIX]{x1D716}^{2}}\bigg\}\bigg)\quad \text{as }\unicode[STIX]{x1D716}{\searrow}0.\end{eqnarray}$$

Proof. We show the assertion for $1/4<H\leqslant 1/3$. We can show it for $1/3<H\leqslant 1/2$ more easily.

Set $p_{\unicode[STIX]{x1D716}}=\mathbf{E}[(1-\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D702}}(\unicode[STIX]{x1D716},w))\unicode[STIX]{x1D6FF}_{a^{\prime }}(y_{1}^{\unicode[STIX]{x1D716}})]$. We take $\unicode[STIX]{x1D702}^{\prime }>0$ arbitrarily and fix it for a while. It is obvious that

$$\begin{eqnarray}0\leqslant p_{\unicode[STIX]{x1D716}}=\mathbf{E}\bigg[\{1-\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D702}}(\unicode[STIX]{x1D716},w)\}\unicode[STIX]{x1D713}\bigg(\frac{|y_{1}^{\unicode[STIX]{x1D716}}-a^{\prime }|^{2}}{{\unicode[STIX]{x1D702}^{\prime }}^{2}}\bigg)\unicode[STIX]{x1D6FF}_{a^{\prime }}(y_{1}^{\unicode[STIX]{x1D716}})\bigg].\end{eqnarray}$$

Set $A(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2},\unicode[STIX]{x1D709}_{3})=1-\unicode[STIX]{x1D713}(\unicode[STIX]{x1D709}_{1})\unicode[STIX]{x1D713}(\unicode[STIX]{x1D709}_{2})\unicode[STIX]{x1D713}(\unicode[STIX]{x1D709}_{3})$ for $\unicode[STIX]{x1D709}=(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2},\unicode[STIX]{x1D709}_{3})\in \mathbf{R}^{3}$ and write $(\unicode[STIX]{x1D709}_{i};i=1,2,3)=(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2},\unicode[STIX]{x1D709}_{3})$. Set $g(u)=u\vee 0$ for $u\in \mathbf{R}$. Then, in the sense of distributional derivative, $g^{\prime \prime }=\unicode[STIX]{x1D6FF}_{0}$. Take a bounded continuous function $C:\mathbf{R}^{n}\rightarrow \mathbf{R}$ such that $C(u_{1},\ldots ,u_{n})=g(u_{1}-a_{1}^{\prime })g(u_{2}-a_{2}^{\prime })\cdots g(u_{n}-a_{n}^{\prime })$ if $|u-a^{\prime }|\leqslant 2\unicode[STIX]{x1D702}^{\prime }$. Then,

$$\begin{eqnarray}\displaystyle p_{\unicode[STIX]{x1D716}}=\mathbf{E}\bigg[A\bigg(\frac{\Vert \unicode[STIX]{x1D70F}_{-\bar{\unicode[STIX]{x1D6FE}}}(\unicode[STIX]{x1D716}\mathbf{w})^{i}\Vert _{(i\unicode[STIX]{x1D6FC},12m/i)\text{-}\text{Bes}}^{12m}}{\unicode[STIX]{x1D702}^{12m}};i=1,2,3\bigg)\unicode[STIX]{x1D713}\bigg(\frac{|y_{1}^{\unicode[STIX]{x1D716}}-a^{\prime }|^{2}}{{\unicode[STIX]{x1D702}^{\prime }}^{2}}\bigg)(\unicode[STIX]{x2202}_{1}^{2}\cdots \unicode[STIX]{x2202}_{n}^{2}C)(y_{1}^{\unicode[STIX]{x1D716}})\bigg]. & & \displaystyle \nonumber\end{eqnarray}$$

Now, we use integration by parts formula for generalized expectations as in [Reference WatanabeWat87, Reference Ikeda and WatanabeIW89] to see that $p_{\unicode[STIX]{x1D716}}$ is equal to a finite sum of the following form;

$$\begin{eqnarray}\displaystyle p_{\unicode[STIX]{x1D716}}=\mathop{\sum }_{j,k}\mathbf{E}\bigg[F_{j,k}(\unicode[STIX]{x1D716},w)\unicode[STIX]{x1D6FB}^{j}A\!\bigg(\frac{\Vert \unicode[STIX]{x1D70F}_{-\bar{\unicode[STIX]{x1D6FE}}}(\unicode[STIX]{x1D716}\mathbf{w})^{i}\Vert _{(i\unicode[STIX]{x1D6FC},12m/i)\text{-}\text{Bes}}^{12m}}{\unicode[STIX]{x1D702}^{12m}};i=1,2,3\bigg)\unicode[STIX]{x1D713}^{(k)}\!\bigg(\frac{|y_{1}^{\unicode[STIX]{x1D716}}-a^{\prime }|^{2}}{{\unicode[STIX]{x1D702}^{\prime }}^{2}}\bigg)\!C(y_{1}^{\unicode[STIX]{x1D716}})\bigg]. & & \displaystyle \nonumber\end{eqnarray}$$

Here $j=(j_{1},j_{2},j_{3})$ and $k$ run over finite subsets of $\mathbf{N}^{3}$ and $\mathbf{N}$, respectively, $\unicode[STIX]{x1D6FB}^{j}A=\unicode[STIX]{x2202}_{1}^{j_{1}}\unicode[STIX]{x2202}_{2}^{j_{2}}\unicode[STIX]{x2202}_{3}^{j_{3}}A$ and $F_{j,k}(\unicode[STIX]{x1D716},w)$ is a polynomial in components of the following (i)–(iv): (i) $y_{1}^{\unicode[STIX]{x1D716}}$ and its derivatives, (ii) $\Vert \unicode[STIX]{x1D70F}_{-\bar{\unicode[STIX]{x1D6FE}}}(\unicode[STIX]{x1D716}\mathbf{w})^{i}\Vert _{(i\unicode[STIX]{x1D6FC},12m/i)\text{-}\text{Bes}}^{12m/i}$ and its derivatives, (iii) $Q^{\unicode[STIX]{x1D716}}$, which is Malliavin covariance matrix of $y_{1}^{\unicode[STIX]{x1D716}}$ and its derivatives, and (iv) $(Q^{\unicode[STIX]{x1D716}})^{-1}$. Note that the derivatives of $(Q^{\unicode[STIX]{x1D716}})^{-1}$ do not appear.

From Propositions 5.1 and 5.3, there exists $\unicode[STIX]{x1D70C}>0$ such that $|(Q^{\unicode[STIX]{x1D716}})^{-1}|=O(\unicode[STIX]{x1D716}^{-\unicode[STIX]{x1D70C}})$ in $L^{r}$ as $\unicode[STIX]{x1D716}{\searrow}0$ for all $1<r<\infty$. (Recall a well-known formula to obtain the inverse matrix $A^{-1}$ with the adjugate matrix of $A$ divided by $\det A$.) Therefore, there exists $\unicode[STIX]{x1D70C}>0$ such that $|F_{j,k}(\unicode[STIX]{x1D716})|=O(\unicode[STIX]{x1D716}^{-\unicode[STIX]{x1D70C}})$ in any $L^{r}$-norm. ($\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}(r)>0$ may change from line to line.) By Hölder’s inequality, we have

(6.1)$$\begin{eqnarray}\displaystyle \qquad p_{\unicode[STIX]{x1D716}} & {\leqslant} & \displaystyle \frac{c}{\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D70C}}}\mathop{\sum }_{j,k}\mathbf{E}\bigg[\bigg|\unicode[STIX]{x1D6FB}^{j}A\bigg(\frac{\Vert \unicode[STIX]{x1D70F}_{-\bar{\unicode[STIX]{x1D6FE}}}(\unicode[STIX]{x1D716}\mathbf{w})^{i}\Vert _{(i\unicode[STIX]{x1D6FC},12m/i)\text{-}\text{Bes}}^{12m}}{\unicode[STIX]{x1D702}^{12m}};i=1,2,3\bigg)\bigg|^{r^{\prime }}\bigg|\unicode[STIX]{x1D713}^{(k)}\bigg(\frac{|y_{1}^{\unicode[STIX]{x1D716}}-a^{\prime }|^{2}}{{\unicode[STIX]{x1D702}^{\prime }}^{2}}\bigg)\bigg|^{r^{\prime }}\bigg]^{1/r^{\prime }}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{c}{\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D70C}}}\mathbf{P}\bigg[\mathop{\bigcup }_{i=1}^{3}\bigg\{\Vert \unicode[STIX]{x1D70F}_{-\bar{\unicode[STIX]{x1D6FE}}}(\unicode[STIX]{x1D716}\mathbf{w})^{i}\Vert _{(i\unicode[STIX]{x1D6FC},12m/i)\text{-}\text{Bes}}^{1/i}\geqslant \frac{\unicode[STIX]{x1D702}}{2^{1/(12m)}}\bigg\}\cap \{|y_{1}^{\unicode[STIX]{x1D716}}-a^{\prime }|\leqslant \unicode[STIX]{x1D702}^{\prime }\}\bigg]^{1/r^{\prime }}.\end{eqnarray}$$

Here, $1/r+1/r^{\prime }=1$ and $c=c(r,r^{\prime },\unicode[STIX]{x1D702},\unicode[STIX]{x1D702}^{\prime })$ is a positive constant, which may change from line to line. Set $U_{\unicode[STIX]{x1D702}^{\prime \prime }}=\bigcap _{i=1}^{3}\{\mathbf{x}\in G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},12m}^{\text{B}}(\mathbf{R}^{d})\mid \Vert \mathbf{x}^{i}\Vert _{(i\unicode[STIX]{x1D6FC},12m/i)\text{-}\text{Bes}}^{1/i}<\unicode[STIX]{x1D702}^{\prime \prime }\}$ for $\unicode[STIX]{x1D702}^{\prime \prime }>0$. Then this forms a fundamental system of open neighborhoods around $(\mathbf{x}^{1},\mathbf{x}^{2},\mathbf{x}^{3})\equiv (0,0,0)$ with respect to $(\unicode[STIX]{x1D6FC},12m)$-Besov topology. By Proposition 3.2, $\unicode[STIX]{x1D70F}_{\bar{\unicode[STIX]{x1D6FE}}}^{-1}(U_{\unicode[STIX]{x1D702}^{\prime \prime }})=\{\mathbf{x}\in G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},12m}^{\text{B}}(\mathbf{R}^{d})\mid \unicode[STIX]{x1D70F}_{\bar{\unicode[STIX]{x1D6FE}}}(\mathbf{x})\in U_{\unicode[STIX]{x1D702}^{\prime \prime }}\}$ is an open neighborhood of $\bar{\boldsymbol{\unicode[STIX]{x1D6FE}}}$ in $(\unicode[STIX]{x1D6FC},12m)$-geometric rough path space. The first set on the most right-hand side of (6.1) can be written as $\{\unicode[STIX]{x1D716}\mathbf{w}\notin \unicode[STIX]{x1D70F}_{\bar{\unicode[STIX]{x1D6FE}}}^{-1}(U_{2^{-1/(12m)}\unicode[STIX]{x1D702}})\}$.

First taking $\limsup _{\unicode[STIX]{x1D716}{\searrow}0}\unicode[STIX]{x1D716}^{2}\log$ and then letting $r^{\prime }{\searrow}1$, we obtain

(6.2)$$\begin{eqnarray}\displaystyle & & \displaystyle \limsup _{\unicode[STIX]{x1D716}{\searrow}0}\unicode[STIX]{x1D716}^{2}\log p_{\unicode[STIX]{x1D716}}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \limsup _{\unicode[STIX]{x1D716}{\searrow}0}\unicode[STIX]{x1D716}^{2}\log \mathbf{P}[w\in \unicode[STIX]{x1D6FA}\mid \unicode[STIX]{x1D716}\mathbf{w}\notin \unicode[STIX]{x1D70F}_{\bar{\unicode[STIX]{x1D6FE}}}^{-1}(U_{2^{-1/(12m)}\unicode[STIX]{x1D702}}),|y_{1}^{\unicode[STIX]{x1D716}}-a^{\prime }|\leqslant \unicode[STIX]{x1D702}^{\prime }]\nonumber\\ \displaystyle & & \displaystyle \quad =\limsup _{\unicode[STIX]{x1D716}{\searrow}0}\unicode[STIX]{x1D716}^{2}\log \hat{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D716}}\big[\big\{(\mathbf{x},l)\in G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},12m}^{\text{B}}(\mathbf{R}^{d})\times \mathbf{R}\langle \unicode[STIX]{x1D706}\rangle \,\Big|\,\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \mathbf{x}\in \unicode[STIX]{x1D70F}_{\bar{\unicode[STIX]{x1D6FE}}}^{-1}(U_{2^{-1/(12m)}\unicode[STIX]{x1D702}})^{\complement },|a+\unicode[STIX]{x1D6F7}(\mathbf{x},\mathbf{l})_{0,1}^{1}-a^{\prime }|\leqslant \unicode[STIX]{x1D702}^{\prime }\big\}\big]\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant -\!\inf \bigg\{\frac{\Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}^{2}}{2}\,\bigg|\,\unicode[STIX]{x1D6FE}\in \mathfrak{H},\boldsymbol{\unicode[STIX]{x1D6FE}}\in \unicode[STIX]{x1D70F}_{\bar{\unicode[STIX]{x1D6FE}}}^{-1}(U_{2^{-1/(12m)}\unicode[STIX]{x1D702}})^{c},|a+\unicode[STIX]{x1D6F7}(\boldsymbol{\unicode[STIX]{x1D6FE}},\mathbf{0})_{0,1}^{1}-a^{\prime }|\leqslant \unicode[STIX]{x1D702}^{\prime }\bigg\}.\end{eqnarray}$$

Here, $\unicode[STIX]{x1D6F7}:G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d+1})\rightarrow G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{n})$ denotes the Lyons–Itô map that corresponds to the coefficient $[\unicode[STIX]{x1D70E},b]$ and we used the embeddings $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},12m}^{\text{B}}(\mathbf{R}^{d})\times \mathbf{R}\langle \unicode[STIX]{x1D706}\rangle {\hookrightarrow}G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},12m}^{\text{B}}(\mathbf{R}^{d+1}){\hookrightarrow}G\unicode[STIX]{x1D6FA}_{p}(\mathbf{R}^{d+1})$ implicitly. In the last inequality we used large deviation upper estimate for a closed set. Notice also that $a+\unicode[STIX]{x1D6F7}(\boldsymbol{\unicode[STIX]{x1D6FE}},0)^{1}=\unicode[STIX]{x1D719}^{0}(\unicode[STIX]{x1D6FE})$.

Now let $\unicode[STIX]{x1D702}^{\prime }$ tend to $0$. As $\unicode[STIX]{x1D702}^{\prime }$ decreases, the most right-hand side of (6.2) decreases. The proof is finished if the limit is strictly smaller than $-\Vert \bar{\unicode[STIX]{x1D6FE}}\Vert _{\mathfrak{H}}^{2}/2$. Assume otherwise. Then, there exists $\{\unicode[STIX]{x1D6FE}_{k}\}_{k=1}^{\infty }\subset \mathfrak{H}$ such that

$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{\unicode[STIX]{x1D6FE}}_{k}\in \unicode[STIX]{x1D70F}_{\bar{\unicode[STIX]{x1D6FE}}}^{-1}(U_{2^{-1/(12m)}\unicode[STIX]{x1D702}})^{c},\quad |a+\unicode[STIX]{x1D6F7}(\boldsymbol{\unicode[STIX]{x1D6FE}}_{k},0)_{0,1}^{1}-a^{\prime }|\leqslant \frac{1}{k}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \liminf _{k\rightarrow \infty }\bigg(-\frac{\Vert \unicode[STIX]{x1D6FE}_{k}\Vert _{\mathfrak{H}}^{2}}{2}\bigg)\geqslant -\frac{\Vert \bar{\unicode[STIX]{x1D6FE}}\Vert _{\mathfrak{H}}^{2}}{2}. & \displaystyle \nonumber\end{eqnarray}$$

In particular, $\{\unicode[STIX]{x1D6FE}_{k}\}$ is bounded in $\mathfrak{H}$. Hence, by goodness of the rate function, the lift $\{\boldsymbol{\unicode[STIX]{x1D6FE}}_{k}\}$ is precompact in $G\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},12m}^{\text{B}}(\mathbf{R}^{d})$. By taking a subsequence if necessary, we may assume $\{\unicode[STIX]{x1D6FE}_{k}\}$ converges to some $\mathbf{z}$ in $(\unicode[STIX]{x1D6FC},12m)$-Besov topology. By the continuity of $\unicode[STIX]{x1D6F7}$, we have $a+\unicode[STIX]{x1D6F7}(\mathbf{z},0)_{0,1}^{1}=a^{\prime }$. Since $\mathbf{z}\in \unicode[STIX]{x1D70F}_{\bar{\unicode[STIX]{x1D6FE}}}^{-1}(U_{2^{-1/(12m)}\unicode[STIX]{x1D702}})^{c}$, $\mathbf{z}\neq \bar{\boldsymbol{\unicode[STIX]{x1D6FE}}}$. From the lower semicontinuity of the rate function, we see that $\mathbf{z}$ is the lift of some $z\in \mathfrak{H}$ and $\Vert z\Vert _{\mathfrak{H}}^{2}/2\leqslant \Vert \bar{\unicode[STIX]{x1D6FE}}\Vert _{\mathfrak{H}}^{2}/2$. This clearly contradicts (A2).◻

6.2 Proof of Theorem 2.3

Now, let us calculate the kernel $p_{t}(a,a^{\prime })$. We see that $p_{\unicode[STIX]{x1D716}^{1/H}}(a,a^{\prime })=\mathbf{E}[\unicode[STIX]{x1D6FF}_{a^{\prime }}(y_{1}^{\unicode[STIX]{x1D716}})]=I_{1}(\unicode[STIX]{x1D716})+I_{2}(\unicode[STIX]{x1D716})$, where

$$\begin{eqnarray}I_{1}(\unicode[STIX]{x1D716})=\mathbf{E}[\unicode[STIX]{x1D6FF}_{a^{\prime }}(y_{1}^{\unicode[STIX]{x1D716}})\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D702}}(\unicode[STIX]{x1D716},w)],\qquad I_{2}(\unicode[STIX]{x1D716})=\mathbf{E}[\unicode[STIX]{x1D6FF}_{a^{\prime }}(y_{1}^{\unicode[STIX]{x1D716}})\{1-\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D702}}(\unicode[STIX]{x1D716},w)\}].\end{eqnarray}$$

As we have shown in Lemma 6.1, the second term $I_{2}(\unicode[STIX]{x1D716})$ on the right-hand side does not contribute to the asymptotic expansion for any $\unicode[STIX]{x1D702}>0$. So, we have only to calculate the first term $I_{1}(\unicode[STIX]{x1D716})$ for some $\unicode[STIX]{x1D702}>0$.

However, the proof of the asymptotic expansion of $I_{1}(\unicode[STIX]{x1D716})$ in the case $H\in (1/4,1/3]$ is essentially the same as in the case $H\in (1/3,1/2]$ (see [Reference InahamaIna16b, Subsections 5.2–5.3]). Therefore, for the sake of brevity, we will give a sketch of the proof only.

Sketch of the proof of Theorem 2.3.

By the Cameron–Martin formula, we have

$$\begin{eqnarray}I_{1}(\unicode[STIX]{x1D716})=\mathbf{E}\bigg[\exp \bigg(-\frac{\Vert \bar{\unicode[STIX]{x1D6FE}}\Vert _{\mathfrak{H}}^{2}}{2\unicode[STIX]{x1D716}^{2}}-\frac{1}{\unicode[STIX]{x1D716}}\langle \bar{\unicode[STIX]{x1D6FE}},w\rangle \bigg)\unicode[STIX]{x1D6FF}_{a^{\prime }}({\tilde{y}}_{1}^{\unicode[STIX]{x1D716}})\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D702}}\bigg(\unicode[STIX]{x1D716},w+\frac{\bar{\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x1D716}}\bigg)\bigg].\end{eqnarray}$$

Moreover, there exists $\bar{\unicode[STIX]{x1D708}}\in \mathbf{R}^{n}$ such that $\langle \bar{\unicode[STIX]{x1D6FE}},w\rangle =\langle \bar{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D719}_{1}^{1}(w,\bar{\unicode[STIX]{x1D6FE}})\rangle$ for all $w$, where the inner product on the right-hand side is a standard one on $\mathbf{R}^{n}$. (In fact, $\bar{\unicode[STIX]{x1D708}}$ is a covector that appears in the Lagrange multiplier method for $\unicode[STIX]{x1D719}_{1}^{0}$ and $\unicode[STIX]{x1D6FE}\mapsto \Vert \unicode[STIX]{x1D6FE}\Vert _{\mathfrak{H}}^{2}/2$ at $\bar{\unicode[STIX]{x1D6FE}}$. Note that $\unicode[STIX]{x1D719}_{1}^{1}$ is a continuous extension of $\unicode[STIX]{x1D6FE}\mapsto D_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D719}_{1}^{0}(\bar{\unicode[STIX]{x1D6FE}})$.) Hence, we have

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D716}^{n}\exp \bigg(\frac{\Vert \bar{\unicode[STIX]{x1D6FE}}\Vert _{\mathfrak{H}}^{2}}{2\unicode[STIX]{x1D716}^{2}}\bigg)I_{1}(\unicode[STIX]{x1D716})\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D716}^{n}\mathbf{E}\bigg[\exp \bigg(-\frac{1}{\unicode[STIX]{x1D716}}\langle \bar{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D719}_{1}^{1}\rangle \bigg)\unicode[STIX]{x1D6FF}_{a^{\prime }}(a^{\prime }+\unicode[STIX]{x1D716}\unicode[STIX]{x1D719}_{1}^{1}+r_{\unicode[STIX]{x1D716},1}^{2})\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D702}}\bigg(\unicode[STIX]{x1D716},w+\frac{\bar{\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x1D716}}\bigg)\bigg]\nonumber\\ \displaystyle & & \displaystyle \quad =\mathbf{E}\bigg[\exp \bigg(-\frac{1}{\unicode[STIX]{x1D716}}\langle \bar{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D719}_{1}^{1}\rangle \bigg)\unicode[STIX]{x1D6FF}_{0}(\unicode[STIX]{x1D719}_{1}^{1}+\unicode[STIX]{x1D716}^{-1}r_{\unicode[STIX]{x1D716},1}^{2})\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D702}}\bigg(\unicode[STIX]{x1D716},w+\frac{\bar{\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x1D716}}\bigg)\bigg]\nonumber\\ \displaystyle & & \displaystyle \quad =\mathbf{E}\bigg[\exp \bigg(\frac{\langle \bar{\unicode[STIX]{x1D708}},r_{\unicode[STIX]{x1D716},1}^{2}\rangle }{\unicode[STIX]{x1D716}^{2}}\bigg)\unicode[STIX]{x1D6FF}_{0}(\unicode[STIX]{x1D719}_{1}^{1}+\unicode[STIX]{x1D716}^{-1}r_{\unicode[STIX]{x1D716},1}^{2})\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D702}}\bigg(\unicode[STIX]{x1D716},w+\frac{\bar{\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x1D716}}\bigg)\bigg]\nonumber\\ \displaystyle & & \displaystyle \quad =\mathbf{E}\bigg[F(\unicode[STIX]{x1D716},w)\unicode[STIX]{x1D6FF}_{0}\bigg(\frac{{\tilde{y}}_{1}^{\unicode[STIX]{x1D716}}-a^{\prime }}{\unicode[STIX]{x1D716}}\bigg)\bigg],\nonumber\end{eqnarray}$$

where

$$\begin{eqnarray}F(\unicode[STIX]{x1D716},w)=\exp \bigg(\frac{\langle \bar{\unicode[STIX]{x1D708}},r_{\unicode[STIX]{x1D716},1}^{2}\rangle }{\unicode[STIX]{x1D716}^{2}}\bigg)\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D702}}\bigg(\unicode[STIX]{x1D716},w+\frac{\bar{\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x1D716}}\bigg)\unicode[STIX]{x1D713}\bigg(\frac{1}{{\unicode[STIX]{x1D702}^{\prime }}^{2}}\bigg|\frac{{\tilde{y}}_{1}^{\unicode[STIX]{x1D716}}-a^{\prime }}{\unicode[STIX]{x1D716}}\bigg|^{2}\bigg)\end{eqnarray}$$

for any positive constant $\unicode[STIX]{x1D702}^{\prime }$. Here we have used $\unicode[STIX]{x1D6FF}_{0}(\bullet )=\unicode[STIX]{x1D713}(|\bullet |^{2}/{\unicode[STIX]{x1D702}^{\prime }}^{2})\unicode[STIX]{x1D6FF}_{0}(\bullet )$.

By a slight modification of Watanabe’s asymptotic expansion theory and uniform non-degeneracy (Proposition 5.4), $\unicode[STIX]{x1D6FF}_{0}(({\tilde{y}}_{1}^{\unicode[STIX]{x1D716}}-a^{\prime })/\unicode[STIX]{x1D716})$ admits the following asymptotic expansion as $\unicode[STIX]{x1D716}{\searrow}0$ in $\tilde{\mathbf{D}}_{-\infty }$-topology as follows: for some $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D708}_{j}}\in \tilde{\mathbf{D}}_{-\infty }$ ($j=1,2,\ldots$),

(6.3)$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{0}\bigg(\frac{{\tilde{y}}_{1}^{\unicode[STIX]{x1D716}}-a^{\prime }}{\unicode[STIX]{x1D716}}\bigg)\sim \unicode[STIX]{x1D6FF}_{0}(\unicode[STIX]{x1D719}_{1}^{1})+\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D708}_{1}}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D708}_{1}}+\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D708}_{2}}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D708}_{2}}+\cdots\end{eqnarray}$$

(see [Reference Ikeda and WatanabeIW89, Theorem 9.3, p. 387]). Recall that $0=\unicode[STIX]{x1D708}_{0}<\unicode[STIX]{x1D708}_{1}<\unicode[STIX]{x1D708}_{2}<\cdots \,$ are all the elements of $\unicode[STIX]{x1D6EC}_{3}$ in increasing order. Moreover, since $\unicode[STIX]{x1D719}_{1}^{1}$ is a Gaussian random vector whose covariance matrix equals $Q(\bar{\unicode[STIX]{x1D6FE}})$, $\unicode[STIX]{x1D6FF}_{0}(\unicode[STIX]{x1D719}_{1}^{1})$ is actually a nontrivial finite measure with its total mass $\mathbf{E}[\unicode[STIX]{x1D719}_{1}^{1}]=(2\unicode[STIX]{x1D70B})^{-n/2}(\det Q(\bar{\unicode[STIX]{x1D6FE}}))^{-1/2}>0$.

Therefore, our problem reduces to showing $F(\unicode[STIX]{x1D716},\bullet )$ belongs to $\tilde{\mathbf{D}}_{\infty }$ and admits asymptotic expansion in $\tilde{\mathbf{D}}_{\infty }$-topology with the index set $\unicode[STIX]{x1D6EC}_{3}^{\prime }$. However, this is highly nontrivial since $\exp (\langle \bar{\unicode[STIX]{x1D708}},r_{\unicode[STIX]{x1D716},1}^{2}\rangle /\unicode[STIX]{x1D716}^{2})$ itself does not have nice integrability. This is why the two technical factors are involved in the definition of $F(\unicode[STIX]{x1D716},\bullet )$.

Let us observe the two technical factors. First, $\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D702}}(\unicode[STIX]{x1D716},w+\bar{\unicode[STIX]{x1D6FE}}/\unicode[STIX]{x1D716})$ and its derivatives vanish outside $\{w~|~\unicode[STIX]{x1D716}\mathbf{w}\in U_{\unicode[STIX]{x1D702}}\}$ and $\unicode[STIX]{x1D713}({\unicode[STIX]{x1D702}^{\prime }}^{-2}|({\tilde{y}}_{1}^{\unicode[STIX]{x1D716}}-a^{\prime })/\unicode[STIX]{x1D716}|^{2})$ and its derivatives vanish outside $\{|r_{\unicode[STIX]{x1D716},1}^{1}/\unicode[STIX]{x1D716}|\leqslant \unicode[STIX]{x1D702}^{\prime }\}$. Second, as $\unicode[STIX]{x1D716}{\searrow}0$,

(6.4)$$\begin{eqnarray}\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D702}}\bigg(\unicode[STIX]{x1D716},w+\frac{\bar{\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x1D716}}\bigg)=1+O(\unicode[STIX]{x1D716}^{M}),\qquad \unicode[STIX]{x1D713}\bigg(\frac{1}{{\unicode[STIX]{x1D702}^{\prime }}^{2}}\bigg|\frac{{\tilde{y}}_{1}^{\unicode[STIX]{x1D716}}-a^{\prime }}{\unicode[STIX]{x1D716}}\bigg|^{2}\bigg)=1+O(\unicode[STIX]{x1D716}^{M})\end{eqnarray}$$

in $\mathbf{D}_{\infty }$-topology for any (large) $M>0$. Therefore, these two Wiener functionals have no influence in the coefficient of the expansion of $F(\unicode[STIX]{x1D716},\bullet )$.

The expansion of $\langle \bar{\unicode[STIX]{x1D708}},r_{\unicode[STIX]{x1D716},1}^{2}\rangle /\unicode[STIX]{x1D716}^{2}=\langle \bar{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D719}_{1}^{2}+r_{\unicode[STIX]{x1D716},1}^{\unicode[STIX]{x1D705}_{3}}/\unicode[STIX]{x1D716}^{2}\rangle$ is indexed by $\unicode[STIX]{x1D6EC}_{2}^{\prime }$. Hence, if the integrability issue were left aside, we would easily have an expansion of the following type:

(6.5)$$\begin{eqnarray}\exp \bigg(\frac{\langle \bar{\unicode[STIX]{x1D708}},r_{\unicode[STIX]{x1D716},1}^{2}\rangle }{\unicode[STIX]{x1D716}^{2}}\bigg)\sim e^{\langle \bar{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D719}_{1}^{2}\rangle }(1+\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D70C}_{1}}\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D70C}_{1}}+\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D70C}_{2}}\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D70C}_{2}}+\cdots \,),\end{eqnarray}$$

where $0=\unicode[STIX]{x1D70C}_{0}<\unicode[STIX]{x1D70C}_{1}<\unicode[STIX]{x1D70C}_{2}<\cdots \,$ are all the elements of $\unicode[STIX]{x1D6EC}_{3}^{\prime }$ in increasing order. Due to (6.4), it also should hold that

(6.6)$$\begin{eqnarray}F(\unicode[STIX]{x1D716},\bullet )\sim e^{\langle \bar{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D719}_{1}^{2}\rangle }(1+\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D70C}_{1}}\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D70C}_{1}}+\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D70C}_{2}}\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D70C}_{2}}+\cdots \,).\end{eqnarray}$$

Once (6.6) is actually shown, then we prove our main theorem (Theorem 2.3). Moreover, $\unicode[STIX]{x1D6FC}_{0}:=\mathbf{E}[e^{\langle \bar{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D719}_{1}^{2}\rangle }\unicode[STIX]{x1D6FF}_{0}(\unicode[STIX]{x1D719}_{1}^{1})]>0$ is finite due to Assumption (A4).

Roughly speaking, the proof of (6.6) goes in the following way. First, Assumption (A4) implies $\mathbf{E}[e^{\langle \bar{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D719}_{1}^{2}\rangle }\unicode[STIX]{x1D6FF}_{0}(\unicode[STIX]{x1D719}_{1}^{1})]<\infty$ since $\langle \bar{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D719}_{1}^{2}\rangle$ belongs to the inhomogeneous Wiener chaos of order 2 whose main term corresponds to the Hessian in (A4). Though it is not at all obvious, the Hessian is actually Hilbert–Schmidt on $\mathfrak{H}\times \mathfrak{H}$.

However, what we want to estimate is something like

$$\begin{eqnarray}\exp \bigg(\langle \bar{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D719}_{1}^{2}\rangle +\frac{\langle \bar{\unicode[STIX]{x1D708}},r_{\unicode[STIX]{x1D716},1}^{\unicode[STIX]{x1D705}_{3}}\rangle }{\unicode[STIX]{x1D716}^{2}}\bigg)\cdot \unicode[STIX]{x1D6FF}_{0}\bigg(\unicode[STIX]{x1D719}_{1}^{1}+\frac{r_{\unicode[STIX]{x1D716},1}^{2}}{\unicode[STIX]{x1D716}}\bigg).\end{eqnarray}$$

Here, thanks to the factor $\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D702}}(\unicode[STIX]{x1D716},w+\bar{\unicode[STIX]{x1D6FE}}/\unicode[STIX]{x1D716})$, we may assume $\unicode[STIX]{x1D716}\mathbf{w}$ stay in the bounded set $U_{\unicode[STIX]{x1D702}}$ in the geometric rough path space. Hence, we can use the deterministic version of Taylor expansion of the Lyons–Itô map (Proposition 4.1) to prove that $\langle \bar{\unicode[STIX]{x1D708}},r_{\unicode[STIX]{x1D716},1}^{\unicode[STIX]{x1D705}_{3}}\rangle /\unicode[STIX]{x1D716}^{2}$ and $r_{\unicode[STIX]{x1D716},1}^{2}/\unicode[STIX]{x1D716}$ are actually so “small” that adding them does not destroy the integrability we need.◻