Weak backward error analysis for Langevin process

Abstract

We consider numerical approximations of stochastic Langevin equations by implicit methods. We show a weak backward error analysis result in the sense that the generator associated with the numerical solution coincides with the solution of a modified Kolmogorov equation up to high order terms with respect to the stepsize. This implies that every measure of the numerical scheme is close to a modified invariant measure obtained by asymptotic expansion. Moreover, we prove that, up to negligible terms, the dynamics associated with the considered implicit scheme is exponentially mixing: The law of the scheme converges exponentially fast to a constant up to an error that we can optimize.

Appendices

Appendix A: Proof of Proposition 2.11 and Lemma 2.12

1.1 Notations, assumptions and result

In all this appendix, the constant $C$ may vary from line to line and we will use the following notations:

Let $\phi \in \fancyscript{C}^{\infty }_{pol}(\mathbb {R}^{2d})$, then, for all $m\in \mathbb {N}$, there exists an integer $r_{m}$ such that $\phi \in \fancyscript{C}^m_{r_{m}}(\mathbb {R}^{2d})$. For a multi-index $\mathbf {k}=(k_1,\ldots ,k_{2d})\in \mathbb {N}^d$, we set $|\mathbf {k}|=k_1+\cdots +k_{2d}$ and for a function $\phi \in C^{\infty }(\mathbb {R}^{2d})$, we set

$$\begin{aligned} D^\mathbf {k}\phi (x)=\frac{\partial ^{|\mathbf {k}|}\phi (x)}{\partial ^{k_1}_{x_1} \ldots \partial ^{k_{d}}_{x_{2d}}},\quad x=(x_1,\ldots ,x_{2d})\in \mathbb {R}^{2d}. \end{aligned}$$

Moreover, we will assume in al the appendix that $\int \phi d\rho =0$. We recall that $u$ is the solution of the Kolmogorov equation (2.16) and it is defined for all $q\in \mathbb {R}^{d}$, $p\in \mathbb {R}^{d}$ and $t\ge 0$ by

$$\begin{aligned} u(t,q,p)=\mathbb {E}\phi (q_q(t),p_p(t)). \end{aligned}$$

The aim of this appendix is to prove the following result (Proposition 2.11):

Proposition 6.1

There exists a strictly positive real number $\lambda _0$ such that for any $m\in \mathbb {N}$, $\mathbf k \in \mathbb {N}^{2d}$ such that $|\mathbf k |=m$ and $0<\lambda <\lambda _0$, there exist an integer $s>2 r_{m+1+d}$ and a strictly positive real number $C$ such that for all $t\ge 0$ and $(q,p)\in \mathbb {R}^{2d}$

$$\begin{aligned} \big | D^\mathbf{k }u(t,q,p)\big |\le C(1+|q|^s+|p|^s)\parallel \phi \parallel _{m+d+1,r_{m+1+d}}\exp (-\lambda t). \end{aligned}$$

To prove this Proposition, we will use the same idea as in [25]. Unlike in [25], we need to know how the estimate depend on $\phi$. Moreover, using an estimate of $u$ described in [17], the proof is easier than in [25].

The proof proceeds as follows. We first show estimates on $u$ and its derivatives in an appropriate space. More precisely, we will show that for any $0<\lambda <2\lambda _0$, $m\in \mathbb {N}$ and $\mathbf k \in \mathbb {N}^{2d}$ such that $|\mathbf k |=m$, there exists a strictly positive real number $C_m$ such that for all $t>0$

$$\begin{aligned} \int |D^\mathbf{k }u(t,q,p)|^2\pi _s(q,p)dqdp\le C_m\exp (-\lambda t), \end{aligned}$$

where the function $\pi _s$ is defined as

$$\begin{aligned} \pi _s=\frac{1}{\Gamma ^s}, \end{aligned}$$

(6.1)

for some integer $s$.

Moreover, we have the following result on $\pi _s$: For all multi-index $\mathbf j$ and integer $s$, there exists a function $\psi _\mathbf{j ,s}\in C^{\infty }$ such that

$$\begin{aligned} \partial ^\mathbf{j }\pi _s(q,p)=\psi _\mathbf{j ,s}(q,p)\pi _s(q,p) \end{aligned}$$

(6.2)

where

$$\begin{aligned} \psi _\mathbf{j ,s}(q,p)\xrightarrow {|(q,p)|\rightarrow \infty }0. \end{aligned}$$

Then, for any $m\in \mathbb {N}$ and $\mathbf k \in \mathbb {N}^{2d}$ such that $|\mathbf k |=m$, it is possible to choose an integer $s_m$ such that we have, for all $t>0$, $s\ge s_m$ and $0<\zeta <2\lambda _0$

$$\begin{aligned} \int _{\mathbb {R}^{2d}}|D^\mathbf{k }\big (u(t,q,p)\pi _s(q,p)\big )|^2dqdp<C_m\exp (-\zeta t). \end{aligned}$$

We then conclude by applying Sobolev imbedding Theorem (see [2]).

We will also use the following notation: for all $s\in \mathbb {N}$, $d\pi _s=\pi _s(q,p)dqdp$.

1.2 Estimates on $u(t)$ and its derivatives in $L^2(\pi _s)=\{f:\mathbb {R}^{2d}\rightarrow \mathbb {R};\int |f|^2d\pi _s<+\infty \}$

By expression (2.13) of $L^{\top }$ and inequality (2.7) on $L\Gamma$, computations lead to

$$\begin{aligned} L^{\top }(\pi _s)&= s\frac{L\Gamma }{\Gamma ^{s+1}}-\frac{sd\sigma ^2}{\Gamma }\pi _s+\frac{s(s+1)\sigma ^2}{2}\frac{|\partial _p\Gamma |^2}{\Gamma ^2}\pi _s+d\gamma \pi _s\\&\le (-a_1 s+\gamma d)\pi _s+\varPhi _s\pi _s, \end{aligned}$$

where $\varPhi _s(q,p)\xrightarrow {|(q,p)|\rightarrow {+\infty }}0$ and $a_1$ is defined by (2.7). Hence, for each $s\in \mathbb {N}^*$, there exists a real number $\nu _s>0$ such that

$$\begin{aligned} L^{\top }(\pi _s)\le \nu _s\pi _s. \end{aligned}$$

(6.3)

We will now show the following proposition then we will use Sobolev inequalities to prove Proposition 6.1:

Proposition 6.2

Let $m\in \mathbb {N}$, $0<\zeta <2\lambda _0$, where $\lambda _0$ is defined in the following, and $\mathbf k \in \mathbb {N}^{2d}$ such that $|\mathbf k |=m$ be fixed. There exists an integer $s_m>2 r_m$ such that, for all $s\ge s_m$, there exists a strictly positive real number $C_{m,s}$ such that we have for all $T>0$,

$$\begin{aligned} \int |D^\mathbf{k } u(T)|^2d\pi _s \le C_{m,s}\parallel \phi \parallel _{m,r_m}^2\exp (-\zeta T), \end{aligned}$$

(6.4)

where $r_m$ is defined at the beginning of this Appendix.

Proposition 6.2 is a corollary of the following result:

Proposition 6.3

Let $\lambda _0$ be defined in the following. For all $m\in \mathbb {N}$, $\mathbf k \in \mathbb {N}^{2d}$ such that $|\mathbf k |=m$, positive polynomial function $Q$ and $0<\zeta <2\lambda _0$, there exists an integer $s_{m}>2 r_m$ such that for all $s\ge s_m$ there exists a strictly positive real numbers $C_{m,Q,s}$ such that we have for all $T>0$

$$\begin{aligned} \int _0^T \exp (\zeta t) \int Q|D^\mathbf{k } u(t)|^2d\pi _{s} dt \le C_{m,Q,s}\parallel \phi \parallel _{m,r_m}^2. \end{aligned}$$

(6.5)

For the convenience of the reading, we will first show Propositions 6.2 and 6.3 for $m=0$. Then, we will show Proposition 6.3 for $m=1$ and explain how to deduce Proposition 6.2 for $m=1$. Finally, we will show Proposition 6.3 by induction and show that Proposition 6.2 is a corollary of Proposition 6.3. The idea to prove Proposition 6.3 for $m\ge 1$ and $\mathbf k \in \mathbb {N}^{2d}$ such that $|\mathbf k |=m$ is the following: We first show the result for $Q|\partial _{p}D^\mathbf{k-1 }.|^2$ and $Q|(\alpha \partial _{p}-\partial _{q})D^\mathbf{k-1 }.|^2$. We can then deduce the result for $Q|\partial _qD^\mathbf{k-1 }.|^2$.

To show this two Propositions for $m=0$, we need a better point-wise estimatie of $u$:

Lemma 6.4

There exists $C=C(r_0)>0$, $\lambda _0=\lambda _0(r_0)>0$ such that, for all $t\ge 0$ and $(q,p)\in \mathbb {R}^{2d}$,

$$\begin{aligned} |u(t,q,p)|\le C\Gamma ^{r_0}(q,p)\exp (-\lambda _0 t)\parallel \phi \parallel _{0,r_0}. \end{aligned}$$

(6.6)

A proof of this result can be found in [17] or [10]. To show this Lemma, the two main ingredients are the property (2.6) on $\Gamma$ and that for $x\in \mathbb {R}^{2d}$, $t>0$ and open $\fancyscript{O}\subset \mathbb {R}^{2d}$, the transition kernel for (2.4) satisfies $Q_t(x,\fancyscript{O})>0$. Under Assumption B, the second property is true (see [17]).

We have the following corollary:

Lemma 6.5

For $s>2r_0$, there exists a strictly positive real number $C_{s}$ such that

$$\begin{aligned} \int |u(t)|^2d\pi _s\le C_s\parallel \phi \parallel _{0,r_0}^2\exp (-2\lambda _0 t)\quad \forall t\ge 0. \end{aligned}$$

(6.7)

Moreover, for $Q$ a positive polynomial function, we have, for all $s>s_{Q}\ge 2r_0$, that there exists a strictly positive real number $C_{Q,s}$ such that

$$\begin{aligned} \int Q|u(t)|^2d\pi _s\le C_{Q,s}\parallel \phi \parallel _{0,r_0}^2 \exp (-2\lambda _0 t)\quad \forall t\ge 0. \end{aligned}$$

(6.8)

Then, the results (6.4) and (6.5) for $m=0$ are a consequence of Lemma 6.5.

The following Lemma is the key of the proof of all the other Lemmas.

Lemma 6.6

Let $A$ be a linear operator and $Q$ a polynomial function. There exists an integer $s_{A,Q}$ such that for all $s\ge s_{A,Q}$, we have for all $\zeta >0$ and $T>0$

$$\begin{aligned}&\exp (\zeta T)\int Q|Au(t)|^2d\pi _s +\sigma ^2\int _0^T\exp (\zeta t)\int Q|\partial _p (Au(t))|^2d\pi _sdt\nonumber \\&\quad \le \int Q|A u(0)|^2d\pi _s+(\zeta +\nu _s)\int _0^T\exp (\zeta t)\nonumber \\&\qquad \times \int Q|Au(t)|^2d\pi _sdt\nonumber \\&\qquad +2\int _0^T\exp (\zeta t)\int Q \langle [A,L]u(t),Au(t)\rangle d\pi _sdt\nonumber \\&\qquad -\int _0^T\exp (\zeta t)\int |Au(t)|^2 LQd\pi _sdt\nonumber \\&\qquad -\sigma ^2\int _0^T\exp (\zeta t)\int \langle \partial _pQ,\partial _p|Au(t)|^2\rangle d\pi _sdt, \end{aligned}$$

(6.9)

where, for $A$ and $B$ two linear operators, $[A,B]=AB-BA$.

Proof

Let $s$ large enough such that $\int Q|Au(0)|^2d\pi _s<\infty$. Using (2.12) and (2.14), we get

$$\begin{aligned}&\frac{d}{dt}[\exp (\zeta t)Q|Au(t)|^2]=\zeta \exp (\zeta t)Q|Au(t)|^2+2\exp (\zeta t)Q\langle ALu(t),Au(t)\rangle \\&\quad =\zeta \exp (\zeta t)Q|Au(t)|^2+2\exp (\zeta t)Q\langle LAu(t),Au(t)\rangle \\&\qquad +2\exp (\zeta t)Q\langle [A,L]u(t),Au(t)\rangle \\&\quad =\zeta \exp (\zeta t)Q|Au(t)|^2+\exp (\zeta t)QL|Au(t)|^2-\sigma ^2\exp (\zeta t)Q|\partial _p(Au(t))|^2\\&\qquad +2\exp (\zeta t)Q\langle [A,L]u(t),Au(t)\rangle \\&\quad =\zeta \exp (\zeta t)Q|Au(t)|^2+\exp (\zeta t)L\big (Q|Au(t)|^2\big )-\exp (\zeta t)|Au(t)|^2LQ\\&\qquad -\sigma ^2\exp (\zeta t)\langle \partial _pQ,\partial _p|Au(t)|^2\rangle -\sigma ^2\exp (\zeta t)Q|\partial _p(Au(t))|^2\\&\qquad +2\exp (\zeta t)Q\langle [A,L]u(t),Au(t)\rangle . \end{aligned}$$

We integrate with respect to $t$,

$$\begin{aligned}&\exp (\zeta T)Q|Au(T)|^2 = Q|Au(0)|^2+\zeta \int _0^T \exp (\zeta t)Q|Au(t)|^2dt\\&\quad +\int _0^T \exp (\zeta t)L(Q|Au(t)|^2)dt-\int _0^T\exp (\zeta t)|Au(t)|^2LQdt\\&\quad -\sigma ^2\int _0^T\exp (\zeta t)\langle \partial _pQ,\partial _p|Au(t)|^2\rangle dt-\sigma ^2\int _0^T \exp (\zeta t)Q|\partial _p Au(t)|^2 dt\\&\quad +2\int _0^T\exp (\zeta t) Q\langle [A,L]u(t),Au(t)\rangle dt. \end{aligned}$$

We integrate with respect to $\pi _s$. Using the inequality (6.3) on $L^{\top }\pi _s$, we have (6.9).$\square$

We will also need the following computations:

Lemma 6.7

Let $k\in \mathbb {N}$, we have for any $(q,p)\in \mathbb {R}^{2d}$

$$\begin{aligned} -LH^k(q,p)\le 2\gamma kH^{k}(q,p) \end{aligned}$$

(6.10)

and for all $t\ge 0$ and linear operator $A$,

$$\begin{aligned} \langle \partial _pH^{k},\partial _p|Au(t)|^2\rangle \le 2kH^{k}|Au(t)|^2+2kH^{k-1}|\partial _pAu(t)|^2. \end{aligned}$$

(6.11)

Proof

For any $(q,p)\in \mathbb {R}^{2d}$ and $k\in \mathbb {N}^*$, we have

$$\begin{aligned} \partial _pH^k(q,p)=kH^{k-1}(q,p)p \end{aligned}$$

and

$$\begin{aligned} -LH^k(q,p)&= \gamma k|p|^2H^{k-1}(q,p)-\frac{k}{2}H^{k-1}(q,p)-\frac{k(k-1)}{2}|p|^2H^{k-2}(q,p)\\&\le \gamma k|p|^2H^{k-1}(q,p)\le 2\gamma kH^{k}(q,p). \end{aligned}$$

We have used the positivity of $V$. Moreover, we have for any $t\ge 0$, $\ell \in \mathbb {N}$ and for each component of $Au$ that we still write $Au$,

$$\begin{aligned} \langle \partial _pH^{2l},\partial _p|A u(t)|^2\rangle&= 4\ell H^{\ell }Au(t)\langle pH^{\ell -1},\partial _pAu(t)\rangle \\&\le 2\ell H^{2\ell }|Au(t)|^2+2\ell |p|^2H^{2\ell -2}|\partial _pAu(t)|^2\\&\le 2\ell H^{2\ell }|Au(t)|^2+4\ell H^{2\ell -1}|\partial _pAu(t)|^2 \end{aligned}$$

and

$$\begin{aligned} \langle \partial _pH^{2l+1},\partial _p|A u(t)|^2\rangle&= 2(2\ell +1) \langle H^{\ell }u(t) p,H^{\ell }\partial _pAu(t)\rangle \\&\le (2\ell +1) H^{2\ell }|p|^2|u(t)|^2+(2\ell +1)H^{2\ell }|\partial _pAu(t)|^2\\&\le 2(2\ell +1) H^{2\ell +1}|u(t)|^2+(2\ell +1) H^{2\ell }|\partial _pAu(t)|^2. \end{aligned}$$

$\square$

We now show the results (6.4) and (6.5) for $m=1$. First, we show the two following preliminary lemmas.

Lemma 6.8

Let $Q$ be a positive polynomial function. Let $s>2 r_0$ large enough and $0<\zeta <2\lambda _0$. There exists a strictly positive real number $C_{Q,s}$ such that for all $T>0$

$$\begin{aligned} \int _0^T \exp (\zeta t)\int Q|\partial _p u(t)|^2d\pi _s dt \le C_{Q,s}\parallel \phi \parallel _{0,r_0}^2\!. \end{aligned}$$

(6.12)

Proof

Let $\zeta >0$. We have $[Id,L]=0$, then, using Lemma 6.6 with $A=I_d$ and $Q=1$, we get for $T>0$

$$\begin{aligned}&\exp (\zeta T)\int |u(t)|^2d\pi _s+\sigma ^2\int _0^T\exp (\zeta t)\int |\partial _p u(t)|^2d\pi _sdt\\&\quad \le \int |u(0)|^2d\pi _s+(\zeta +\nu _s)\int _0^T\exp (\zeta t)\int |u(t)|^2d\pi _sdt. \end{aligned}$$

We choose $0<\zeta <2\lambda _0$ and, using (6.7), we bound the last term. Then, we have (6.12) for $Q=1$.

Let $Q$ a positive polynomial function. Using Lemma 6.6 for $A=I_d$, we get for $s$ large enough, $\zeta >0$ and $T>0$

$$\begin{aligned}&\exp (\zeta T)\int Q|u(t)|^2d\pi _s +\sigma ^2\int _0^T\exp (\zeta t)\int Q|\partial _p (u(t))|^2d\pi _sdt\nonumber \\&\quad \le \int Q|u(0)|^2d\pi _s+(\zeta +\nu _s)\int _0^T\exp (\zeta t)\int Q|u(t)|^2d\pi _sdt\nonumber \\&\qquad -\int _0^T\exp (\zeta t)\int |u(t)|^2 LQd\pi _sdt\nonumber \\&\qquad -\sigma ^2\int _0^T\exp (\zeta t)\int \langle \partial _pQ,\partial _p|u(t)|^2\rangle d\pi _s dt, \end{aligned}$$

(6.13)

As $V\in \fancyscript{C}^{\infty }_{pol}(\mathbb {R}^{d})$, there exists a positive polynomial function $Q_1$ such that $|LQ|\le Q_1$, then we have for any $T>0$ and $s$ large enough

$$\begin{aligned}&-\int _0^T\exp (\zeta t)\int |u(t)|^2 LQd\pi _sdt -\sigma ^2\int _0^T\exp (\zeta t)\int \langle \partial _pQ,\partial _p|u(t)|^2\rangle d\pi _s dt\\&\quad \le \int _0^T\exp (\zeta t)\Big (2Q_1+\sigma ^2|\partial _pQ|^2\Big )|u(t)|^2 d\pi _s dt\\&\qquad +\sigma ^2\int _0^T\exp (\zeta t)\int |\partial _p u(t)|^2d\pi _sdt. \end{aligned}$$

We choose $0<\zeta <2\lambda _0$ and use (6.8) and (6.12) with $Q=1$ to have the result (6.12) for any positive polynomial function $Q$.$\square$

Lemma 6.9

Let $s>2 r_1$ large enough and $0<\zeta <2\lambda _0$. There exists a strictly positive real number $\alpha _s$ such that for $\alpha >\alpha _s$, there exists a positive real number $C_s$ such that, for all $T>0$,

$$\begin{aligned}&\int _0^T\exp (\zeta t)\int |\alpha \partial _p u(t)-\partial _q u(t)|^2d\pi _s dt+\int _0^T \exp (\zeta t)\int |\partial _p(\alpha \partial _p u(t) \nonumber \\&\quad -\partial _q u(t))|^2d\pi _s dt\le C_s\parallel \phi \parallel _{1,r_1}^2\!. \end{aligned}$$

(6.14)

Let $k>0$ an integer. For any $0<\zeta <2\lambda _0$ and $s>2r_1$ large enough, there exists a strictly positive real number $\alpha _k$ such that $\forall \alpha \ge \alpha _k$ there exists a strictly positive real number $C_{k,s}$ such that, for all $T>0$,

$$\begin{aligned}&\exp (\zeta T)\int H^k|\alpha \partial _p u(T)-\partial _q u(T)|^2d\pi _s+\int _0^T \exp (\zeta t)\int H^k|\partial _p(\alpha \partial _p u(t)\nonumber \\&\quad -\partial _q u(t))|^2d\pi _s dt \le C_{k,s}\parallel \phi \parallel _{1,r_1}^2. \end{aligned}$$

(6.15)

Proof

Let $\alpha >0$ and $i\in \{1,\ldots ,d\}$. We use the following notation: $A_1=\alpha \partial _{p_i}-\partial _{q_i}$. We have for any function $\psi \in C^{\infty }(\mathbb {R}^{2d})$

$$\begin{aligned}{}[A_1,L]\psi =-\alpha A_1\psi +\alpha (\alpha -\gamma )\partial _{p_i}\psi +\langle \partial _{q_i}\partial _qV,\partial _p\psi \rangle . \end{aligned}$$

Using the polynomial growth of $\partial _{q_i}\partial _q V$, we have that there exist a positive polynomial function $Q_1$, $\varepsilon _1>0$ and $\varepsilon _2>0$ such that for any function $\psi \in C^{\infty }(\mathbb {R}^{2d})$

$$\begin{aligned}&2\langle [A_1,L]\psi ,A_1 \psi \rangle =-2\alpha |A_1\psi |^2+2A_1 \psi \langle \partial _{q_i}\partial _qV,\partial _{p} \psi \rangle +2\alpha (\alpha -\gamma )\partial _{p_i}\psi A_1\psi \nonumber \\&\quad \le (\varepsilon _1+\varepsilon _2-2\alpha )|A_1 \psi |^2+\frac{Q_1}{\varepsilon _1}|\partial _p \psi |^2+\frac{\alpha (\alpha -\gamma )^2}{\varepsilon _2}|\partial _{p_i}\psi |^2. \end{aligned}$$

(6.16)

Then, choosing $s$ large enough and using Lemma 6.6 with $Q=1$, we get for all $T>0$, $\varepsilon _1$, $\varepsilon _2$ and $\zeta >0$

$$\begin{aligned}&\exp (\zeta T)\int |A_1 u(T)|^2d\pi _s+\sigma ^2\int _0^T \exp (\zeta t)\int |\partial _p(A_1 u(t))|^2d\pi _s dt\\&\quad \le \int |A_1 u(0)|^2d\pi _s+\frac{1}{\varepsilon _1}\int _0^T \exp (\zeta t) \int Q_1|\partial _p u(t)|^2d\pi _s dt\\&\qquad +\frac{(\alpha (\alpha -\gamma ))^2}{\varepsilon _2}\int _0^T \exp (\zeta t) \int |\partial _{p_i} u(t)|^2d\pi _s dt\\&\qquad +(\nu _s+\zeta +\varepsilon _1+\varepsilon _2-2\alpha )\int _0^T \exp (\zeta t)\int |A_1 u(t)|^2d\pi _s dt. \end{aligned}$$

We choose $0<\zeta <2\lambda _0$ and $\alpha$, $\varepsilon _1$ and $\varepsilon _2$ such that $\nu _s+\zeta +\varepsilon _1+\varepsilon _2-2\alpha <0$. We then use (6.12) to prove (6.14).

We now prove (6.15) by recursion on $k$. We have proved the case $k=0$. Let us assume (6.15) is true for $k-1$. We want to obtain it for $k$.

Using computations for $k=0$, (6.16), (6.10), (6.11) and Lemma 6.6, we get for $s$ large enough, $\varepsilon _1>0$, $\varepsilon _2>0$, $\zeta >0$ and $T>0$

$$\begin{aligned}&\exp (\zeta T)\int H^{k}|\alpha \partial _pu(t) -\partial _qu(t)|^2d\pi _s\\&\qquad +\sigma ^2\int _0^T \exp (\zeta t)\int H^{k}|\partial _p\big (\alpha \partial _pu(t)-\partial _qu(t)\big )|^2d\pi _sdt\\&\quad \le \int H^k|\alpha \partial _pu(0)-\partial _qu(0)|^2d\pi _s\\&\qquad +(\zeta +\varepsilon _2+\varepsilon _1-2\alpha +2k\sigma ^2+2k\gamma +\nu _s)\int _0^T \exp (\zeta t)\\&\qquad \times \int H^{k}|\alpha \partial _pu(t)-\partial _qu(t)|^2d\pi _sdt\\&\qquad +2k\sigma ^2\int _0^T\exp (\zeta t)\int H^{k-1}|\partial _p(\partial _p u(t)-\partial _q u(t))|^2d\pi _sdt\\&\qquad +\left( \frac{Q_1}{\varepsilon _1}+\frac{\alpha ^2(\alpha +\gamma )^2}{\varepsilon _2}\right) \int _0^T \exp (\zeta t)\\&\qquad \times \int H^{k}|\partial _pu(t)|^2d\pi _s dt, \end{aligned}$$

where $Q_1$ is a positive polynomial function. We take $\zeta <2\lambda$ and choose $\alpha$, $\varepsilon _2$ and $\varepsilon _1$ such that $\varepsilon _2+\varepsilon _1-2\alpha +2k\sigma ^2+2k\gamma +\zeta +\nu _s\le 0$. Then, using the induction hypothesis on $k$, the polynomial growth of H and (6.12), we obtain (6.15) for $k$.$\square$

Remark 6.10

Using the fact that $q^2\le C H(q,p)$ and $p^2\le 2 H(q,p)$, (6.14) and (6.15), we have for $0<\zeta <2\lambda _0$, $s$ large enough and $Q$ a positive polynomial function that there exist real positive numbers $\alpha _s$ and $C_{Q,s}$ depending also of $\alpha _s$, such that, for $T>0$,

$$\begin{aligned}&\int _0^T\exp (\zeta t)\int Q|\alpha \partial _p u(t)-\partial _q u(t)|^2d\pi _s dt \le C_{Q,s}\parallel \phi \parallel _{1,r_1}^2. \end{aligned}$$

(6.17)

Using above Remark, we can show the following lemma:

Lemma 6.11

Let $Q$ be a positive polynomial function. Let $s>2 r_1$ large enough. For all $0<\zeta <2\lambda _0$, there exists a strictly positive real number $C_{Q,s}$ such that for all $T>0$

$$\begin{aligned} \int _0^T \exp (\zeta t)\int Q|\partial _q u(t)|^2d\pi _s dt \le C_{Q,s}\parallel \phi \parallel _{1,r_1}^2\!. \end{aligned}$$

(6.18)

Proof

Let $0<\zeta <2\lambda _0$. We have the following inequality: for any function $\psi \in C^{\infty }(\mathbb {R}^{2d})$ and $\alpha >0$

$$\begin{aligned} |\partial _q\psi |^2\le |(\partial _q -\alpha \partial _p) \psi |^2+\alpha ^2|\partial _p \psi |^2. \end{aligned}$$

Then, using (6.17) and (6.12), we have (6.18). $\square$

Using (6.12) and (6.18), we obtain (6.5) for $m=1$. We will now show (6.4) for $m=1$.

Lemma 6.12

For $s> 2r_1$ large enough and $0<\zeta <2\lambda _0$, there exists a strictly positive real number $C_s$ such that for $T>0$,

$$\begin{aligned} \int |\partial _p u(T)|^2d\pi _s\le C_s\parallel \phi \parallel _{1,r_1}^2\exp (-\zeta T). \end{aligned}$$

Proof

We have for $t\ge 0$

$$\begin{aligned} 2\langle [\partial _{p_i},L]u(t),\partial _{p_i} u(t)\rangle&= 2\partial _{q_i}u(t)\partial _{p_i} u(t)-2\gamma |\partial _{p_i} u(t)|^2\\&\le |\partial _{q_i}u(t)|^2+(1-2\gamma )|\partial _{p_i}u(t)|^2. \end{aligned}$$

Then, using Lemma 6.6, we get for $T>0$ and $\zeta >0$

$$\begin{aligned}&\exp (\zeta t)\int |\partial _{p} u(t)|^2d\pi _s\le \int |\partial _{p}u(0)|^2d\pi _s +(\zeta +\nu _s+1-2\gamma )\nonumber \\&\quad \times \int _0^T\exp (\zeta t)\int |\partial _pu(t)|^2d\pi _sds\!+\!\int _0^T\exp (\zeta t)\int |\partial _qu(t)|^2d\pi _sds.\qquad \end{aligned}$$

(6.19)

We choose $0<\zeta <2\lambda _0$ and use (6.12) and (6.18) to conclude.$\square$

Lemma 6.13

For $s>2 r_1$ large enough and $0<\zeta <2\lambda _0$, there exists a strictly positive real number $C_s$ such that for all $T>0$,

$$\begin{aligned} \int |\partial _q u(T)|^2d\pi _s\le C_s \parallel \phi \parallel _{1,r_1}^2\exp (-\zeta T). \end{aligned}$$

Proof

We have for $t\ge 0$

$$\begin{aligned} 2\langle [\partial _{q_i},L]u(t),\partial _{q_i} u(t)\rangle =2\langle \partial _{q_i}DV,\partial _{p} u(t)\rangle \partial _{q_i}u(t)\le Q|\partial _{p}u(t)|^2+|\partial _{q_i}u(t)|^2, \end{aligned}$$

where $Q$ is a positive polynomial function such that $|\partial _{q_i}DV|^2<Q$. Then, we use Lemma 6.6, $0<\zeta <2\lambda _0$, (6.12) and (6.18) to conclude. $\square$

We have shown the result (6.4) for $m=1$.

We will now show equation (6.5) by induction on $m$. We already proved the case $m=1$. We suppose that (6.5) holds up to $m$ and we want to obtain it for $m+1$. First, we show a result on $\partial _pD^\mathbf{k } u(t)$ where $\mathbf k \in \mathbb {N}^{2d}$ such that $|\mathbf k |=m$.

Lemma 6.14

Let us assume that induction hypothesis (6.5) holds up to $m$. For $s>2 r_m$ large enough, $0<\zeta <2\lambda _0$ and $\mathbf k \in \mathbb {N}^{2d}$ such that $|\mathbf k |=m$, there exists a strictly positive real number $C_{m,s}$ such that for all $T>0$,

$$\begin{aligned} \int _0^T \exp (\zeta t)\int |\partial _pD^\mathbf{k }u(t)|^2d\pi _s dt \le C_{m,s}\parallel \phi \parallel _{m,r_m}^2. \end{aligned}$$

(6.20)

Proof

By induction, we can show for any function $\psi$

$$\begin{aligned} \,[D^\mathbf{k },L]\psi \le \sum _\mathbf{i \in \mathbb {N}^{2d}, |\mathbf i |\le m}P_\mathbf{i }|D^\mathbf{i }\psi |, \end{aligned}$$

where $P_\mathbf{i }$ is a positive polynomial function which depends on the polynomial growth of $V$ and its derivatives. Then, we get for $t>0$

$$\begin{aligned} 2\langle [D^\mathbf{k },L]u(t),D^\mathbf{k }u(t)\rangle \le \sum _\mathbf{i \in \mathbb {N}^{2d}, |\mathbf i |\le m}P_\mathbf{i }^2|D^\mathbf{i } u(t)|^2+C|D^\mathbf{k }u(t)|^2, \end{aligned}$$

(6.21)

where $C$ is a positive real number. Using Lemma 6.6, we get for $T>0$ and $\zeta >0$

$$\begin{aligned}&\exp (\zeta T)\int |D^\mathbf{k }u(t)|^2d\pi _s +\sigma ^2\int _0^T\exp (\zeta t)\int |\partial _p (D^\mathbf{k }u(t))|^2d\pi _sdt\\&\quad \le \int |D^\mathbf{k } u(0)|^2d\pi _s+(\zeta +\nu _s+C)\\&\qquad \times \int _0^T\exp (\zeta t)\int |D^\mathbf{k }u(t)|^2d\pi _sdt\\&\qquad +\sum _\mathbf{i \in \mathbb {N}^{2d}, |\mathbf i |\le m}\int _0^T\exp (\zeta t)\int P_\mathbf{i }^2|D^\mathbf{i }u(t)| d\pi _sdt, \end{aligned}$$

We choose $0<\zeta <2\lambda _0$ and use induction hypothesis (6.5) to conclude. $\square$

Lemma 6.15

Let us assume that induction hypothesis (6.5) holds. For $0<\zeta <2\lambda _0$, $s>2 r_m$ large enough, $\ell \in \mathbb {N}$ and $\mathbf k \in \mathbb {N}^{2d}$ such that $|\mathbf k |=m$, there exists a strictly positive real number $C_{m,\ell ,s}$ such that

$$\begin{aligned} \int _0^T \exp (\zeta t)\int H^{\ell }|\partial _pD^\mathbf{k }u(t)|^2d\pi _s dt \le C_{m,\ell ,s}\parallel \phi \parallel _{m,r_m}^2, \end{aligned}$$

(6.22)

for all $T>0$.

Proof

We proceed by induction on $\ell$.

We also have the result for $\ell =0$ [see (6.20)]. Suppose that the induction hypothesis (6.22) holds for $\ell -1$. Using computations for $\ell =0$ (6.21), (6.10), (6.11) and Lemma 6.6, we get for $s$ large enough, $\varepsilon _1>0$, $\varepsilon _2>0$, $\zeta >0$ and $T>0$

$$\begin{aligned}&\exp (\zeta T)\int H^{\ell }|D^\mathbf{k }u(t)|^2d\pi _s +\sigma ^2\int _0^T \exp (\zeta t)\int H^{\ell }|\partial _pD^\mathbf{k }u(t)|^2d\pi _sdt\\&\quad \le \int H^{\ell }|D^\mathbf{k }u(0)|^2d\pi _s\\&\qquad +(\zeta +C+2\ell \sigma ^2+2\ell \gamma +\nu _s)\int _0^T \exp (\zeta t)\int H^{\ell }|D^\mathbf{k }u(t)|^2d\pi _sdt\\&\qquad +2\ell \sigma ^2\int _0^T\exp (\zeta t)\int H^{\ell -1}|\partial _pD^\mathbf{k }u(t)|^2d\pi _sdt\\&\qquad +\sum _{\mathbf{i }\in {\mathbb {N}}^{2d}, |\mathbf{i }|\le m}\int _0^T\exp (\zeta t)\int H^{\ell } P_\mathbf{i }^2|D^\mathbf{i }u(t)| d\pi _sdt, \end{aligned}$$

where $C$ and $P_\mathbf{i }$ are defined in (6.21). We take $0<\zeta <2\lambda _0$. Then, using the induction hypothesis on $\ell$, the polynomial growth of H and induction hypothesis on $m$, we obtain (6.15) for $\ell$.$\square$

Remark 6.16

Using the same ideas as in Remark 6.10, we prove the following result. Let $Q$ a positive polynomial function. For $s>2 r_m$ large enough, $0<\zeta <2\lambda _0$ and $\mathbf k \in \mathbb {N}^{2d}$ such that $|\mathbf k |=m$, there exists a strictly positive real number $C_{m,Q,s}$ such that for all $T>0$

$$\begin{aligned}&\exp (\zeta T)\int Q |D^\mathbf{k } u(T)|^2d\pi _s+\int _0^T \exp (\zeta t)\int Q|\partial _pD^\mathbf{k }u(t)|^2d\pi _s dt\nonumber \\&\quad \le C_{m,Q,s}\parallel \phi \parallel _{m,r_m}^2. \end{aligned}$$

(6.23)

Lemma 6.17

Let us assume that induction hypothesis (6.5) holds. Let $D^{m}_q u(t)$ denoted an arbitrary partial derivative of $u(t)$ of the type $\partial _{q_{i_1}} \ldots \partial _{q_{i_m}}$. For $s>2 r_{m+1}$ large enough and $0<\zeta <2\lambda _0$, there exists a strictly positive real number $\tilde{\alpha }_0$ such that for all $\alpha \ge \tilde{\alpha }_0$, there exists a strictly positive real number $C$ depending of $m$, $s$ and $\alpha$ such that for all $T>0$,

$$\begin{aligned} \int _0^T \exp (\zeta t)\int |\alpha \partial _p\big (D^{m}_q u(t)\big )-\partial _q\big (D^{m}_q u(t)\big )|^2d\pi _s dt \le C\parallel \phi \parallel _{m+1,r_{m+1}}^2\!. \end{aligned}$$

(6.24)

Proof

Let $\alpha >0$, $i\in \{1,\ldots ,d\}$ and $A_1=\alpha \partial _p-\partial _q$. For any function $\psi \in C^{\infty }(\mathbb {R}^{2d})$, we have

$$\begin{aligned} \,[A_1D^m_q,L]\psi =A_1[D^m_q,L]\psi +[A_1,L](D^m_q\psi ). \end{aligned}$$

Moreover, by induction an $m$, we can show for any function $\psi \in C^{\infty }(\mathbb {R}^{2d})$

$$\begin{aligned} A_1[D^m_q,L]\psi \le \sum _\mathbf{i \in \mathbb {N}^{2d}, |\mathbf i |\le m}P_\mathbf{i }|\partial _pD^\mathbf{i }\psi |, \end{aligned}$$

where $P_\mathbf{i }$ is a positive polynomial function which depends on the polynomial growth of $V$ and its derivatives. Then, we get for $t>0$

$$\begin{aligned} 2\langle A_1[D^m_q,L]u(t),A_1D^m_q u(t)\rangle \le \sum _\mathbf{i \in \mathbb {N}^{2d}, |\mathbf i |\le m}P_\mathbf{i }^2|\partial _pD^\mathbf{i } u(t)|^2+C|A_1D^{m}_q u(t)|^2, \end{aligned}$$

(6.25)

where $C$ is a positive real number. Using (6.25) and (6.16), we have that there exist a positive polynomial function $Q_1$, $\varepsilon _1>0$ and $\varepsilon _2>0$ such that

$$\begin{aligned}&\langle [A_1D^m_q,L]u(t),A_1D^m_q u(t)\rangle \\&\quad \le \sum _\mathbf{i \in \mathbb {N}^{2d}, |\mathbf i |\le m}P_\mathbf{i }^2|\partial _pD^\mathbf{i } u(t)|^2 +(\varepsilon _1+\varepsilon _2-2\alpha +C)|A_1 D^m_qu(t)|^2\\&\qquad +\frac{Q_1}{\varepsilon _1}|\partial _p D^m_qu(t)|^2+\frac{\alpha (\alpha -\gamma )^2}{\varepsilon _2}|\partial _{p_i}D^m_qu(t)|^2. \end{aligned}$$

To conclude, we proceed as Lemma 6.9: We choose $s$ large enough and use Lemma 6.6. We the choose $\zeta <2\lambda$ and $\alpha$, $\varepsilon _1$ and $\varepsilon _2$ such that $\nu _s+\zeta +\varepsilon _1+\varepsilon _2-2\alpha +C<0$ and use (6.23).$\square$

Lemma 6.18

Let us assume that induction hypothesis (6.5) holds. Let $Q$ a positive polynomial function. For $s>2 r_{m+1}$ large enough and $0<\zeta <2\lambda _0$, there exists a strictly positive real number $\tilde{\alpha }_s$ such that for all $\alpha \ge \tilde{\alpha }_s$, there exists a strictly positive real number $C_{m,s}$ which also depends on $\alpha$ and $\zeta$ such that for all $T>0$,

$$\begin{aligned} \int _0^T \exp (\zeta t)\int Q|\alpha \partial _p\big (D^{m}_q u(t)\big )-\partial _q\big (D^{m}_q u(t)\big )|^2d\pi _s dt \le C_{m,s}\parallel \phi \parallel _{m+1,r_{m+1}}^2\!. \end{aligned}$$

(6.26)

Proof

First, we prove this result by induction on $k$ for $H^k$. We use same arguments and computations used to prove (6.15) and (6.22), then we have the result for $Q$ (see Remark 6.10). $\square$

Remark 6.19

Using (6.23) and (6.26), we prove that inequality (6.5) holds up to $m+1$. Indeed, if there is a derivative in a direction $p_1$,...,$p_d$, then we use the result (6.23). In the other case, we use (6.26), (6.23) and $\partial _qD^{m}_q=-\big (\alpha \partial _{p_i}D^{m}_{q}-\partial _{q_i}D^{m}_q\big )+\alpha \partial _{p_i}D^m_q$, for $i\in \{1,\ldots ,d\}$.

We have shown that (6.5) is true for all $m$, then we have shown Proposition 6.3. We now prove that (6.4) is true for the derivatives of order $m$, where $m\in \mathbb {N}^*$.

Lemma 6.20

Let $m\in \mathbb {N}^*$ be fixed. For $s>2 r_m$ large enough, $0<\zeta <2\lambda _0$ and $\mathbf k \in \mathbb {N}^{2d}$ such that $|\mathbf k |=m$, there exists a strictly positive real number $C_{m,s}$ such that for all $T>0$,

$$\begin{aligned} \int |D^\mathbf{k } u(T)|^2d\pi _s \le C_{m,s}\parallel \phi \parallel _{m,r_{m}}^2\exp (-\zeta T). \end{aligned}$$

Proof

We proceed as to prove Lemma 6.11 and Lemma 6.12 and we use Lemma 6.6, (6.21) and Proposition 6.3. $\square$

We have shown the result (6.4).

1.3 Point-wise estimates of $u$ and its derivatives

Proof of Proposition 6.1

Using (6.4) and (6.2), we get : For all $m\in \mathbb {N}$, $n\in \mathbb {N}$, $\mathbf k \in \mathbb {N}^{2d}$ such that $|\mathbf k |=m$, $\mathbf l \in \mathbb {N}^{2d}$ such that $|\mathbf l |=m$, $0<\zeta <2\lambda _0$ and $s$ large enough, there exists a strictly positive real number $C_{m,n,s}$ such that for all $t>0$

$$\begin{aligned} \int \Big |D^\mathbf{l }\big (D^\mathbf{k }u(t,q,p)\pi _s(q,p)\big )\Big |^2dqdp\le C_{m,n,s}\parallel \phi \parallel _{m+n,r_{m+n}}^2\exp (-\zeta t). \end{aligned}$$

Then, using Sobolev embedding theorem [2], we get, for $n=d+1$

$$\begin{aligned} |D^\mathbf{k }u(t,q,p)|^2\pi _s^2(q,p) \le C_{m,d+1,s}\parallel \phi \parallel _{m+d+1,r_{m+d+1}}^2\exp (-\zeta t), \end{aligned}$$

for all $t>0$ and $(q,p)\in \mathbb {R}^{2d}$. The conclusion follows.$\square$

1.4 Proof of the Lemma 2.12

The two following results are consequences of Proposition 6.1:

Corollary 6.21

Let $g\in \fancyscript{C}^{\infty }_{pol}(\mathbb {R}^{2d})$ such that $\int _{\mathbb {R}^{2d}}g(q,p)\rho (q,p)dqdp=0$, then there exists a unique function $h\in \fancyscript{C}^{\infty }_{pol}(\mathbb {R}^{2d})$ such that

$$\begin{aligned} Lh=g\quad \text {and }\;\int _{\mathbb {R}^{2d}}h(q,p)\rho (q,p)dqdp=0. \end{aligned}$$

Proof

It is know that the unique solution $h$ of $Lh=g$ which verifies $\int _{\mathbb {R}^{2d}}h(q,p)\rho (q,p)dqdp=0$ is defined by

$$\begin{aligned} h(q,p)=\int _0^{+\infty }\mathbb {E}(g(q_q(t),p_p(t)))dt. \end{aligned}$$

Then the regularity of $h$ is a consequence of Proposition 6.1.

Remark 6.22

$L$ and its formal adjoint in $L^2(\rho )$ have the same behavior. Indeed, we have for any function $\phi \in C^{\infty }(\mathbb {R}^{2d})$ and $(q,p)\in \mathbb {R}^{2d}$

$$\begin{aligned} L^*(q,p;\partial _p,\partial _q)\phi (q,p)&= -\langle p,\partial _q\phi (q,p)\rangle +\langle \partial _qV,\partial _p\phi (q,p)\rangle -\gamma \langle p,\partial _p\phi (q,p)\rangle \\&+\frac{\sigma ^2}{2}\sum _{i=1}^d\frac{\partial ^2}{\partial _{p_i}\partial _{p_i}}\phi (q,p)\\&= L(q,-p;\partial _p,\partial _q)\phi (q,-p). \end{aligned}$$

Then we have the Lemma 2.12:

Let $g\in \fancyscript{C}^{\infty }_{pol}(\mathbb {R}^{2d})$ such that $\int _{\mathbb {R}^{2d}}g(q,p)\rho (q,p)dqdp=0$. Then there exists a unique function $h\in \fancyscript{C}^{\infty }_{pol}(\mathbb {R}^{2d})$ such that

$$\begin{aligned} L^*h=g\quad \text {and }\;\int _{\mathbb {R}^{2d}}h(q,p)\rho (q,p)dqdp=0, \end{aligned}$$

where $L^*$ is the adjoint of $L$ with respect to the $L^2(\rho )$ product.

Appendix B: Proof of Proposition 3.2

1.1 The case of the implicit split-step scheme (2.9)

Let $0<\delta <\delta _0$ be fixed. Let us recall that we have

$$\begin{aligned} q_1&= q_0+\delta p^*_0\\ p^*_0&= p_0-\delta \gamma p^*_0-\delta \partial _qV(q_1)\\ p_1&= p^*_0+\sqrt{\delta }\sigma \eta _0. \end{aligned}$$

Before proving Proposition 3.2, we need an asymptotic expansion for $q_1$, $p_1$ and $p_0^*$.

We define the function $\varPsi _{\delta }$ which associates to $(q,p)$ the solution $z$ of $(1+\gamma \delta )z=(1+\gamma \delta )q+\delta p-\delta ^2 \partial _qV(z)$. As the implicit split-step scheme (2.9) is well-defined (see [17]), we also have that the function $\varPsi _{\delta }$ is well-defined. We can remark that $q_1=\varPsi _{\delta }(q_0,p_0)$. Moreover, we have that $(\delta ,q,p)\mapsto \varPsi _{\delta }(q,p)$ is $C^{\infty }$ on $]0,\frac{\gamma +\sqrt{\gamma ^2+4\theta } }{2\theta }[\times \mathbb {R}^{d}\times \mathbb {R}^{d}$: Let $\varOmega _1=]0,\frac{\gamma +\sqrt{\gamma ^2+4\theta } }{2\theta }[\times \mathbb {R}^{d}\times \mathbb {R}^{d}\times \mathbb {R}^{d}$ and the function $f\in C^{\infty }$ defined on $\varOmega _1$ by

$$\begin{aligned} f(\delta ,q,p,z)=-(1+\delta \gamma )z+ (1+\delta \gamma )q+\delta p-\delta ^2 \partial _qV(z). \end{aligned}$$

Using the semi-convexity of $V$, we have that, for all $(\delta ,q,p,z)\in \varOmega _1$, $\partial _zf(\delta ,q,p,z)$ is invertible. By implicit function Theorem, we obtain that the function defined by $(\delta ,q,p)\mapsto \varPsi _{\delta }(q,p)=z$ is $C^{\infty }$ on a neighborhood of each point of $]0,\frac{\gamma +\sqrt{\gamma ^2+4\theta } }{2\theta }[\times \mathbb {R}^{d}\times \mathbb {R}^{d}$.

We have the following asymptotic expansion:

Lemma 7.1

Let $\delta _0=\min (\frac{1}{\gamma },\frac{ \gamma \beta }{4\theta })$ and $(q,p)\in \mathbb {R}^{2d}$ such that $q_0=q$ and $p_0=p$ be fixed. We have for $0<\delta <\delta _0$ and $N\in \mathbb {N}$,

$$\begin{aligned} q_1=\varPsi _{\delta }(q,p)=q+\sum _{k=1}^N\delta ^kd_k(q,p)+\delta ^{N+1}R_{N+1}(q,p,\delta ), \end{aligned}$$

(7.1)

where, for all $k\ge 0$, $d_k\in \fancyscript{C}^{\infty }_{pol}(\mathbb {R}^{2d})$ is defined for all $(x,y)\in \mathbb {R}^{2d}$ by

$$\begin{aligned} d_1(x,y)&= y,\text { and for}\,\, k\ge 2\\ d_{k}(x,y)&= (-1)^{k-1}\gamma ^{k-2}\big (\gamma y+\partial _qV(x)\big ) +\sum _{j=2}^{k-1}(-1)^{j-1}\gamma ^{j-2}\sum _{n=1}^{k-j}\frac{1}{n!}\sum \limits _{\begin{array}{c} k_1+ \cdots +k_n=k-j-n,\\ 0\le k_i\le N \end{array}}\\&\times \partial _q^{n+1}V(x)(d_{k_1+1}(x,y),\ldots ,d_{k_n+1}(x,y)). \end{aligned}$$

and $R_{N+1}$ verifies: There exist $C>0$ and $\ell _N\in \mathbb {N}$ such that for any $(x,y)\in \mathbb {R}^{2d}$ and $\delta <\delta _0$

$$\begin{aligned} |R_{N+1}(x,y,\delta )|\le C(1+|x|^{\ell _N}+|y|^{\ell _N}). \end{aligned}$$

(7.2)

Proof

Let $0<\delta <\delta _0$. We have previously shown that $(\delta ,q,p)\mapsto \varPsi _{\delta }(q,p)$ is $C^{\infty }$ on $]0,\frac{\gamma +\sqrt{\gamma ^2+4\theta } }{2\theta }[\times \mathbb {R}^{2d}$. Then, for $(q,p)\in \mathbb {R}^{2d}$ fixed, we have that $d_k(q,p)$ is the $k^{th}$ term of the Taylor expansion of $\delta \mapsto \varPsi _{\delta }(q,p)$ and we can write (7.1). We now search an expression for $d_k$.

Let $(q,p)\in \mathbb {R}^{2d}$ such that $q_0=q$ and $p_0=p$. Let $\ell \in \mathbb {N}$. We use the temporary notation, for all $0\le k\le \ell -1$, $g_{k,\ell }=d_{k+1}$ and $g_{\ell ,\ell }=R_{\ell +1}$. Hence, we have

$$\begin{aligned} q_1&= q+\sum _{k=1}^{\ell }\delta ^kd_k(q,p)+\delta ^{\ell +1}R_{\ell +1}(q,p,\delta )\\&= q+\delta z_{1,\ell }, \end{aligned}$$

where $z_{1,\ell }=\sum _{k=0}^{\ell }g_{k,\ell }(q,p)$. Using Taylor expansion, we obtain

$$\begin{aligned} \partial _qV(q_1)&= \partial _{q}V(q+\delta z_{1,l})=\partial _qV(q)+\sum _{n=1}^{\ell }\frac{1}{n!}\partial _q^{n+1}V(q)(\delta z_{1,l},\ldots ,\delta z_{1,l})+\delta ^{\ell +1}\theta _{\ell }(q,p)\\&= \partial _qV(q)+\sum _{n=1}^{\ell }\frac{1}{n!}\delta ^n\partial _q^{n+1}V(q)(\sum _{k=0}^{\ell }\delta ^kg_{k,\ell }(q,p),\ldots ,\sum _{k=0}^{\ell }\delta ^kg_{k,\ell }(q,p))+\delta ^{\ell +1}\theta _{\ell }(q,p)\\&= \partial _qV(q)+\sum _{n=1}^{\ell }\delta ^n\frac{1}{n!}\sum _{m= 0}^{n\ell }\delta ^m\sum \limits _{\begin{array}{c} k_1+\cdots +k_n=m,\\ 0\le k_i\le \ell \end{array}}\partial _q^{n+1}V(q)(g_{k_1,\ell }(q,p),\ldots ,g_{k_n,\ell }(q,p))\\&+\delta ^{\ell +1}\theta _{\ell }(q,p)\\&= \partial _qV(q)+I_{1,\ell }(q,p)+\delta ^{\ell +1}I_{2,\ell }(q,p)+\delta ^{\ell +1}\theta _{\ell }(q,p), \end{aligned}$$

where

$$\begin{aligned} \theta _{\ell }(q,p)&= \int _0^1\frac{(1-t)^{\ell }}{\ell !}\partial _q^{\ell +2}V(q+t\delta z_1)(z_1,\ldots ,z_1)dt,\\ I_{1,\ell }(q,p)&= \sum _{n=1}^{\ell }\frac{1}{n!}\delta ^n\sum _{m= 0}^{j-n}\delta ^m\sum \limits _{\begin{array}{c} k_1+\cdots +k_n=m,\\ 0\le k_i\le \ell \end{array}}\partial _q^{n+1}V(q)(g_{k_1,\ell }(q,p),\ldots ,g_{k_n,\ell }(q,p)),\\&= \sum _{j=1}^{\ell }\delta ^j\sum _{n=1}^{j}\frac{1}{n!}\sum \limits _{\begin{array}{c} k_1+\cdots +k_n=\ell -n,\\ 0\le k_i\le \ell -1 \end{array}}\partial _q^{n+1}V(q)(d_{k_1+1}(q,p),\ldots ,d_{k_n+1}(q,p)),\\ I_{2,\ell }(q,p)&= \sum _{n=1}^{\ell }\frac{1}{n!}\delta ^n\sum _{m= \ell -n+1}^{n\ell }\delta ^{m-\ell -1}\sum \limits _{\begin{array}{c} k_1+\cdots +k_n=m,\\ 0\le k_i\le \ell \end{array}}\partial _q^{n+1}V(q)(g_{k_1,\ell }(q,p),\ldots ,g_{k_n,\ell }(q,p)). \end{aligned}$$

Let $N$ be fixed, using the above computations, we have

$$\begin{aligned} q_1&= q+(1+\gamma \delta )^{-1}\big (\delta p-\delta ^2 \partial _qV(q_1)\big )\nonumber \\&= q+\sum _{k=0}^{N-1}(-\gamma )^k\delta ^{k+1} p+\delta ^{N+1}g(q,p)\nonumber \\&+\sum _{k=0}^{N-2}(-\gamma )^k\delta ^{k+2}\Big (\partial _qV(q)-\delta ^2I_{1,N-k-2}(q,p)\nonumber \\&-\delta ^{N-k-1}\big (I_{2,N-k-2}(q,p)+\theta _{N-k-2}(q,p)\big )\Big )\nonumber \\&= q+\delta p+\sum _{k=2}^{N}(-1)^{k-1}\gamma ^{k-2}\delta ^k \big (\gamma p+\partial _qV(q)\big )+J(q,p)+\delta ^{N+1}G(q,p),\nonumber \\ \end{aligned}$$

(7.3)

where

$$\begin{aligned} J(q,p)&= -\sum _{k=0}^{N-2}(-\gamma )^{k}I_{1,N-k-2}(q,p)\delta ^{k+2}\\&= \sum _{k=2}^N(-1)^{k-1}\gamma ^{k-2}\sum _{j=1}^{N-k}\delta ^{j+k}\sum _{n=1}^j\frac{1}{n!}\\&\times \sum \limits _{\begin{array}{c} k_1+\cdots +k_n=j-n,\\ 0\le k_i\le N \end{array}}\partial _q^{n+1}V(q)(d_{k_1+1}(q,p),\ldots ,d_{k_n+1}(q,p))\\&= \sum _{k=3}^N\delta ^k\sum _{j=2}^{k-1}(-1)^{j-1}\gamma ^{j-2}\sum _{n=1}^{k-j}\frac{1}{n!}\\&\times \sum \limits _{\begin{array}{c} k_1+\cdots +k_n=k-j-n,\\ 0\le k_i\le N \end{array}}\partial _q^{n+1}V(q)(d_{k_1+1}(q,p),\ldots ,d_{k_n+1}(q,p)),\\ G(q,p)&= g(q,p)-\sum _{k=0}^{N-1}(-\gamma )^kI_{2,N-k-2}(q,p),\\ g(q,p)&= -\frac{(-\gamma )^{N-1}}{1+\delta \gamma }(\gamma p+ \partial _qV(q_1)). \end{aligned}$$

Identifying the terms in the two expansions (7.3) and (7.1), we get

$$\begin{aligned} q_1=q+\sum _{k=1}^N\delta ^kd_k(q,p)+\delta ^{N+1}R_{N+1}(\delta ,q,p), \end{aligned}$$

where, for all $k\ge 0$, $d_k$ is defined for all $(q,p)\in \mathbb {R}^{2d}$ by

$$\begin{aligned} d_1(q,p)&= p, \text { and for} k\ge 2,\\ d_{k}(q,p)&= (-1)^{k-1}\gamma ^{k-2}\big (\gamma p+\partial _qV(q)\big )+\sum _{j=2}^{k-1}(-1)^{j-1}\gamma ^{j-2}\\&\times \sum _{n=1}^{k-j}\frac{1}{n!}\sum \limits _{\begin{array}{c} k_1+\cdots +k_n=k-j-n,\\ 0\le k_i \end{array}}\partial _q^{n+1}V(q)(d_{k_1+1}(q,p),\ldots ,d_{k_n+1}(q,p)), \end{aligned}$$

moreover, by induction, we have, for all $k\in \mathbb {N}$, $d_k\in \fancyscript{C}^{\infty }_{pol}(\mathbb {R}^{2d})$.

The above identification does not give an easy expression of $R_{N}$, then we have not immediately (7.2). To show this result, we will use that, for $N$, $q\in \mathbb {R}^{d}$ and $p\in \mathbb {R}^{d}$ fixed, $R_N(q,p,.)$ is the remainder of order $N$ of $\delta \mapsto \varPsi _{\delta }(q,p)$. Therefore, if we show that, for any $n\in \mathbb {N}$, there exist $C>0$ and $\ell _n\in \mathbb {N}$ such that for any $0<\delta <\delta _0$ and $(q,p)\in \mathbb {R}^{2d}$,

$$\begin{aligned} |\partial _{\delta }^n\varPsi _{\delta }(q,p)|^2\le C (1+|q|^{\ell _n}+|p|^{\ell _n}), \end{aligned}$$

(7.4)

then we show (7.2) and Lemma 7.1 is shown.

Let us show the result (7.4) by induction on $n$. Let $0<\delta <\delta _0$ and $(q,p)\in \mathbb {R}^{2d}$ be fixed. We have, by definition of $\varPsi _{\delta }$,

$$\begin{aligned} \varPsi _{\delta }(q,p)(1+\delta \gamma )+\delta ^2 \partial _qV(\varPsi _{\delta }(q,p)) =\delta p+(1+\delta \gamma )q. \end{aligned}$$

We multiply this equation by $\varPsi _{\delta }(q,p)$ and use dissipativity inequality (2.3). We get

$$\begin{aligned}&\langle p,\varPsi _{\delta }(q,p)\rangle \delta +\langle q ,\varPsi _{\delta }(q,p)\rangle (1+\delta \gamma )\\&\quad =(1+\delta \gamma )|\varPsi _{\delta }(q,p)|^2+\delta ^2\langle \partial _qV(\varPsi _{\delta }(q,p)),\varPsi _{\delta }(q,p)\rangle \\&\quad \ge (1+\delta \gamma +\beta _1\delta ^2)|\varPsi _{\delta }(q,p)|^2-\kappa \delta ^2. \end{aligned}$$

Since

$$\begin{aligned} 2\langle a,b\rangle \le \varepsilon |a|^2+\frac{1}{\varepsilon }|b|^2\quad \text {for all} \varepsilon >0, a\in \mathbb {R}^{d}\hbox {and } b\in \mathbb {R}^{d}, \end{aligned}$$

(7.5)

we get, for any positive constants $\varepsilon _1$ and $\varepsilon _2$:

$$\begin{aligned} |\varPsi _{\delta }(q,p)|^2&\Big [1+\delta \gamma +\beta _1\delta ^2-\frac{\varepsilon _1}{2}-\frac{\varepsilon _2}{2}\Big ]\le \kappa \delta ^2+\frac{\delta }{2\varepsilon _2}|p|^2+\frac{1+\delta \gamma }{2\varepsilon _1}|q|^2. \end{aligned}$$

We choose $\varepsilon _1=\varepsilon _2=\frac{1}{2}$, then we have that there exists a positive constant $C$, which is independent of $\delta$, such that

$$\begin{aligned} |\varPsi _{\delta }(q,p)|^2\le C(1+|q|^2+|p|^2). \end{aligned}$$

This proves (7.4) for $n=0$ and $\ell _0=2$.

Let us assume the result (7.4) is true for all $0\le j<n$ and let us show it for $n$. Let $0<\delta <\delta _0$ and $(q,p)\in \mathbb {R}^{2d}$ be fixed. We have

$$\begin{aligned} (1+\delta \gamma )\partial ^n_{\delta }\varPsi _{\delta }(q,p)&= -n\gamma \partial ^{n-1}_{\delta }\varPsi _{\delta }(q,p)+(\gamma q+p)\mathbf {1}_{\{n=1\}}\\&-2n\partial ^{n-1}_{\delta }\Big (\partial _qV(\varPsi _{\delta }(q,p))\Big )-\delta ^2\partial _{\delta }^n\Big (\partial _qV(\varPsi _{\delta }(q,p))\Big )\\&-2\left( {\begin{array}{c}n\\ 2\end{array}}\right) \delta \partial _{\delta }^{n-2}\Big (\partial _qV(\psi _{\delta }(q,p))\Big )\\&=: B_1(\delta ,q,p)-2nB_2(\delta ,q,p)-\delta ^2 B_3(\delta ,q,p)-\delta B_4(\delta ,q,p). \end{aligned}$$

By induction hypothesis, we have that $B_1$ has polynomial growth in $(q,p)$. Moreover, using Faà di Bruno’s formula, we get

$$\begin{aligned} B_2(\delta ,q,p)&= \sum \frac{(n-1)!}{m_1!m_2!(2!)^{m_2}\ldots m_{n-1}!((n-1)!)^{m_{n-1}}}\partial _q^{m_1+\cdots +m_{n-1}+1}V(\varPsi _{\delta }(q,p))\\&\times \prod _{j=1}^{n-1}\big (\partial _{\delta }^j\varPsi _{\delta }(q,p)\big )^{m_j}, \end{aligned}$$

where $m_1+2m_2+\cdots +(n-1)m_{n-1}=n-1$. Using the polynomial growth of $V$ and its derivatives (B-1) and induction hypothesis, we have that $B_2$ has polynomial growth in $(q,p)$. Using the same method, we have that $B_4$ has polynomial growth in $(q,p)$. Using Faà di Bruno’s formula, we also have

$$\begin{aligned} B_3(\delta ,q,p)&= \sum \frac{n!}{m_1!m_2!(2!)^{m_2}\ldots m_{n}!(n!)^{m_{n}}}\partial _q^{m_1+\cdots +m_{n}+1}V(\varPsi _{\delta }(q,p))\\&\times \prod _{j=1}^{n}\big (\partial _{\delta }^j\varPsi _{\delta }(q,p)\big )^{m_j}\\&= \delta \partial _q^2V(q+\delta \varPsi _{\delta }(q,p))\partial _{\delta }^{n}\varPsi _{\delta }(q,p)+ B_5(\delta ,q,p), \end{aligned}$$

where $m_1+2m_2+\cdots +nm_{n}=n$ and

$$\begin{aligned} B_5(\delta ,q,p)&= \sum \frac{n!}{k_1!k_2!(2!)^{k_2}\ldots k_{n-1}!((n-1)!)^{k_{n-1}}}\partial _q^{k_1+\cdots +k_{n-1}+1}V(\varPsi _{\delta }(q,p))\\&\times \prod _{j=1}^{n-1}\big (\partial _{\delta }^j\varPsi _{\delta }(q,p)\big )^{k_j},k_1+2k_2+\cdots +(n-1)k_{n-1}=n. \end{aligned}$$

Then we have that $B_1$, $B_2$, $B_4$ and $B_5$ have polynomial growth in $(q,p)$ and we get

$$\begin{aligned} \big ((1+\delta \gamma )I+\delta ^2 \partial _q^2V(q+\delta \varPsi _{\delta }(q,p))\big )\partial ^n_{\delta }\varPsi _{\delta }(q,p)=B_6(\delta ,q,p) \end{aligned}$$

where $B_6:=-n\gamma B_1-2nB_2-\delta B_4-\delta ^2 B_5$ has polynomial growth in $(q,p)$.

Multiplying by $\delta ^n_{\delta }\varPsi _{\delta }(q,p)$, and using (7.5) on $|\langle B_6(\delta ,q,p),\partial ^n_{\delta }\varPsi _{\delta }(q,p)\rangle |$ and semi-convexity assumption B-3, we get

$$\begin{aligned} |\partial ^n_{\delta }\varPsi _{\delta }(q,p)|^2(1+\gamma \delta -\theta \delta ^2-\frac{1}{2})\le \frac{1}{2}B_6^2(\delta ,q,p), \end{aligned}$$

where $\theta$ is the constant of semi-convexity and $B_6$ has polynomial growth in $(q,p)$. Since $\delta <\frac{\gamma }{\theta }$, we have that there exist constants $C$ and $\ell _n\in \mathbb {N}$ such that

$$\begin{aligned} |\partial _{\delta }^n\varPsi _{\delta }(q,p)|^2\le C (1+|q|^{\ell _n}+|p|^{\ell _n}), \end{aligned}$$

which proves (7.4).$\square$

Corollary 7.2

Let $\delta _0=\min (\frac{1}{\gamma },\frac{ \gamma \beta }{4\theta })$ and $(q^0,p^0)\in \mathbb {R}^{2d}$ such that $q_0=q^0$ and $p_0=p^0$ be fixed. We have for $0<\delta <\delta _0$ and $N\in \mathbb {N}$,

$$\begin{aligned}q_1&=q^0+\sum _{k=1}^N\delta ^kd_{k}(q^0,p^0)+\delta ^{N+1}R_{N+1}(q^0,p^0,\delta ),\\ p_1&=\sum _{k=0}^N\delta ^kd_{k+1}(q^0,p^0)+\delta ^{N+1}R_{N+2}(q^0,p^0,\delta )+\sqrt{\delta }\sigma \eta _0 \end{aligned}$$

where, for all $k\ge 0$, $d_k\in \fancyscript{C}^{\infty }_{pol}(\mathbb {R}^{2d})$ is defined for all $(x,y)\in \mathbb {R}^{2d}$ by

$$\begin{aligned} d_1(x,y)&= y,\text { and for} k\ge 2\\ d_{k}(x,y)&= (-1)^{k-1}\gamma ^{k-2}\big (\gamma y+\partial _qV(x)\big )\\&+\sum _{j=2}^{k-1}(-1)^{j-1}\gamma ^{j-2}\sum _{n=1}^{k-j}\frac{1}{n!}\\&\times \sum \limits _{\begin{array}{c} k_1+\cdots +k_n=k-j-n,\\ 0\le k_i \end{array}}\partial _q^{n+1}V(x)(d_{k_1+1}(x,y),\ldots ,d_{k_n+1}(x,y)). \end{aligned}$$

and $R_{N+1}$ verifies: There exist $C>0$ and $\ell _N\in \mathbb {N}$ such that for any $(x,y)\in \mathbb {R}^{2d}$ and $\delta <\delta _0$

$$\begin{aligned} |R_{N+1}(x,y,\delta )|\le C(1+|x|^{\ell _N}+|y|^{\ell _N}). \end{aligned}$$

Proof

[Proof of Proposition 3.2 in the case of the implicit split-step scheme (2.9)] Let $N$ fixed. We have, with the notation of Corollary 7.2, for all $N_0\in \mathbb {N}$

$$\begin{aligned} q_1&= q+\sum _{k=1}^{N_0}\delta ^k d_{k}(q,p)+\delta ^{N_0+1} R_{N_0+1}(q,p,\delta )=q+\delta R_{1}(q,p,\delta ),\\ p_1&= \sum _{k=0}^{N_0}\delta ^kd_{k+1}(q,p)+\delta ^{N_0+1}\mathbb {R}_{N_0+2}(q,p,\delta )+\sqrt{\delta }\sigma \eta _0\\&= p+\sqrt{\delta }\sigma \eta _0+\delta R_2(q,p,\delta )=z+\sqrt{\delta }\sigma \eta _0. \end{aligned}$$

Let $\phi \in \fancyscript{C}^{\infty }_{pol}(\mathbb {R}^{2d})$, for any $n\in \mathbb {N}$, there exists an integer $r_{n}$ such that $\phi \in \fancyscript{C}^{n}_{r_n}(\mathbb {R}^{2d})$. Using Taylor expansion, we get

$$\begin{aligned} \phi (q_1,p_1)&= \phi (q_1,z+\sqrt{\delta }\sigma \eta _0)\\&= \phi (q_1,z)+\sum _{k=1}^{2N+1}\frac{1}{k!}\delta ^{k/2}\sigma ^{k}\partial _p^k\phi (q_1,z)(\eta _0,\ldots \eta _0)\\&+\int _0^1\frac{(1-t)^{2N+1}}{(2N+1)!}\delta ^{N+1}\sigma ^{2N+2}\partial _p^{2N+2}\phi (q_1,z+t\sqrt{\delta }\sigma \eta _0)(\eta _0,\ldots \eta _0)dt. \end{aligned}$$

Let $\lfloor .\rfloor$ denotes the integer part. Using Taylor expansion on $\partial _p^k\phi (q_1,z)$ and computations done in the proof of Lemma 7.1, we obtain

$$\begin{aligned} \phi (q_1,p_1)=I_1(x,\eta _0)+I_2(x,\eta _0)+I_3(x,\eta _0)+I_4(x,\eta _0), \end{aligned}$$

where $x=(q,p)$ and

$$\begin{aligned} I_1(x,\eta _0)&= \sum _{k=0}^{2N+1}\frac{1}{k!}\sigma ^k\Big (\delta ^{k/2}\partial ^k_p\phi (x)(\eta _0,\ldots ,\eta _0)+\sum _{n=1}^{\ell _k}\frac{1}{n!}\sum _{\ell =0}^{n}\left( {\begin{array}{c}n\\ \ell \end{array}}\right) \sum _{m=n}^{\ell _k}\delta ^{m+k/2}\\&\times \sum \limits _{\begin{array}{c} k_1+\cdots +k_{\ell }\\ +\tilde{k}_1+\cdots +\tilde{k}_{n-\ell }=m \\ 0<k_i\le \ell _k\\ 0<\tilde{k}_i\le \ell _k \end{array}}\partial _q^{\ell }\partial _p^{n-\ell +k}\phi (x)(d_{k_1}(x),\ldots ,d_{k_{\ell }}(x),\\&\times d_{\tilde{k}_1+1}(x),\ldots ,d_{\tilde{k}_{n-\ell }+1}(x),\eta _0,\ldots ,\eta _0)\Big ),\\ I_2(x,\eta _0)&= \sum _{k=0}^{2N+1}\frac{1}{k!}\sigma ^k\sum _{n=0}^{\ell _k}\frac{1}{n!}\sum _{\ell =0}^{n}\left( {\begin{array}{c}n\\ \ell \end{array}}\right) \sum _{m=\ell _k+1}^{n(\ell _k+1)}\delta ^{m+k/2} B_{m,n,\ell ,k}(x_1,\eta _0) ,\\ I_3(x,\eta _0)&= \sum _{k=0}^{2N+1}\frac{1}{k!}\sigma ^k\delta ^{k/2}\delta ^{\ell _k+1}\int _0^1\frac{(1-t)^{\ell _k}}{(l_k)!}\sum _{\ell =0}^{\ell _k+1}\left( {\begin{array}{c}\ell _k+1\\ \ell \end{array}}\right) \partial _q^{\ell }\partial _p^{\ell _k+1-\ell +k}\\&\times \phi (q+t\delta R_{1}(x,\delta ),p+t\delta R_{2}(x,\delta ))\\&\times (R_{1}(x,\delta ),\ldots ,R_{1}(x,\delta ),R_{2}(x,\delta ),\ldots ,R_{2}(x,\delta ),\eta _0,\ldots ,\eta _0)dt,\\ I_4(x,\eta _0)&= \delta ^{N+1}\sigma ^{2N+2}\int _0^1\frac{(1-t)^{2N+1}}{(2N+1)!}\partial _p^{2N+2}\\&\times \phi (q+\delta R_{1}(x,\delta ),z+t\sqrt{\delta }\eta _0)(\eta _0,\ldots ,\eta _0)dt,\\ \ell _k&= N-\Big \lfloor \frac{k+1}{2}\Big \rfloor \end{aligned}$$

and, with the temporary notation: for all $0\le k\le 2N+1$ and $1\le i\le \ell _k$, $\tilde{g}_{k,i}=d_{i+1}$, $g_{k,i}=d_{i}$, $\tilde{g}_{k,\ell _k+1}=R_{\ell _k+2}$ and $g_{k,\ell _k+1}=R_{\ell _k+1}$,

$$\begin{aligned} B_{m,n,\ell ,k}(x,\eta _0)&= \sum \limits _{\begin{array}{c} k_1+\cdots +k_{\ell }\\ +\tilde{k}_1+\cdots +\tilde{k}_{n-\ell }=m \\ 0<k_i\le \ell _k+1\\ 0<\tilde{k}_i\le \ell _k+1 \end{array}}\partial _q^{\ell }\partial _p^{n-\ell +k}\phi (x)(g_{k_1}(x),\ldots ,g_{k_{\ell }}(x),\\&\times \tilde{g}_{\tilde{k}_1}(x),\ldots ,\tilde{g}_{\tilde{k}_{n-\ell }}(x),\eta _0,\ldots ,\eta _0). \end{aligned}$$

We have, for all $\ell \in \mathbb {N}$, $\mathbb {E}(\eta _{0,i}^{2\ell +1})=0$ and $\eta _{0,i}$ are independent of $q$, $p$ and $\eta _{0,j}$ for $j\ne i$, then the expectation of all the odd term in $k$ in $I_1$, $I_2$ and $I_3$ vanish. Hence we have

$$\begin{aligned} \mathbb {E}(I_2(x,\eta _0))&= \sum _{k=0}^{N}\frac{1}{(2k)!}\sigma ^{2k}\sum _{n=0}^{N-k}\frac{1}{n!}\sum _{\ell =0}^{n}\left( {\begin{array}{c}n\\ \ell \end{array}}\right) \sum _{m=N-k+1}^{n(N-k+1)}\delta ^{m+k} \mathbb {E}(B_{m,n,\ell ,2k}(x,\eta _0)),\\ \mathbb {E}(I_3(x,\eta _0))&= \sum _{k=0}^{N}\frac{1}{(2k)!}\delta ^{N+1}\sigma ^{2k}\int _0^1\frac{(1-t)^{N-k}}{(N-k)!}\sum _{\ell =0}^{N-k+1}\left( {\begin{array}{c}N-k+1\\ \ell \end{array}}\right) \\&\times \mathbb {E}\Big (\partial _q^{\ell }\partial _p^{N-k+1-\ell +k}\phi (q+t\delta R_{1}(x,\delta ),p+t\delta R_{2}(x,\delta ))\\&\times (R_{1}(x,\delta ),\ldots ,R_{1}(x,\delta ),R_{2}(x,\delta ),\ldots ,R_{2}(x,\delta ),\eta _0,\ldots ,\eta _0)\Big )dt. \end{aligned}$$

In $\mathbb {E}(I_2(x,\eta _0))$, we have $m+k\ge N+1$, then we can factor $\delta ^{N+1}$. Moreover, if $n=\ell =0$ then $B_{m,n,\ell ,2k}=0$, hence, in each term of $\mathbb {E}(I_2(x,\eta _0))$, we have at least one derivative of $\phi$. Using $\phi \in \fancyscript{C}^{2N+2}_{r_{2N+2}}(\mathbb {R}^{2d})$ and the polynomial growth of $d_j$, $\tilde{R}_{N_0+1}$ and $R_{N_0+1}$ for $j\in \mathbb {N}^*$ and $N_0\in \mathbb {N}$, we get that there exist integers $n_1$, $n_2$ and $n_3$ such that

$$\begin{aligned} | \mathbb {E}(I_2(q,p,\eta _0))|&\le C_N\delta ^{N+1}(1+|p|^{n_1}+|q|^{n_1})|\phi |_{2N,r_{2N}}\!,\\ | \mathbb {E}(I_4(q,p,\eta _0))|&\le C_N\delta ^{N+1}(1+|p|^{n_2}+|q|^{n_2})\parallel D^{2N+2}\phi \parallel _{0,r_{2N+2}}\\ | \mathbb {E}(I_3(q,p,\eta _0))|&\le C_N\delta ^{N+1}(1+|p|^{n_3}+|q|^{n_3})\parallel D^N\phi \parallel _{N+1,r_{2N+1}}\!. \end{aligned}$$

Hence, we have that there exists $k_1\in \mathbb {N}^*$ such that

$$\begin{aligned} | \mathbb {E}\phi (q_1,p_1)-\mathbb {E}(I_1)|\le C_N\delta ^{N+1}(1+|p|^{k_1}+|q|^{k_1})|\phi |_{2N+2,r_{2N+2}}, \end{aligned}$$

where

$$\begin{aligned} \mathbb {E}(I_1)&= \sum _{j=0}^{N}\frac{1}{(2j)!}\sigma ^{2j}\delta ^j\mathbb {E}\Big (\partial ^{2j}_p\phi (q,p)(\eta _0,\ldots ,\eta _0)\Big )\\&+\sum _{j=0}^N\frac{1}{(2j)!}\sigma ^{2j}\sum _{n=1}^{N-j}\frac{1}{n!}\sum _{\ell =0}^n\left( {\begin{array}{c}n\\ \ell \end{array}}\right) \sum _{m=n}^{N-j}\delta ^{m+j}\sum \limits _{\begin{array}{c} k_1+\cdots +k_{\ell }\\ +\tilde{k}_1+\cdots +\tilde{k}_{n-\ell }=m \\ \tilde{k}_i, k_i\ge 1 \end{array}}\mathbb {E}\Big (\partial _q^{\ell }\partial _p^{n-{\ell }+2j}\phi (q,p)\\&\times (d_{k_1}(q,p),\ldots ,d_{k_{\ell }}(q,p),d_{\tilde{k}_1+1}(q,p),\ldots ,d_{\tilde{k}_{n-\ell }+1}(q,p),\eta _0,\ldots ,\eta _0)\Big ),\\&= \sum _{k=0}^{N}\delta ^k A_k(q,p)\phi (q,p), \end{aligned}$$

and

$$\begin{aligned} A_k(q,p)\phi (q,p)&= \frac{\sigma ^{2k}}{(2k)!}\mathbb {E}\big (\partial _p^{2k}\phi (q,p)(\eta _0,\ldots ,\eta _0)\big )\!+\!\sum _{j=0}^{k-1}\frac{1}{(2j)!}\sigma ^{2j}\sum _{n=1}^{k-j}\frac{1}{n!}\sum _{\ell =0}^n\left( {\begin{array}{c}n\\ \ell \end{array}}\right) \\&\times \sum \limits _{\begin{array}{c} k_1+\cdots +k_{\ell }\\ +\tilde{k}_1+\cdots +\tilde{k}_{n-\ell }=k-j \\ 0<k_i,\tilde{k}_i \end{array}}\mathbb {E}\Big (\partial _q^l\partial _p^{n-\ell +2j}\phi (q,p)(d_{k_1}(q,p),\ldots ,\\&\times d_{k_{\ell }}(q,p),d_{\tilde{k}_1+1}(q,p),\ldots ,d_{\tilde{k}_{n-\ell }+1}(q,p),\eta _0,\ldots ,\eta _0)\Big ) \end{aligned}$$

Using property of $\eta _0$, we get that $A_0=I$,

$$\begin{aligned} A_1\phi (q,p)=\frac{\sigma ^2}{2}\sum _{i=1}^d\frac{\partial ^2}{\partial _{p_i}\partial _{p_i}}\phi (q,p)+\langle \partial _p\phi ,d_2(q,p)\rangle +\langle \partial _q\phi (q,p),d_1(q,p)\rangle =L \end{aligned}$$

and $A_k$ is an operator of order $2k$. $\square$

1.2 The case of the implicit Euler scheme (2.10)

The proof of Propostition 3.2 in the case of the scheme (2.10) uses the same arguments than in the case of the scheme (2.9). Let $0<\delta <\delta _0=\min (\frac{1}{\gamma },\frac{\gamma \beta }{4\theta })$ be fixed. We need asymptotic expansions for $q_1=q+\delta p_1$ and $p_1=p-\delta \gamma p_1-\partial _qV(q_1)\delta +\sqrt{\delta }\sigma \eta _0$. We use the local notation $\alpha =\sqrt{\delta }$. We define the function $\psi _\alpha$ which associates to $(q,p)$ the solution $z$ of $(1+\gamma \alpha ^2)z=(1+\gamma \alpha ^2)q+\alpha ^2 p- \alpha ^4 \partial _qV(z)+\alpha \sigma \eta _0$. As the implicit Euler scheme (2.10) is well-defined (see [17]), we also have that the function $\psi _{\alpha }$ is well-defined. Moreover, using same arguments as for the scheme (2.9), we can show that $(\alpha ,q,p)\mapsto \psi _{\alpha }(q,p)$ is $C^\infty$ on $]0,\sqrt{\delta _0}[\times \mathbb {R}^{d}\times \mathbb {R}^{d}$.

We have the following lemma:

Lemma 7.3

Let $\delta _0=\min (\frac{1}{\gamma },\frac{ \gamma \beta }{4\theta })$ and $(q^0,p^0)\in \mathbb {R}^{2d}$ such that $q_0=q^0$ and $p_0=p^0$ be fixed. For $0<\delta <\delta _0$ and the local notation $\alpha =\sqrt{\delta }$, we have

$$\begin{aligned} \forall N_0\in \mathbb {N}, \quad&q_1=\psi _{\alpha }(q^0,p^0)=q^0+\sum _{k=2}^{2N_0+1}\delta ^{\frac{k}{2}} d_k(q^0,p^0,\eta _0)\\&\qquad \quad +\delta ^{N_0+1} R_{N_0+1}(q^0,p^0,\delta ,\eta _0),\\&p_1=\sum _{k=0}^{2N_0+1}\delta ^{\frac{k}{2}} d_{k+2}(q,p,\eta _0)+\delta ^{N_0+1} R_{N_0+2}(q^0,p^0,\delta ,\eta _0), \end{aligned}$$

where, $\forall k\ge 2$, $d_k$ is defined for all $(q,p)\in \mathbb {R}^{2d}$ by

$$\begin{aligned} d_2(q,p,\eta _0)=p, d_3(q,p,\eta _0)=\sigma \eta _0 \end{aligned}$$

and $\forall k\ge 2$

$$\begin{aligned} d_{2k}(q,p,\eta _0)=\,&(-1)^{k-1}\gamma ^{k-2}(\gamma p+\partial _qV(q)) +\sum _{i=2}^{k-1}(-1)^{i+1}\gamma ^i\\&\times \sum _{n=1}^{k-i}\frac{1}{n!}\sum \limits _{\begin{array}{c} k_1+\cdots +k_{n}=2(k-i)\\ k_s\ge 2 \end{array}} \partial _q^{n} V(q)(d_{k_1}(q,p,\eta _0),\ldots ,d_{k_i}(q,p,\eta _0)),\\ d_{2k+1}(q,p,\eta _0)=\,&(-1)^{k-1}\gamma ^{k-1}\sigma \eta _0 +\sum _{i=2}^{k}(-1)^{i+1}\gamma ^i\\&\times \sum _{n=1}^{k-i}\!\frac{1}{n!}\!\sum \limits _{\begin{array}{c} k_1+\cdots \!+\!k_{n}\!=\!2(k-i)+1\\ k_s\ge 2 \end{array}} \!\partial _q^{n} V(q)(d_{k_1}(q,p,\eta _0),\ldots ,d_{k_i}(q,p,\eta _0)). \end{aligned}$$

Moreover, we have that for any $k\ge 2$, $\mathbb {E}(d_k)\in \fancyscript{C}^{\infty }_{pol}(\mathbb {R}^{2d})$,

$$\begin{aligned} \mathbb {E}(d_{2k+1}(q,p,\eta _0))=0 \end{aligned}$$

(7.6)

and, for any $N\in \mathbb {N}$, $R_{N}$ verifies: There exist $C>0$ and $\ell _{N}\in \mathbb {N}$ such that for any $(x,y)\in \mathbb {R}^{2d}$ and $\delta <\delta _0$

$$\begin{aligned} |\mathbb {E}(R_{N}(q,p,\delta ,\eta _0))|\le C(1+|q|^{\ell _{N}}+|p|^{\ell _{N}}). \end{aligned}$$

Proof

To prove this Lemma, we use the same ideas as in Lemma 7.1. We first compute $d_k$ for all $k$. By induction, we rewrite $d_k$ only in terms of $d_1$, $d_2$ and the derivatives of $V$ evaluate in $q$. Using the independence of $\eta _0$ with $(q,p)$, we can show (7.6).

To prove that $\mathbb {E}(R_{N})$ has polynomial growth, we show that, for $n\in \mathbb {N}$, there exist $C_n>0$ and $k_n\in \mathbb {N}$ such that, for $(q,p)\in \mathbb {R}^{2d}$, $\delta <\delta _0$ and the local notation $\alpha =\sqrt{\delta }$, we have

$$\begin{aligned} \mathbb {E}(|\partial _{\alpha }^n\psi _{\alpha }(q,p)|^2)\le C_n(1+|q|^{k_n}+|p|^{k_n}). \end{aligned}$$

$\square$

The proof of Proposition 3.2 in the case of the scheme (2.10) is similar to the case of the scheme (2.9), but we must use an asymptotic expansion of $\partial _p^k\phi$ to a larger order ($2N+1-k$ instead of $N-\lfloor (k+1)/2\rfloor$).