Stochastic Wasserstein Hamiltonian Flows

Abstract

In this paper, we study the stochastic Hamiltonian flow in Wasserstein manifold, the probability density space equipped with $L^2$-Wasserstein metric tensor, via the Wong–Zakai approximation. We begin our investigation by showing that the stochastic Euler–Lagrange equation, regardless it is deduced from either the variational principle or particle dynamics, can be interpreted as the stochastic kinetic Hamiltonian flows in Wasserstein manifold. We further propose a novel variational formulation to derive more general stochastic Wasserstein Hamiltonian flows, and demonstrate that this new formulation is applicable to various systems including the stochastic Schrödinger equation, Schrödinger equation with random dispersion, and Schrödinger bridge problem with common noise.

A Appendix

Proof of Lemma 2.2

The local existence of (2.4) and (2.1) is ensured thanks to the local Lipschitz condition of f and $\sigma$. To obtain a global solution, a priori bound on $H_0(x,p)$ is needed. Denote the solutions of (2.1) and (2.4) with same initial condition $(x_0,p_0)$ by $(x_t^{\delta },p_t^{\delta }), \delta >0$ and $x_t^{0},p_t^{0}$, respectively. Applying the chain rule to $H_0(x_t^{\delta },p_t^{\delta })$ for (2.4) and (2.1), we get that

$$\begin{aligned} H_0(x_t^{\delta },p_t^{\delta })&=H_0(x_0,p_0)+\int _{0}^t \eta \nabla _p H_0(x_s^{\delta },p_s^{\delta }) \cdot \nabla _x \sigma (x_s) \dot{\xi }_{\delta }(s)ds\\ H_0(x_t,p_t)&=H_0(x_0,p_0)+\int _0^{\tau } \eta \nabla _p H_0(x_s,p_s) \cdot \nabla _x \sigma (x_s) dB_s\\&\quad +\frac{1}{2}\int _0^\tau \eta ^2 \nabla _{pp} H_0(x_s,p_s)\cdot (\nabla _x \sigma (x_s),\nabla \sigma (x_s))ds. \end{aligned}$$

By applying growth condition (2.3) and taking expectation on the second equation, we derive that

$$\begin{aligned} H_0(x_t^{\delta },p_t^{\delta })&\le (H_0(x_0,p_0)+\eta C_1T)\exp \Big (\int _0^tc_1\eta |\dot{\xi }_{\delta }(s)|ds\Big ),\\ {\mathbb {E}}\Big [H_0(x_t,p_t)\Big ]&\le \Big ({\mathbb {E}}\Big [H_0(x_0,p_0)\Big ]+\frac{\eta ^2}{2} C_1 T\Big ) \exp \Big (\int _0^{\tau }c_1 \frac{\eta ^2}{2} ds\Big ). \end{aligned}$$

The first inequality leads to $H_0(x_t^{\delta },p_t^{\delta })<\infty$ since $\dot{\xi }_{\delta }(s)=\frac{B_{t_{k+1}}-B_{t_k}}{\delta },$ if $s\in [t_k,t_{k+1}].$ Furthermore, taking expectation on the first inequality, applying Fernique’s theorem (see, e.g. [27]) for Gaussian variable and independent increments of $B_t$, we get that

$$\begin{aligned} {\mathbb {E}}\Big [H_0(x_t^{\delta },p_t^{\delta })\Big ]&\le C(T,\eta ,c_1)(2^{[\frac{t}{\delta }]}({\mathbb {E}}\Big [H_0(x_0,p_0)\Big ]+1), \end{aligned}$$

where [w] is the integer part of the real number w. The second inequality yield that $H_0(x_t,p_t)<\infty , a.s,$ and the global existence of the strong solution of (2.4). Similarly, for $p\ge 2$, we have that

$$\begin{aligned} {\mathbb {E}}\Big [H_0^p(x_t^{\delta },p_t^{\delta })\Big ]&\le C(T,\eta ,c_1,C_1, p)2^{p[\frac{t}{\delta }]}\Big ({\mathbb {E}}\Big [H_0^p(x_0,p_0)\Big ]+1\Big ),\\ {\mathbb {E}}\Big [H_0^p(x_t,p_t)\Big ]&\le C(T,\eta ,c_1,p)\Big ({\mathbb {E}}\Big [H_0^p(x_0,p_0)\Big ]+1\Big ). \end{aligned}$$

Furthermore, applying the above bounded moment estimate, we obtain that for $s\le t$,

$$\begin{aligned} {\mathbb {E}}\Big [|x(t)-x(s)|^{2p}+|p(t)-p(s)|^{2p}\Big ]&\le C(T,\eta ,c_1,C_1,c_0,C_1,p,x_0,p_0) |t-s|^p\\ {\mathbb {E}}\Big [|x^{\delta }(t)-x^{\delta }(s)|^{2p}+|p(t)-p(s)|^{2p}\Big ]&\le C(T,\eta ,c_1,C_1,c_0,C_1,p,x_0,p_0) 2^{[\frac{t}{\delta }]} |t-s|^p. \end{aligned}$$

However, the above estimate of $x^{\delta }$ is too rough and exponentially depending on $\frac{1}{\delta }$. As a consequence, we can not expect any convergence result. A delicate estimate of $(x^{\delta },p^{\delta })$ is needed.

Assume that $t\in [t_k,t_{k+1}],$ $t_k=k\delta .$ Then by using the expansion of (2.1), we have that

$$\begin{aligned} H_0(x_t^{\delta },p_t^{\delta })&=H_0(x_0,p_0)-\sum _{j=0}^{k-1}\int _{t_j}^{t_{j+1}}\eta \nabla _p H_0(x_s^{\delta },p_s^{\delta })\cdot \nabla _x \sigma (x_s^{\delta }) d\xi _{\delta }(s)~ \\&\quad -\int _{t_k}^t \eta \nabla _p H_0(x_s^{\delta },p_s^{\delta })\cdot \nabla _x \sigma (x^{\delta }_s) d\xi _{\delta }(s)\\&=H_0(x_0,p_0)-\sum _{j=0}^{k-1}\int _{t_j}^{t_{j+1}}\eta \nabla _p H_0(x_{t_j}^{\delta },p_{t_j}^{\delta })\cdot \nabla _x \sigma (x^{\delta }_{t_j}) d\xi _{\delta }(s)\\&\quad -\int _{t_k}^t \eta \nabla _p H_0(x_{t_k}^{\delta },p_{t_k}^{\delta })\cdot \nabla _x \sigma (x^{\delta }_{t_k}) d\xi _{\delta }(s)\\&\quad -\sum _{j=0}^{k-1}\int _{t_j}^{t_{j+1}}\eta \Big (\int _{t_j}^s \nabla _{pp} H_0(x_{r}^{\delta },p_{r}^{\delta }) \cdot (\nabla _x \sigma (x^{\delta }_{r}),\\&\quad -\eta \nabla _x \sigma (x^{\delta }_r) \dot{\xi }_{\delta }(r)) dr \dot{\xi }_{\delta }(s) \\&\quad + \int _{t_j}^s\nabla _{pp} H_0(x_{r}^{\delta },p_{r}^{\delta }) \cdot ( \nabla _x \sigma (x^{\delta }_{r}), -\frac{1}{2}(p_r^{\delta })^{\top }d_xg^{-1}(x)p_r^{\delta }\\&\quad -\nabla _x f(x^{\delta }_s)) dr \dot{\xi }_{\delta }(s)\\&\quad + \int _{t_j}^s\nabla _{p} H_0(x_{r}^{\delta },p_{r}^{\delta }) \cdot \nabla _{xx} \sigma (x^{\delta }_{r})g^{-1}(x^{\delta }_r)p^{\delta }_r dr \dot{\xi }_{\delta }(s)\\&\quad +\int _{t_j}^s \nabla _{px} H_0(x_r^{\delta },p^{\delta })\cdot (\nabla _x \sigma (x_r^{\delta })\dot{\xi }_{\delta }(s), g^{-1}(x_r^{\delta })p^{\delta }_r)dr\Big )ds \\&\quad -\int _{t_k}^t\eta \Big (\int _{t_k}^s \nabla _{pp} H_0(x_{r}^{\delta },p_{r}^{\delta }) \cdot (\nabla _x \sigma (x^{\delta }_{r}),-\eta \nabla _x \sigma (x^{\delta }_r) \dot{\xi }_{\delta }(r)) dr \dot{\xi }_{\delta }(s) \\&\quad + \int _{t_k}^s\nabla _{pp} H_0(x_{r}^{\delta },p_{r}^{\delta }) \cdot (\nabla _x \sigma (x^{\delta }_{r}), -\frac{1}{2}(p_r^{\delta })^{\top }d_xg^{-1}(x^{\delta }_r)p^{\delta }_r\\&\quad -\nabla _x f(x^{\delta }_s)) dr \dot{\xi }_{\delta }(s)\\&\quad + \int _{t_k}^s\nabla _{p} H_0(x_{r}^{\delta },p_{r}^{\delta }) \cdot \nabla _{xx} \sigma (x^{\delta }_{r})g^{-1}(x^{\delta _r})p^{\delta }_r dr \dot{\xi }_{\delta }(s)\\&\quad + \int _{t_k}^{s} \nabla _{px} H_0(x_r^{\delta },p^{\delta })\cdot (\nabla _x \sigma (x_r^{\delta })\dot{\xi }_{\delta }(s)),g^{-1}(x_r^{\delta })p_r^{\delta })dr \Big )ds\\&=: H_0(x_0,p_0) +\sum _{j=0}^{k-1}I_{j}^1+I^1_k(t) \\&\quad +\sum _{j=0}^{k-1} (I_{j}^{21}+I_{j}^{22}+I_j^{23}+I_j^{24})+I_{k}^{21}(t)+I_{k}^{22}(t)+I_k^{23}(t)+I_k^{24}(t). \end{aligned}$$

Making use of the growth condition (2.3), we have that

$$\begin{aligned}&\sum _{j=0}^{k-1} (I_{j}^{21}+I_{j}^{22}+I_j^{23}+I_j^{24})+I_{k}^{21}(t)+I_{k}^{22}(t)+I_k^{23}(t)+I_k^{24}(t)\\&\quad \le \sum _{j=0}^{k-1}\int _{t_j}^{t_{j+1}}(C_1+c_1 H_0(x_s^{\delta },p_s^{\delta }))|\dot{\xi }_{\delta }(s)|^2\delta ds\\&\qquad +\sum _{j=0}^{k-1} \int _{t_j}^{t_{j+1}} (C_1+c_1 H_0(x_s^{\delta },p_s^{\delta }))|\dot{\xi }_{\delta }(s)| \delta ds\\&\qquad +\int _{t_k}^{t}(C_1+c_1 H_0(x_s^{\delta },p_s^{\delta }))|\dot{\xi }_{\delta }(s)|^2\delta ds + \int _{t_k}^{t} (C_1+c_1 H_0(x_s^{\delta },p_s^{\delta }))|\dot{\xi }_{\delta }(s)| \delta ds\\&\quad =\int _{0}^{t}(C_1+c_1 H_0(x_s^{\delta },p_s^{\delta })) |\dot{\xi }_{\delta }(s)|^2\delta ds + \int _{0}^{t} (C_1+c_1 H_0(x_s^{\delta },p_s^{\delta }))|\dot{\xi }_{\delta }(s)| \delta ds. \end{aligned}$$

By using the Gronwall–Bellman inequality, we obtain that

$$\begin{aligned} H_0(x_t^{\delta },p_t^{\delta })&\le \exp (\int _0^t c_1(|\dot{\xi }_{\delta }(s)|^2 +|\dot{\xi }_{\delta }(s)|)\delta ds)(H_0(x_0,p_0)+CT+|\sum _{j=0}^{k-1}I_{j}^1+I^1_k(t)|). \end{aligned}$$

For simplicity, assume that $T=K\delta .$ Denote $[t]_{\delta }=t_k=k\delta$ if $t\in [t_k,t_{k+1}).$ The definition of $\xi _{\delta }(s)$ yields that $s\in [t_j,t_{j+1}]$

$$\begin{aligned} |\dot{\xi }_{\delta }(s)|^2\delta +|\dot{\xi }_{\delta }(s)|\delta =\left| \frac{B(t_{j+1})-B(t_j)}{\delta }|^2\delta +\right| B(t_{j+1})-B(t_j)|. \end{aligned}$$

Define a stopping time $\tau _R=\inf \{t\in [0,T]| \int _0^{[t]_{\delta }}|\dot{\xi }_{\delta }|^2 \delta ds \ge R\}.$ The stopping time is well-defined since the quadratic variation process of Brownian motion is bounded in [0, T]. Then taking $t\le \tau _{R}$ and using Hölder’s inequality, then it holds that

$$\begin{aligned} H_0(x_t^{\delta },p_t^{\delta })&\le \exp (\int _{[t]}^t c_1(|\dot{\xi }_{\delta }(s)|^2+|\dot{\xi }_{\delta }| ds)\exp (C(R+T))(H_0(x_0,p_0)\nonumber \\&\quad +CT+|\sum _{j=0}^{k-1}I_{j}^1+I^1_k(t)|)\nonumber \\&\le \exp \left( \int _{[t]}^t c_1(\frac{3}{2}|\dot{\xi }_{\delta }(s)|^2) ds\right) \exp (C(R+T)) H_0(x_0,p_0)\nonumber \\&\quad + \exp (C(R+T)) \exp \left( \int _{[t]}^t c_1\frac{3}{2}|\dot{\xi }_{\delta }(s)|^2 ds\right) \left| \int _0^{[t]}\right. \nonumber \\&\quad \left. -\eta \nabla _p H_0(x_{[s]_\delta }^{\delta },p_{[s]_{\delta }}^{\delta })\cdot \nabla _x \sigma (x_{[s]_{\delta }}) dB(s)\right| \nonumber \\&\quad \displaystyle + \exp (C(R+T)) \exp (\int _{[t]}^t (c_1\frac{3}{2}|\dot{\xi }_{\delta }(s)|^2 ds) \left| \int _{[t]}^t\right. \nonumber \\&\quad \displaystyle \left. -\eta \nabla _p H_0(x_{[s]_\delta }^{\delta },p_{[s]_{\delta }}^{\delta })\cdot \nabla _x \sigma (x_{[s]_{\delta }}) \dot{\xi }_{\delta }(s) ds\right| . \end{aligned}$$

(A.1)

Similarly, one could obtain a analogous estimate of (A.1) with the integral over $[t_{k-1},t_{k}]$, where $t_k$, $k\le K$, $t_K\le \tau _R.$ By the Cauchy inequality and taking expectation on both sides of (A.1), applying the Burkholder–Davis–Gundy inequality (see e.g, [35]) and using the independent increments of Brownian motion, we get

$$\begin{aligned}&{\mathbb {E}}[H_0^2(x_{t_k}^{\delta },p_{t_k}^{\delta })] \\&\quad \le 3{\mathbb {E}}\Big [\exp (\int _{t_{k-1}}^{t_k} (3c_1|\dot{\xi }_{\delta }(s)|^2 ds)\Big ]\exp (2C(R+T)) {\mathbb {E}}\Big [H_0^2(x_0,p_0)\Big ]\\&\qquad + 3\exp (2C(R+T)) {\mathbb {E}}\Big [\exp \Big (\int _{t_{k-1}}^{t_k} 3c_1|\dot{\xi }_{\delta }(s)|^2 ds\Big )\Big ] {\mathbb {E}}\Big [\Big |\int _0^{t_{k-1}}\\&\qquad -\eta \nabla _p H_0(x_{[s]_\delta }^{\delta },p_{[s]_{\delta }}^{\delta })\cdot \nabla _x \sigma (x_{[s]_{\delta }}) dB(s)\Big |^2\Big ]\\&\qquad + 3\exp (2C(R+T)) {\mathbb {E}}\Big [\exp \Big (\int _{t_{k-1}}^{t_k} 3c_1|\dot{\xi }_{\delta }(s)|^2 ds\Big ) |B(t_{k})-B(t_{k-1})|^2 \\&\quad \times \big |\eta \nabla _p H_0(x_{t_{k-1}}^{\delta },p_{t_{k-1}}^{\delta })\cdot \nabla _x \sigma (x_{t_{k-1}}) \big |^2\Big ]\\&\quad \le 3{\mathbb {E}}\Big [\exp (\int _{t_{k-1}}^{t_k} (3c_1|\dot{\xi }_{\delta }(s)|^2 ds)\Big ]\exp (2C(R+T)) {\mathbb {E}}\Big [H_0^2(x_0,p_0)\Big ]\\&\qquad + 3\exp (2C(R+T)) {\mathbb {E}}\Big [\exp \Big (\int _{t_{k-1}}^{t_k} 3c_1|\dot{\xi }_{\delta }(s)|^2 ds\Big )\Big ]\\&\qquad {\mathbb {E}}\Big [\int _0^{t_{k-1}}(C_1+c_1 H_0(x_{[s]_{\delta }}^{\delta },p_{[s]_{\delta }}^{\delta }))^2 ds\Big ]\\&\qquad + 3\exp (2C(R+T)) {\mathbb {E}}\Big [\exp \Big (\int _{t_{k-1}}^{t_k} 3c_1|\dot{\xi }_{\delta }(s)|^2 ds\Big ) |B(t_{k})-B(t_{k-1})|^2\Big ] \\&\quad \times {\mathbb {E}}\Big [(C_1+c_1 H_0^2(x_{t_{k-1}}^{\delta },p_{t_{k-1}}^{\delta }))\Big ]. \end{aligned}$$

Applying the Fernique theorem and choosing sufficient small $\delta$ such that $12c_1\delta <1,$ then we have that

$$\begin{aligned}&{\mathbb {E}}\Big [\exp \Big (\int _{t_{k-1}}^{t_k} 3c_1|\dot{\xi }_{\delta }(s)|^2 ds\Big )\Big ]\le C,\\&{\mathbb {E}}\Big [\exp (\int _{t_{k-1}}^{t_k} 3c_1|\dot{\xi }_{\delta }(s)|^2 ds\Big ) |B(t_{k})-B(t_{k-1})|^2\Big ]\\&\quad \le \sqrt{{\mathbb {E}}\Big [\exp \Big (\int _{t_{k-1}}^{t_k} 6c_1|\dot{\xi }_{\delta }(s)|^2 ds\Big )\Big ]}\sqrt{{\mathbb {E}}\Big [|B(t_{k})-B(t_{k-1})|^4\Big ]} \le C\delta . \end{aligned}$$

The above estimation gives

$$\begin{aligned} {\mathbb {E}}[H_0^2(x_{t_k}^{\delta },p_{t_k}^{\delta })]&\le 3 \exp (2C(R+T))C {\mathbb {E}}[H_0^2(x_0,p_0)]\\&\quad +6 \exp (2C(R+T))C\int _{0}^{t_{k-1}}{\mathbb {E}}\Big [(C_1^2+c_1^2H_0^2(x_{[s]_{\delta }}^{\delta },p_{[s]_{\delta }}^{\delta }))\Big ]ds\\&\quad +6 \exp (2C(R+T)C\delta {\mathbb {E}}\Big [C_1^2+c_1^2H_0^2(x_{t_{k-1}}^{\delta },p_{t_{k-1}}^{\delta })\Big ]. \end{aligned}$$

Then the Grownall’s inequality yield that

$$\begin{aligned} {\mathbb {E}}[H_0^2(x_{t_k}^{\delta },p_{t_k}^{\delta })]&\le \exp (6TCc_1^2\exp (2C(R+T))) \\&\quad \times \Big (3\exp (2C(R+T))C{\mathbb {E}}[H_0^2(x_0,p_0)] + 6C_1^2T C \exp (2C(R+T))\Big ) \end{aligned}$$

Combining the above estimates with (A.1) and the Burkholder–Davis–Gundy inequality, we conclude that

$$\begin{aligned}&\sup _{t\in [0,\tau ^R)}{\mathbb {E}}[H_0^2(x_{t}^{\delta },p_{t}^{\delta })] \le (\exp (6TCc_1^2\exp (2C(R+T)))+C) \\&\quad \times \Big (3\exp (2C(R+T))C{\mathbb {E}}[H_0^2(x_0,p_0)] + 6C_1^2T C \exp (2C(R+T))\Big )\\&\quad =:C_R. \end{aligned}$$

Similarly, by choosing sufficient small $\delta$, we have that for $t\in [0,\tau ^R)$,

$$\begin{aligned} {\mathbb {E}}[H_0^{p}(x_{t}^{\delta },p_{t}^{\delta })] \le C_{R,p}<\infty . \end{aligned}$$

As a consequence, by again using (A.1), we obtain that

$$\begin{aligned} {\mathbb {E}}\Big [\sup _{t\in [0,\tau ^R)}H_0^p(x_{t}^{\delta },p_{t}^{\delta })\Big ]&\le C_{R,p}<\infty . \end{aligned}$$

Next we show the convergence in probability of the solution of (2.1) to that of (2.4). Introduce another stopping time $\tau _{R_1}:=\inf \{t\in [0,T]| |x_t|+|p_t|\ge R_1, |x^{\delta }_{[t]_{\delta }}|+|p^{\delta }_{[t]_{\delta }}|\ge R_1 \}.$ Let $t\in [0,\tau _{R}\wedge \tau _{R_1}).$ By using the polynomial growth condition of $f,\sigma$ and the fact that $\sigma$ is independent of p, we obtain that

$$\begin{aligned}&|x^{\delta }(t)-x(t)|^2\\&\quad = |x^\delta (0)-x(0)|^2 +\int _{0}^t2\langle x^{\delta }(s)-x(s),g^{-1}(x^{\delta }(s)) p^{\delta }(s)-g^{-1}(x(s))p(s)\rangle ds\\&\quad \le |x^{\delta }(0)-x(0)|^2+\int _0^t C_g(1+|p(s)|)(|x^{\delta }(s)-x(s)|^2+|p^{\delta }(s)-p(s)|^2) ds,\\&|p^{\delta }(t)-p(t)|^2\\&\quad = \int _0^t \langle - (p^{\delta }(s))^{\top } d_x g^{-1}(x^\delta (s)) p^{\delta }(s) + p(s)^{\top } d_x g^{-1}(x(s)) p(s) ,p^{\delta }(s)-p(s)\rangle ds \\&\qquad +\int _0^t2\langle -\nabla _x f(x^{\delta }(s))+\nabla _x f(x(s)),p^{\delta }(s)-p(s)\rangle ds\\&\qquad -\int _0^t2\eta \langle p^{\delta }(s)-p(s), \nabla _x \sigma (x^{\delta }(s)) d\xi _{\delta }(s)-\nabla _x\sigma (x(s))dB_t\rangle \\&\quad \le C_g \int _0^t (1+|x^{\delta }(s)|)(|p^{\delta }(s)|^2+|p(s)|^2)(|p^{\delta }(s)-p(s)|^2+|x^{\delta }(s)-x(s)|^2) ds\\&\qquad + C_f \int _0^t (1+|x(s)|^{p_f}+|x^{\delta }|^{p_f})(|p^{\delta }(s)-p(s)|^2+|x^{\delta }(s)-x(s)|^2)ds\\&\qquad -\int _0^t2\eta \langle p^{\delta }(s)-p(s), \nabla _x \sigma (x^{\delta }(s)) d\xi _{\delta }(s)-\nabla _x\sigma (x(s))dB_s\rangle , \end{aligned}$$

where $C_g$ and $C_f$ are constants depending on f and g. To deal with the last term, we split it as follows,

$$\begin{aligned}&\int _0^t2\eta \langle p^{\delta }(s)-p(s), \nabla _x \sigma (x^{\delta }(s)) d\xi _{\delta }(s)-\nabla _x\sigma (x(s))dB_s\rangle \\&\quad =2\eta \int _0^t \langle p^{\delta }([s]_{\delta })-p([s]_{\delta }), \nabla _x \sigma (x^{\delta }(s)) d\xi _{\delta }(s)-\nabla _x\sigma (x(s))dB_s\rangle \\&\qquad +2\eta \int _0^t \langle p^{\delta }(s)-p(s)-p^{\delta }([s]_{\delta })+p([s]_{\delta }), \nabla _x \sigma (x^{\delta }(s)) d\xi _{\delta }(s)-\nabla _x\sigma (x(s))dB_s\rangle \\&\quad =2\eta \int _0^t \langle p^{\delta }([s]_{\delta })-p([s]_{\delta }), \nabla _x \sigma (x^{\delta }([s]_{\delta })) d\xi _{\delta }([s]_{\delta })-\nabla _x\sigma (x([s]_{\delta }))dB_s\rangle \\&\qquad +2\eta \int _0^t \langle p^{\delta }([s]_{\delta })-p([s]_{\delta }), (\nabla _x \sigma (x^{\delta }(s))- \nabla _x \sigma (x^{\delta }([s]_{\delta })))d\xi _{\delta }(s)\\&\qquad -(\nabla _x\sigma (x(s))-\nabla _x\sigma (x([s]_{\delta })))dB_s\rangle \\&\qquad + 2\eta \int _0^t \langle p^{\delta }(s)-p(s)-p^{\delta }([s]_{\delta })+p([s]_{\delta }), \\&\qquad \nabla _x \sigma (x^{\delta }([s]_{\delta })) d\xi _{\delta }(s)-\nabla _x\sigma (x([s]_{\delta })dB_s\rangle \\&\qquad + 2\eta \int _0^t \langle p^{\delta }(s)-p(s)-p^{\delta }([s]_{\delta })+p([s]_{\delta }), (\nabla _x \sigma (x^{\delta }(s)-\nabla _x \sigma (x^{\delta }([s]_{\delta })) ) d\xi _{\delta }(s)\\&\qquad -(\nabla _x\sigma (x(s)-\nabla _x\sigma (x([s]_{\delta }))dB_s\rangle \\&\quad =: II^1+II^2+II^3+II^4. \end{aligned}$$

Taking expectation on $II^1$, using the property of the discrete martingale, the a prior estimates for $H_0(x_t,p_t)$ and $H_0(x_t^{\delta },p_t^{\delta })$ and Hölder’s inequality, we have that

$$\begin{aligned} {\mathbb {E}}[II^1]&=0,\\ {\mathbb {E}}[II^2]&\le 2\eta \int _0^t {\mathbb {E}}\Big [\langle p^{\delta }([s]_{\delta })-p([s]_{\delta }), \\&\quad \int _{[s]_{\delta }}^{s} (\nabla _{xx} \sigma (x^{\delta }(r))\cdot (g^{-1}(x^{\delta }(r))p^{\delta }(r)) dr d\xi _{\delta }(s)\rangle \Big ]\\&\quad -2\eta \int _0^t {\mathbb {E}}\Big [\langle p^{\delta }([s]_{\delta })-p([s]_{\delta }),\\&\quad \int _{[s]_{\delta }}^{s} (\nabla _{xx} \sigma (x(r))\cdot (g^{-1}(x^{\delta }(r)p^{\delta }(r)) dr dB_s\rangle \Big ]\\&\le C(R_1) \delta ^{\frac{1}{2}}. \end{aligned}$$

Similar arguments lead to ${\mathbb {E}}[II^4]\le C(R_1)\delta ^{\frac{1}{2}}.$ For the term $II^3$, applying the continuity estimate of $x_t$ and $x^{\delta }_t,$ as well as independent increments of the Brownian motion, we get

$$\begin{aligned}&{\mathbb {E}}[II^3]\\&\le C(R_1)\delta ^{\frac{1}{2}} +2\eta ^2{\mathbb {E}}\Big [\int _0^{[t]_{\delta }} \langle \int _{[s]_{\delta }}^s \nabla _x \sigma (x_{[r]_{\delta }}^{\delta }) d\xi _{\delta }(r) -\int _{[s]_{\delta }}^s \nabla _x \sigma (x_{[r]_{\delta }}) dB_r, \\&\quad \nabla _x \sigma (x^{\delta }([s]_{\delta })) d\xi _{\delta }(s)-\nabla _x\sigma (x([s]_{\delta })dB_s\rangle \Big ]\\&\le C(R_1)\delta ^{\frac{1}{2}} +2\eta ^2{\mathbb {E}}\Big [\int _0^{[t]_{\delta }} |\nabla _x \sigma (x^{\delta }_{[s]_{\delta }})|^2\frac{s-[s]_{\delta }}{\delta ^2}(B([s]_{\delta }+\delta )-B([s]_{\delta }))^2ds\Big ]\\&\quad -2\eta ^2{\mathbb {E}}\Big [\int _0^{[t]_{\delta }} \langle \nabla _x \sigma (x^{\delta }_{[s]_{\delta }}), \nabla _x \sigma (x_{[s]_{\delta }})\rangle \frac{s-[s]_{\delta }}{\delta ^2}(B([s]_{\delta }+\delta )-B([s]_{\delta }))^2ds \Big ]\\&\quad -2\eta ^2 {\mathbb {E}}\Big [\int _0^{[t]_{\delta }} \langle \nabla _x \sigma (x^{\delta }_{[s]_{\delta }}), \nabla _x \sigma (x_{[s]_{\delta }})\rangle \frac{ B([s]_{\delta }+\delta )-B([s]_{\delta })}{\delta } (B(s)-B([s]_{\delta }))ds\Big ]\\&\quad +2\eta ^2 {\mathbb {E}}\Big [\int _0^{[t]_{\delta }} \langle \nabla _x \sigma (x_{[s]_{\delta }}), \nabla _x \sigma (x_{[s]_{\delta }})\rangle \frac{ B([s]_{\delta }+\delta )-B([s]_{\delta })}{\delta } (B(s)-B([s]_{\delta }))ds\Big ]\\&\le C(R_1)\delta ^{\frac{1}{2}}+2\eta ^2 \int _0^{[t]_{\delta }}{\mathbb {E}}\Big [|\nabla _x \sigma (x_{[s]_{\delta }}^{\delta })-\nabla _x \sigma (x_{[s]_{\delta }})|^2\Big ]ds\\&\le C(R_1)\delta ^{\frac{1}{2}}+ \int _0^{t}C(R_1){\mathbb {E}}\Big [| x_s^{\delta }-x_s|^2\Big ]ds, \end{aligned}$$

where $C(R_1)>0$ is monotone with $R_1.$ Combining the above estimates, we achieve that

$$\begin{aligned} {\mathbb {E}}[|x^{\delta }(t)-x(t)|^2]&\le \int _0^t C_g(1+C_{R_1}) ({\mathbb {E}}[|x^{\delta }(s)-x(s)|^2]\\&\quad + {\mathbb {E}}[|p^{\delta }(s)-p(s)|^2])ds \\ {\mathbb {E}}[|p^{\delta }(t)-p(t)|^2]&\le \int _0^t (C_g+C_f)(1+C_{R_1})({\mathbb {E}}[|x^{\delta }(s)-x(s)|^2]\\&\quad + {\mathbb {E}}[|p^{\delta }(s)-p(s)|^2])ds +C(R_1)\delta ^{\frac{1}{2}}. \end{aligned}$$

Then the Gronwall’s inequality implies that

$$\begin{aligned} {\mathbb {E}}[|x^{\delta }(t)-x(t)|^2]+{\mathbb {E}}[|p^{\delta }(t)-p(t)|^2]&\le \exp (2(C_g+C_f)(1+C_{R_1})T)C(R_1)\delta ^{\frac{1}{2}}. \end{aligned}$$

(A.2)

By making use of (A.2) and Chebshev’s inequality, we conclude that

$$\begin{aligned}&{\mathbb {P}}(|x^{\delta }(t)-x(t)|+|p^{\delta }(t)-x(t)|\ge \epsilon )\\&\quad \le {\mathbb {P}}(\{|x^{\delta }(t)-x(t)|+|p^{\delta }(t)-x(t)|\ge \epsilon \}\cap \{t<\tau _R\}\cap \{t<\tau _{R_1}\}) \\&\qquad + {\mathbb {P}}(\{|x^{\delta }(t)-x(t)|+|p^{\delta }(t)-x(t)|\ge \epsilon \}\cap \{t\ge \tau _R\})\\&\qquad +{\mathbb {P}}(\{|x^{\delta }(t)-x(t)|+|p^{\delta }(t)-x(t)|\ge \epsilon \}\cap \{t< \tau _R\}\cap \{t\ge \tau _{R_1}\})\\&\qquad \le 2\frac{{\mathbb {E}}\Big [|x^{\delta }(t)-x(t)|^2+|p^{\delta }(t)-x(t)|^2\Big ]}{\epsilon ^2} \\&\qquad + \frac{{\mathbb {E}}\Big [\int _0^{t}|\dot{\xi }_{\delta }(s)|^2\delta ds\Big ]}{R}+\frac{{\mathbb {E}}\Big [|x(t)|+|p(t)|+|x^{\delta }(t)|+|p^{\delta }(t)|\Big ]}{R_1}\\&\quad \le \frac{2}{\epsilon ^2}\exp (2(C_g+C_f)(1+C_{R_1})T)C(R_1)\delta ^{\frac{1}{2}}+ \frac{C}{R}+C\frac{1+C_R}{R_1}. \end{aligned}$$

Here, ${\mathbb {E}}[|x(t)|+|p(t)|+|x^\delta (t)|+|p^\delta (t)|]<C(1+C_R)$ is ensured by ${\mathbb {E}}[\sup \limits _{t\in [0,\tau ^R)} H_0^{2}(x_{t}^{\delta },p_{t}^{\delta })] \le C_{R}.$ Taking limit on $\delta \rightarrow 0,$ $R_1\rightarrow \infty$, and $R \rightarrow \infty$ leads to

$$\begin{aligned} \lim _{\delta \rightarrow 0}{\mathbb {P}}(|x^{\delta }(t)-x(t)|+|p^{\delta }(t)-p(t)|> \epsilon )=0. \end{aligned}$$

Similarly, one could utilize the properties of martingale and obtain the following estimate, for large enough $q>0,$

$$\begin{aligned} {\mathbb {E}}[|x^{\delta }(t)-x(t)|^{q}]+{\mathbb {E}}[|p^{\delta }(t)-p(t)|^q]&\le C_q \exp (C_q(C_g+C_f)(1+C_{R_1})T)C(R_1)\delta ^{\frac{q}{2} -1}. \end{aligned}$$

This implies that for large enough $q>4$,

$$\begin{aligned}&{\mathbb {E}}\Big [\sup _{k\le K}\sup _{t\in [t_{k-1},t_{k}]}|x^{\delta }(t)-x(t)|^{q}\Big ]+{\mathbb {E}}\Big [\sup _{k\le K}\sup _{t\in [t_{k-1},t_{k}]}|p^{\delta }(t)-p(t)|^q\Big ]\\&\quad \le \sum _{k=0}^{K-1}{\mathbb {E}}\Big [\sup _{t\in [t_{k-1},t_{k}]}|x^{\delta }(t)-x(t)|^{q}\Big ]+{\mathbb {E}}\Big [\sup _{t\in [t_{k-1},t_{k}]}|p^{\delta }(t)-p(t)|^q\Big ]\\&\quad \le C_q K \exp (C_q(C_g+C_f)(1+C_{R_1})T)C(R_1)\delta ^{\frac{q}{2} -1}\\&\quad \le C_q \exp (C_q(C_g+C_f)(1+C_{R_1})T)C(R_1)\delta ^{\frac{q}{2} -2}. \end{aligned}$$

Combining the above estimate and applying the Chebshev’s inequality, we further obtain

$$\begin{aligned} \lim _{\delta \rightarrow 0}{\mathbb {P}}\left( \sup _{t\in [0,T]}|x^{\delta }(t)-x(t)|+\sup _{t\in [0,T]}|p^{\delta }(t)-p(t)|> \epsilon \right) =0. \end{aligned}$$

$\square$