x\Vert^{2}+V(x)+\frac{\gamma}{2}\langle x,y\rangle+\frac{c}{2}\Vert y\Vert^{2}+1,$$ where $x,y\in\mathbb{R}^{r}$ and $c>1$. Under Assumption \[assu:v\_boundedness\](a), we immediately obtain $$\begin{aligned}
\Gamma_{c}(x,y) & \ge\frac{1}{6}\left\Vert \gamma x+\frac{3}{2}y\right\Vert ^{2}+\frac{\gamma^{2}}{12}\Vert x\Vert^{2}+\frac{1}{8}\Vert y\Vert^{2}+1\nonumber \\
                & \ge\frac{\gamma^{2}}{12}\Vert x\Vert^{2}+\frac{1}{8}\Vert y\Vert^{2}+1,\label{eq:lemma_gamma_c_lower_bound}\end{aligned}$$ and $$\Gamma_{c}(x,y)\le\frac{3\gamma^{2}}{8}\Vert x\Vert^{2}+V(x)+\frac{c+1}{2}\Vert y\Vert^{2}+1.\label{eq:lemma_gamma_c_upper_bound}$$ Next, note that $V(x_{1})-V(x_{2})=\langle\nabla V(x_{2}),x_{1}-x_{2}\rangle+\int_{0}^{1}(1-s)(x_{1}-x_{2})^{\mathrm{T}}\nabla^{2}(x_{2}+s(x_{1}-x_{2}))(x_{1}-x_{2})\mathrm{d}s$. By Assumption \[assu:v\_boundedness\](b), we get $$\begin{aligned}
    & \langle\nabla V(x_{2}),x_{1}-x_{2}\rangle-\frac{\nu}{2}\Vert x_{1}-x_{2}\Vert^{2}\le V(x_{1})-V(x_{2})\nonumber \\
\le & \langle\nabla V(x_{2}),x_{1}-x_{2}\rangle+\frac{\nu}{2}\Vert x_{1}-x_{2}\Vert^{2}.\label{eq:lemma_v_diff}\end{aligned}$$ Also due to $\Vert\nabla^{2}V(x)\Vert\le\nu$, we have $$\Vert\nabla V(x)\Vert\le\Vert\nabla V(0)\Vert+\Vert\nabla V(x)-\nabla V(0)\Vert\le\Vert\nabla V(0)\Vert+\nu\Vert x\Vert.\label{eq:lemma_gradv_bound}$$

Let $\tau_{k}=(1-\gamma\delta)\rho_{k}-\delta\cdot\nabla V(\xi_{k})$, and for $t\in[k\delta,(k+1)\delta]$ define $R_{k}(t)=\tau_{k}+\sqrt{2\gamma}(B(t)-B(k\delta))$. Clearly $\rho_{k+1}=\tau_{k}+\sqrt{2\gamma\delta}\eta_{k}=R_{k}((k+1)\delta)$. We then present the following two lemmas.

\[lem:gamma\_c\_contraction\]Let $D=\gamma^{4}C_{\beta}/\nu^{2}$, $\delta\le\min\{1/\gamma,\gamma/(2\nu),(D+1-\sqrt{D^{2}+1})/\gamma\}$, and $c=(1-\gamma\delta/2)/(1-\gamma\delta)$. Then there exist constants $\varepsilon=\varepsilon(\delta)>0$ and $C_{1}=C_{1}(\delta)>0$ such that for all $k\ge0$, $$\Gamma_{c}(\xi_{k+1},\tau_{k})\le(1-\varepsilon\gamma\delta)\Gamma_{c}(\xi_{k},\rho_{k})+C_{1}.$$

\[lem:gamma\_c\_l\]Fix any integer $l>0$. Then there exists a constant $C_{2}=C_{2}(l,\delta)>0$ such that for all $k\ge0$ and $t\in[k\delta,(k+1)\delta]$, $$\begin{aligned}
& \mathbb{E}\left[\left\{ \Gamma_{c}(\xi_{k+1},R_{k}(t))\right\} ^{l}\right] \le \\
& \mathbb{E}\left[\left\{ \Gamma_{c}(\xi_{k+1},\tau_{k})\right\} ^{l}\right]+C_{2}\sum_{i=0}^{l-1}(t-k\delta)^{l-i}\mathbb{E}\left[\left\{ \Gamma_{c}(\xi_{k+1},\tau_{k})\right\} ^{i}\right].\end{aligned}$$

For the Langevin diffusion process $W(t)$, let $\mathcal{A}$ be its generator, defined by $$\begin{aligned}
(\mathcal{A}\phi)(w)= & \ \langle y,\nabla_{x}\phi(w)\rangle-\langle\nabla V(x)+\gamma y,\nabla_{y}\phi(w)\rangle \\
                      & +\gamma\mathrm{tr}(\nabla_{y}^{2}\phi(w)),\end{aligned}$$ where $\phi:\mathbb{R}^{2r}\to\mathbb{R}$ is any twice differentiable function, $w=(x^{\mathrm{T}},y^{\mathrm{T}})^{\mathrm{T}}$, $\nabla_{x}\phi=\partial\phi(w)/\partial x$, $\nabla_{y}\phi=\partial\phi(w)/\partial y$, and $\nabla_{y}^{2}\phi=\partial^{2}\phi(w)/\partial y\partial y^{\mathrm{T}}$. Consider the functional equation $\mathcal{A}\psi=\varphi-\bar{\varphi}$, which is called the Poisson equation, and we have the following lemma for the solution $\psi$.

\[lem:existence\_poisson\]Assume that $\varphi\in\mathscr{C}_{poly}^{r+5}$, and the conditions in Theorem 