In this paper, we provide bounds in Wasserstein and total variation distances between the distributions of the successive iterates of two functional autoregressive processes with isotropic Gaussian noise of the form $Y_{k+1} = \mathrm{T}_\gamma(Y_k) + \sqrt{\gamma\sigma^2} Z_{k+1}$ and $\tilde{Y}_{k+1} = \tilde{\mathrm{T}}_\gamma(\tilde{Y}_k) + \sqrt{\gamma\sigma^2} \tilde{Z}_{k+1}$. More precisely, we give non-asymptotic bounds on $\rho(\mathcal{L}(Y_{k}),\mathcal{L}(\tilde{Y}_k))$, where $\rho$ is an appropriate weighted Wasserstein distance or a $V$-distance, uniformly in the parameter $\gamma$, and on $\rho(\pi_{\gamma},\tilde{\pi}_{\gamma})$, where $\pi_{\gamma}$ and $\tilde{\pi}_{\gamma}$ are the respective stationary measures of the two processes. The class of considered processes encompasses the Euler-Maruyama discretization of Langevin diffusions and its variants. The bounds we derive are of order $\gamma$ as $\gamma \to 0$. To obtain our results, we rely on the construction of a discrete sticky Markov chain $(W_k^{(\gamma)})_{k \in \mathbb{N}}$ which bounds the distance between an appropriate coupling of the two processes. We then establish stability and quantitative convergence results for this process uniformly on $\gamma$. In addition, we show that it converges in distribution to the continuous sticky process studied in previous work. Finally, we apply our result to Bayesian inference of ODE parameters and numerically illustrate them on two particular problems.

翻译：本文给出了两类具有各向同性高斯噪声的函数型自回归过程迭代分布之间的Wasserstein距离和全变差距离的界，其形式为 $Y_{k+1} = \mathrm{T}_\gamma(Y_k) + \sqrt{\gamma\sigma^2} Z_{k+1}$ 和 $\tilde{Y}_{k+1} = \tilde{\mathrm{T}}_\gamma(\tilde{Y}_k) + \sqrt{\gamma\sigma^2} \tilde{Z}_{k+1}$。具体而言，我们建立了关于 $\rho(\mathcal{L}(Y_{k}),\mathcal{L}(\tilde{Y}_k))$ 的非渐近界（其中 $\rho$ 为适当的加权Wasserstein距离或$V$-距离，且对参数 $\gamma$ 一致成立），以及关于 $\rho(\pi_{\gamma},\tilde{\pi}_{\gamma})$ 的界（其中 $\pi_{\gamma}$ 和 $\tilde{\pi}_{\gamma}$ 分别为两过程的平稳测度）。所考虑的过程类别涵盖Langevin扩散的欧拉-丸山离散化及其变体。当 $\gamma \to 0$ 时，我们推导的界阶数为 $\gamma$。为获得结果，我们构造了一条离散粘性马尔可夫链 $(W_k^{(\gamma)})_{k \in \mathbb{N}}$，该链限制了两个过程适当耦合之间的距离，并建立了该过程对 $\gamma$ 一致成立的稳定性与量化收敛结果。此外，我们证明该链在分布上收敛至先前工作中研究的连续粘性过程。最后，将结果应用于常微分方程参数的贝叶斯推断，并通过两个具体问题进行数值验证。