  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.
