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This paper provides an introduction to the Stein method framework in the context of steady-state diffusion approximations. The framework consists of three components: the Poisson equation and gradient bounds, generator coupling, and moment…

We study contractions of Markov chains on general metric spaces with respect to some carefully designed distance-like functions, which are comparable to the total variation and the standard $L^p$-Wasserstein distances for $p \ge 1$. We…

In this manuscript, we provide a non-asymptotic process level control between the telegraph process and the Brownian motion with suitable diffusivity constant via a Wasserstein distance with quadratic average cost. In addition, we derive…

Contraction in Wasserstein 1-distance with explicit rates is established for generalized Hamiltonian Monte Carlo with stochastic gradients under possibly nonconvex conditions. The algorithms considered include splitting schemes of kinetic…

We prove a bound for the Wasserstein distance between vectors of smooth complex random variables and complex Gaussians in the framework of complex Markov diffusion generators. For the special case of chaotic eigenfunctions, this bound can…

We present a framework that allows for the non-asymptotic study of the $2$-Wasserstein distance between the invariant distribution of an ergodic stochastic differential equation and the distribution of its numerical approximation in the…

The convergence rate in Wasserstein distance is estimated for empirical measures of ergodic Markov processes, and the estimate can be sharp in some specific situations. The main result is applied to subordinations of typical models excluded…

In the case of diffusions on $\mathbb R^d$ with constant diffusion matrix, without assuming reversibility nor hypoellipticity, we prove that the contractivity of the deterministic drift is equivalent to the constant rate contraction of…

Let $M$ be a $d$-dimensional connected compact Riemannian manifold with boundary $\partial M$, let $V\in C^2(M)$ such that $\mu(dx):=e^{V(x)} d x$ is a probability measure, and let $X_t$ be the diffusion process generated by…

We introduce a unified framework to estimate the convergence of Markov chains to equilibrium in Wasserstein distance. The framework can provide convergence bounds with rates ranging from polynomial to exponential, all derived from a…

The Wasserstein distance is a metric on a space of probability measures that has seen a surge of applications in statistics, machine learning, and applied mathematics. However, statistical aspects of Wasserstein distances are bottlenecked…

Under natural conditions, we proved the exponential ergodicity in Wasserstein distance of two-type continuous-state branching processes in L\'evy random environments with immigration. Furthermore, we expressed accurately the parameters of…

Several issues in machine learning and inverse problems require to generate discrete data, as if sampled from a model probability distribution. A common way to do so relies on the construction of a uniform probability distribution over a…

For a Markov process associated with a diffusion type Dirichlet form an upper bound is shown for the law of the finite dimensional distributions of the process. Under some more assumptions on the underlaying space this is also shown for the…

In this work, we consider rather general and broad class of Markov chains, Ito chains, that look like Euler-Maryama discretization of some Stochastic Differential Equation. The chain we study is a unified framework for theoretical analysis.…

We provide convergence guarantees in Wasserstein distance for a variety of variance-reduction methods: SAGA Langevin diffusion, SVRG Langevin diffusion and control-variate underdamped Langevin diffusion. We analyze these methods under a…

The exponential contraction in $L^1$-Wasserstein distance and exponential convergence in $L^q$-Wasserstein distance ($q\geq 1$) are considered for stochastic differential equations with irregular drift. When the irregular drift drift is…

We estimate contrasts $\int_0 ^1 \rho(F^{-1}(u)-G^{-1}(u))du$ between two continuous distributions $F$ and $G$ on $\mathbb R$ such that the set $\{F=G\}$ is a finite union of intervals, possibly empty or $\mathbb{R}$. The non-negative…

This work studies finite sample approximations of the exact and entropic regularized Wasserstein distances between centered Gaussian processes and, more generally, covariance operators of functional random processes. We first show that…

We derive a multidimensional Stein's method for asymptotic independence in the case of a general target $\mu$ with a density, being invariant measure of a diffusion process. It allows us to give a general bound in Wasserstein distance…