Related papers: General diffusion processes as the limit of time-s…

In this work, we are concerned with existence and uniqueness of invariant measures for path-dependent random diffusions and their time discretizations. The random diffusion here means a diffusion process living in a random environment…

Superslow diffusion, i.e., the long-time diffusion of particles whose mean-square displacement (variance) grows slower than any power of time, is studied in the framework of the decoupled continuous-time random walk model. We show that this…

In this article we consider a family of real-valued diffusion processes on the time interval $[0,1]$ indexed by their prescribed initial value $x \in \mathbb{R}$ and another point in space, $y \in \mathbb{R}$. We first present an…

Assume that one observes the $k$th, $2k$th$,\ldots,nk$th value of a Markov chain $X_{1,h},\ldots,X_{nk,h}$. That means we assume that a high frequency Markov chain runs in the background on a very fine time grid but that it is only observed…

The fact that a Markov diffusion semi-group on $\mathbb R^d$ contracts the $L^p$ Wasserstein distance, which has been extensively used to establish uniform-in-time stability estimates (e.g. with respect to numerical discretization errors),…

We bound the rate of convergence to stationarity for a signed generalization of the Bernoulli-Laplace diffusion model; this signed generalization is a Markov chain on the homogeneous space (Z_2 \wr S_n) / (S_r \times S_{n-r}). Specifically,…

We consider a Markov chain on $\mathbb{R}^d$ with invariant measure $\mu$. We are interested in the rate of convergence of the empirical measures towards the invariant measure with respect to various dual distances, including in particular…

The aim of this article is to prove that diffusion processes in $\mathbb{R}^d$ with a drift can be approximated by suitable Markov chains on $n^{-1}\mathbb{Z}^d$. Moreover, we investigate sufficient conditions on the conductances which…

Ordinary differential equations obtained as limits of Markov processes appear in many settings. They may arise by scaling large systems, or by averaging rapidly fluctuating systems, or in systems involving multiple time-scales, by a…

Let $(X_n)_{n=0}^\infty$ denote a Markov chain on a Polish space that has a stationary distribution $\varpi$. This article concerns upper bounds on the Wasserstein distance between the distribution of $X_n$ and $\varpi$. In particular, an…

By using the spectrum of the underlying symmetric diffusion operator, the convergence in $L^p$-Wasserstein distance $\mathbb W_p (p\ge 1)$ is characterized for the empirical measure $\mu_t$ of non-symmetric subordinated diffusion processes…

By using the coupling technique, we present sufficient conditions for the exponential ergodicity of general continuous-state nonlinear branching processes in both the $L^1$-Wasserstein distance and the total variation norm, where the drift…

We consider contractivity for diffusion semigroups w.r.t. Kantorovich ($L^1$ Wasserstein) distances based on appropriately chosen concave functions. These distances are inbetween total variation and usual Wasserstein distances. It is shown…

We study ergodic properties of a class of Markov-modulated general birth-death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that…

There is a long history of establishing central limit theorems for Markov chains. Quantitative bounds for chains with a spectral gap were proved by Mann and refined later. Recently, rates of convergence for the total variation distance were…

For Markov processes evolving on multiple time-scales a combination of large component scalings and averaging of rapid fluctuations can lead to useful limits for model approximation. A general approach to proving a law of large numbers to a…

We discuss some applications of the Mittag-Leffler function and related probability distributions in the theory of renewal processes and continuous time random walks. In particular we show the asymptotic (long time) equivalence of a generic…

We propose a discrete time discrete space Markov chain approximation with a Brownian bridge correction for computing curvilinear boundary crossing probabilities of a general diffusion process on a finite time interval. For broad classes of…

We survey recent results of normal and anomalous diffusion of two types of random motions with long memory in ${\Bbb R}^d$ or ${\Bbb Z}^d$. The first class consists of random walks on ${\Bbb Z}^d$ in divergence-free random drift field,…

We establish subgeometric bounds on convergence rate of general Markov processes in the Wasserstein metric. In the discrete time setting we prove that the Lyapunov drift condition and the existence of a "good" $d$-small set imply…