Convergence to quasi-stationarity through Poincaré inequalities and Bakry-Émery criteria

Abstract

This paper aims to provide some tools coming from functional inequalities to deal with quasi-stationarity for absorbed Markov processes. First, it is shown how a Poincaré inequality related to a suitable Doob transform entails exponential convergence of conditioned distributions to a quasi-stationary distribution in total variation and in 1-Wasserstein distance. A special attention is paid to multi-dimensional diffusion processes, for which the aforementioned Poincaré inequality is implied by an easierto-check Bakry-Émery condition depending on the right eigenvector for the sub-Markovian generator, which is not always known. Under additional assumptions on the potential, it is possible to bypass this lack of knowledge showing that exponential quasi-ergodicity is entailed by the classical Bakry-Émery condition.

Uniform convergence of conditional distributions for one-dimensional diffusion processes

“…Here, • T V is the total variation distance, which is usually used to quantify the weak convergence (1.4) (see, e.g., [2,3,5,12,24]), defined as follows:…”

“…First, the equality (2.5) naturally suggests the use of the Doob's h-transform to study quasi-stationarity of the original absorbed process X. This method has been used successfully in, for example, [9,12,23,24,27]. Besides, another useful piece of information is that the spectrum is invariant under Doob's h-transform (see [25, Chapter 4, Sections 3 and 10]).…”

A central limit and Berry-Esseen theorem for continuous-time Markov processes conditioned not to be absorbed

“…Assumption 1 is also satisfied for general strongly Feller processes, as shown in [15], and for some degenerate diffusion processes, as studied in [3,21]. We refer the reader to [7,14,27,2,25] for alternative criteria ensuring Assumption 1.…”

“…The inequality (11) tells that the moments of even order converge to the ones of a Gaussian law at speed 1/t. Gathering (23), ( 24) and (25), one obtains 1/ √ t as speed for the convergence of the moments of odd order. Using a similar reasoning as used in the proof of Theorem 1 (in the last section), a Berry-Esseen theorem could be expected for continuous-time Markov processes satisfying (4), which would better the result stated in [22,Theorem 1.5].…”

On the Stability of Positive Semigroups

“…We also refer the reader to [174] for related Lipschitz norm techniques as well as the more recent articles [12,41,100,112] in the context of positivity preserving operators arising in particle absorption models and quasi-invariant measure literature. Functional inequalities, including Poincaré inequalities and Bakry-Emery criteria approaches are discussed in [153].…”