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 easier-to-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.
Highlights
Consider a time-homogeneous Markov process (Xt)t≥0 defined on a metric state space (E ∪ {∂}, d), where the element ∂ ∈ E is a cemetery point for the process X, which means that
In [9], necessary and sufficient conditions for the uniform-in-law exponential convergence in total variation are provided, where we recall that the total variation distance of two probability measures μ, ν is defined by μ − ν T V := sup |μ(f ) − ν(f )|
Where (Bt)t≥0 is a d-dimensional Brownian motion and V is a C2-function on Rd
Summary
Consider a time-homogeneous Markov process (Xt)t≥0 defined on a metric state space (E ∪ {∂}, d), where the element ∂ ∈ E is a cemetery point for the process X, which means that. In [9], necessary and sufficient conditions for the uniform-in-law exponential convergence in total variation are provided, where we recall that the total variation distance of two probability measures μ, ν is defined by μ − ν T V := sup |μ(f ) − ν(f )|. Some papers dealing with the use of Poincaré inequalities for quasi-stationarity have been already written, in particular for Markov processes living on discrete state spaces ([13, 17, 18]). Where (Bt)t≥0 is a d-dimensional Brownian motion and V is a C2-function on Rd. In the non-absorbed framework, it is well-known that the reversible probability measure γ(dx) := Z−1e−V (x)dx (Z is the renormalization constant) satisfies a Poincaré inequality when the condition. A particular attention will be paid on processes coming down from infinity, for which it will be shown that the rate of convergence κ provided by the Bakry-Émery condition (1.4) can be bettered (see Theorems 3.6 and 3.11)
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