Uniform Conditional Ergodicity and Intrinsic Ultracontractivity

Abstract

We study two properties of semigroups of sub-Markov kernels, namely uniform conditional ergodicity and intrinsic ultracontractivity. In this paper we investigate the relationship between these two properties and we provide sufficient criteria as well as characterisations of them. In particular, our considerations show that, under suitable assumptions, the second property implies the first one. We also introduce a property called compact domination and show how this property and the parabolic boundary Harnack principle are related to the aforementioned properties. Furthermore, we apply these results in some special cases.

Ultracontractivity for Markov Semigroups and Quasi-Stationary Distributions

“…However, the existence of a Yaglom limit does not always imply the uniqueness of QSDs. In [8], it is proved that the intrinsic ultracontractivity (see Definition 2.5 below) is a sufficient condition for the uniqueness of QSDs. We will Downloaded by [Purdue University] at 07:43 29 August 2014 Miura give another proof of this fact for symmetric Markov processes.…”

“…SIU) was introduced by Davies and Simon [3] (resp. Banuelos and Davis [1]), and investigated extensively because of its important consequences (see [1,3,8,10,12] and references therein). The IU implies the SIU trivially.…”

Criteria for Exponential Convergence to Quasi-Stationary Distributions and Applications to Multi-Dimensional Diffusions

“…Convergence of conditioned diffusion processes have been already obtained for diffusions in domains of R d , mainly using spectral theoretic arguments (see for instance [3,19,23,24,14,5] for d = 1 and [4,18,12] for d ≥ 2). Among these references, [18,12] give the most general criteria for diffusions in dimension 2 or more. Using two-sided estimates and spectral properties of the infinitesimal generator of X, Knobloch and Partzsch [18] proved that (1.1) holds for a class of diffusion processes evolving in R d (d ≥ 3) with C 1 diffusion coefficient, drift in a Kato class and C 1,1 domain.…”

“…Among these references, [18,12] give the most general criteria for diffusions in dimension 2 or more. Using two-sided estimates and spectral properties of the infinitesimal generator of X, Knobloch and Partzsch [18] proved that (1.1) holds for a class of diffusion processes evolving in R d (d ≥ 3) with C 1 diffusion coefficient, drift in a Kato class and C 1,1 domain. In [12], the authors obtain (1.1) for diffusions with global Lipschitz coefficients (and additional local regularity near the boundary) in a domain with C 2 boundary.…”

Quasi-stationarity and quasi-ergodicity of general Markov processes

“…When the state space E of the Markov process is countable, quasi-stationarity and quasi-ergodicity have been thoroughly studied, see, for instance, [7,8,10,23]. For Markov processes on general state spaces, Breyer and Roberts [2] established the existence and uniqueness of quasi-ergodic distributions under the assumption that the Markov process is positive λ-recurrent for some constant λ ≤ 0 (see also the recent paper [3,18]). For a survey on quasi-stationary distributions, see [24].…”