Ultracontractivity for Markov Semigroups and Quasi-Stationary Distributions

Abstract

We prove the existence and uniqueness of quasi-stationary distributions for symmetric Markov processes. In particular, we show that if its Markov semigroup is intrinsic ultracontractive, then there exists a unique quasi-stationary distribution. We apply our results to one-dimensional diffusion processes.

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

“…They proved that (1.1) holds for a class of diffusion processes evolving in R d (d ≥ 3), assuming continuity of the transition density, existence of ground states and the existence of a two-sided estimate involving the ground states of the generator. Similar results were obtained in the one-dimensional case in [24].…”

“…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.…”

Uniform convergence of conditional distributions for absorbed one-dimensional diffusions

“…Following the scale function trick of the last section, our results actually cover all SDEs of the form ∞)) (see [13,Chapter 23]). Existence of quasi-stationary distributions of diffusion processes have already been studied in [11], [14], [21], and [26]. The question of convergence of conditional distributions was also studied in [2] and [17].…”

“…Hence, convergence of conditioned diffusion processes have been obtained up to now using spectral theoretic arguments (self-adjoint operators, Sturm-Liouville theory, and so on), which most often require regularity of the coefficients (see, for example, [2], [11], [14], [17], [21], and [26]; see also Subsection 4.5.2 below) and are specific to time-homogeneous onedimensional diffusion processes. The question of uniform convergence with respect to the initial distribution was not studied in the above references.…”

On quasi-ergodic distribution for one-dimensional diffusions

“…According to the proof of [2, Theorem 8.2] and [11,Theorem 3.4], we know that the hypotheses (H) and [IU] are satisfied for X respectively. So it follows from the previous section that there exists a unique quasi-ergodic distribution for Y.…”