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

Abstract

We consider general Markov processes with absorption and provide criteria ensuring the exponential convergence in total variation of the distribution of the process conditioned not to be absorbed. The first one is based on two-sided estimates on the transition kernel of the process and the second one on gradient estimates on its semigroup. We apply these criteria to multi-dimensional diffusion processes in bounded domains of $\mathbb {R}^d$ or in compact Riemannian manifolds with boundary, with absorption at the boundary.

Appendix: Proof of (5.2)

Let us assume that Condition (A) is satisfied. For all t ≥ 0 and all probability measure π on E, let . In the proof of [6, Corollary 2.2], it is proved that, for all probability measures π ₁, π ₂ on E

$$\displaystyle \begin{aligned} \left\|\mathbb{P}_{\pi_1}(X_t\in\cdot\mid t<\tau_\partial)-\mathbb{P}_{\pi_2}(X_t\in\cdot\mid t<\tau_\partial)\right\|{}_{TV} &\leq \frac{(1-c_1c_2)^{\lfloor t/t_0\rfloor}}{c_t(\pi_1)\vee c_t(\pi_2)}\|\pi_1-\pi_2\|{}_{TV}. \end{aligned} $$

But

$$\displaystyle \begin{aligned} \inf_{t\geq 0}c_t(\pi_1)\vee c_t(\pi_2)\geq (\inf_{t\geq 0} c_t(\pi_1))\vee (\inf_{t\geq 0} c_t(\pi_2))=c(\pi_1)\vee c(\pi_2). \end{aligned} $$

This ends the proof of (5.2).