Abstract
Abstract In this paper we study the quasi-stationary behavior of absorbed one-dimensional diffusions. We obtain necessary and sufficient conditions for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. An important tool is provided by one-dimensional strict local martingale diffusions coming down from infinity. We prove, under mild assumptions, that their expectation at any positive time is uniformly bounded with respect to the initial position. We provide several examples and extensions, including the sticky Brownian motion and some one-dimensional processes with jumps.
Highlights
This article studies the quasi-stationary behaviour of general one-dimensional diffusion processes in an interval E of R, absorbed at its finite bound
Our goal is to give conditions ensuring the existence of a quasi-stationary distribution α on E such that, for all probability measures μ on E and all t ≥ 0, Pμ(Xt ∈ · | t < τ∂) − α T V ≤ Ce−γt, (1.1)
We study the long time behavior of absorption probabilities and the exponential ergodicity in total variation of the Q-process associated to X, defined as the diffusion X conditioned never to hit ∂
Summary
This article studies the quasi-stationary behaviour of general one-dimensional diffusion processes in an interval E of R, absorbed at its finite bound-.
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