Absorption properties of stochastic equations with H\"older diffusion coefficients

TL;DR

This paper investigates how the absorption behavior of certain stochastic differential equations depends on the H"older continuity of their diffusion coefficients near singular points, revealing regimes where quasi-stationary distributions exist or not.

Contribution

It characterizes the absorption properties of stochastic equations with H"older diffusion coefficients, identifying regimes for quasi-stationary distributions and their absence.

Findings

Quasi-stationary distributions exist for alpha < 3/4.

No quasi-stationary distributions for alpha in (3/4, 1).

Processes tend to almost sure exponential convergence to the absorbing state when conditioned on not being absorbed.

Abstract

In this article, we address the absorption properties of a class of stochastic differ- ential equations around singular points where both the drift and diffusion functions vanish. According to the H\"older coefficient alpha of the diffusion function around the singular point, we identify different regimes. Stability of the absorbing state, large deviations for the absorption time, existence of stationary or quasi-stationary distributions are discussed. In particular, we show that quasi-stationary distributions only exist for alpha < 3/4, and for alpha in the interval (3/4, 1), no quasi-stationary distribution is found and numerical simulations tend to show that the process conditioned on not being absorbed initiates an almost sure exponential convergence towards the absorbing state (as is demonstrated to be true for alpha = 1). Applications of these results to stochastic bifurcations are discussed.