Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift

TL;DR

This paper introduces a new approximation method for the Yaglom limit of multi-dimensional diffusions with unbounded drift, using interacting particle systems, and demonstrates its effectiveness through numerical examples in biological models.

Contribution

The paper develops a novel particle system-based approach to approximate Yaglom limits, enabling numerical computation of quasi-stationary distributions for complex diffusions.

Findings

The method successfully approximates Yaglom limits in high-dimensional diffusions.

Numerical results demonstrate applicability to biological population models.

The approach provides a practical alternative to spectral theory methods.

Abstract

We study the existence and the exponential ergodicity of a general interacting particle system, whose components are driven by independent diffusion processes with values in an open subset of $\mathds{R}^d$, $d\geq 1$. The interaction occurs when a particle hits the boundary: it jumps to a position chosen with respect to a probability measure depending on the position of the whole system. Then we study the behavior of such a system when the number of particles goes to infinity. This leads us to an approximation method for the Yaglom limit of multi-dimensional diffusion processes with unbounded drift defined on an unbounded open set. While most of known results on such limits are obtained by spectral theory arguments and are concerned with existence and uniqueness problems, our approximation method allows us to get numerical values of quasi-stationary distributions, which find applications to many disciplines. We end the paper with numerical illustrations of our approximation method for stochastic processes related to biological population models.