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 $\mathbb{R}^d$, $d\geq1$. 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.
Highlights
Let D ⊂ Rd be an open set with a regular boundary
The first part of this paper is devoted to the study of interacting particle systems (X1,...,XN ), whose components Xi evolve in D as diffusion processes and jump when they hit the boundary ∂D
The particles Xi evolve as independent diffusion processes with values in D defined by dXt(i) = dBti + qi(N)(Xt(i))dt, X0(i) ∈ D, (1)
Summary
Let D ⊂ Rd be an open set with a regular boundary (see Hypothesis 1). The first part of this paper is devoted to the study of interacting particle systems (X1,...,XN ), whose components Xi evolve in D as diffusion processes and jump when they hit the boundary ∂D. We identify the limit of the family of empirical stationary distributions (X N )N≥2 This leads us to an approximation method of limiting conditional distributions of diffusion processes absorbed at the boundary of an open set of Rd, studied by Cattiaux and Meleard in [7] and defined as follows. In the case of a Brownian motion absorbed at the boundary of a bounded open set (i.e. q = 0), Burdzy et al conjectured in [4] that the unique limiting measure of the sequence (X N )N∈N is the Yaglom limit ν∞ This has been confirmed in the Brownian motion case (see [5], [18] and [26]) and proved in [16] for some Markov processes defined on discrete spaces.
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