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Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift. (English) Zbl 1244.82052
Summary: 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.

##### MSC:
 82C22 Interacting particle systems in time-dependent statistical mechanics 65C50 Other computational problems in probability (MSC2010) 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J60 Diffusion processes
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