Uniform convergence of proliferating particles to the FKPP equation

Abstract

In this paper we consider a system of Brownian particles with proliferation whose rate depends on the empirical measure. The dependence is more local than a mean field one and has been called moderate interaction by Oelschläger [17], [18]. We prove that the empirical process converges, uniformly in the space variable, to the solution of the Fisher-Kolmogorov-Petrowskii-Piskunov equation. We use a semigroup approach which is new in the framework of these systems and is inspired by some literature on stochastic partial differential equations.

Uniform Approximation of 2 Dimensional Navier--Stokes Equation by Stochastic Interacting Particle Systems

“…Note that this probability measure is more regular than S N t , and its nicer properties allow us to obtain better convergence results. To prove the latter, we follow the new approach presented in [7] and then in [9,8,29], based on semigroup theory. Our source of inspiration has been the works of Oelschläger [25] and Jourdain and Méléard [18], where stochastic approximations of PDE's are investigated.…”

A law of large numbers for interacting diffusions via a mild formulation

“…Studying the action of an analytic semigroup in the evolution of an interacting particle system has been recently proposed in similar settings; we refer to [13,14] and references therein. This method is referred to as the semigroup approach.…”

Quantitative particle approximation of nonlinear Fokker-Planck equations with singular kernel

“…In the moderate interaction setting, one introduces a smoothing of the interaction kernel. This has been successfully applied in the aforementioned works [5], [21], as well as by Méléard [37] and Méléard and Roelly-Coppoletta [38], and more recently with a new semigroup approach developed by Flandoli et al [18].…”

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