Approximation of non-linear SPDEs with additive noise via weighted interacting particles systems: the stochastic McKean-Vlasov equation

TL;DR

This paper develops a method to approximate non-linear SPDEs, specifically the stochastic McKean-Vlasov equation with additive noise, using weighted interacting particle systems with evolving weights and common noise.

Contribution

It introduces a novel approach using weighted empirical measures with time-evolving weights to approximate the stochastic McKean-Vlasov equation as a limit of interacting particle systems.

Findings

Weighted empirical measures converge to the SMKV equation

Method extends existing particle approximation techniques

Provides new insights into SPDE representation via particle systems

Abstract

This paper is devoted to the problem of approximating non-linear Stochastic Partial Differential Equations (SPDEs) via interacting particle systems. In particular, we consider the Stochastic McKean-Vlasov equation, which is the McKean-Vlasov (MKV) PDE, perturbed by additive trace class noise. As is well-known, the MKV PDE can be obtained as mean field limit of the empirical measure of a stochastic system of interacting particles, where particles are subject to independent sources of noise. There is now a natural question, which is the one we consider and answer in this paper: can we obtain the SMKV equation, i.e. additive perturbations of the MKV PDE, as limit of interacting particle systems? It turns out that, in order to obtain the SMKV equation, one needs to study weighted empirical measures of particles, where the particles evolve according to a system of SDEs with independent noise, while the weights are time evolving and subject to common noise. The work of this manuscript therefore complements and contributes to various streams of literature, in particular: i) much attention in the community is currently devoted to obtaining SPDEs as scaling limits of appropriate dynamics; this paper contributes to a complementary stream, which is devoted to obtaining representations of SPDE through limits of empirical measures of interacting particle systems; ii) since the literature on limits of weighted empirical measures is often constrained to the case of static (random or deterministic) weights, this paper contributes to further expanding this line of research to the case of time-evolving weights.