It{\^o}'s formula for the flow of measures of Poisson stochastic integrals and applications

TL;DR

This paper extends Itô's formula to the flow of measures for jump processes driven by Poisson random measures, and applies it to derive PDEs and propagation of chaos results for mean-field systems with stable noise.

Contribution

It introduces an Itô's formula for measure flows of Poisson-driven jump processes and applies it to PDEs and propagation of chaos in mean-field models.

Findings

Derived backward Kolmogorov PDE on measure space.

Proved new quantitative weak propagation of chaos results.

Applied to systems driven by α-stable processes.

Abstract

We prove It{\^o}'s formula for the flow of measures associated with a jump process defined by a drift, an integral with respect to a Poisson random measure and with respect to the associated compensated Poisson random measure. We work in $\mathcal{P}_{\beta}(\mathbb{R}^d)$, the space of probability measures on $\mathbb{R}^d$ having a finite moment of order $\beta \in (0, 2]$. As an application, we exhibit the backward Kolmogorov partial differential equation stated on $[0,T] \times \mathcal{P}_{\beta}(\mathbb{R}^d)$ associated with a McKean-Vlasov stochastic differential equation driven by a Poisson random measure. It describes the dynamics of the semigroup associated with the McKean-Vlasov stochastic differential equation, under regularity assumptions on it. Finally, we use the semigroup and the backward Kolmogorov equation to prove new quantitative weak propagation of chaos results for a mean-field system of interacting Ornstein-Uhlenbeck processes driven by i.i.d. $\alpha$-stable processes with $\alpha \in (1,2)$.