Regularity and Sensitivity for McKean-Vlasov SPDEs

TL;DR

This paper develops sensitivity analysis techniques for McKean-Vlasov stochastic partial differential equations, providing precise growth estimates of solutions and derivatives, which are crucial for understanding mean-field games with common noise.

Contribution

It introduces a novel sensitivity analysis framework for McKean-Vlasov SPDEs, including growth estimates and a method to handle equations with random coefficients.

Findings

Derived growth estimates for solutions and derivatives

Extended sensitivity analysis to equations with unbounded noise

Applied stochastic characteristics to connect stochastic and non-stochastic equations

Abstract

In the first part of the paper we develop the sensitivity analysis for the nonlinear McKean-Vlasov diffusions stressing precise estimates of growth of solutions and their derivatives with respect to the initial data, under rather general assumptions on the coefficients. The exact estimates become particularly important when treating the extension of these equations having random coefficient, since the noise is usually assumed to be unbounded. The second part contains our main results dealing with the sensitivity of stochastic McKean-Vlasov diffusions. By using the method of stochastic characteristics, we transfer these equations to the non-stochastic equations with random coefficients thus making it possible to use the estimates obtained in the first part. The motivation for studying sensitivity of McKean-Vlasov SPDEs arises naturally from the analysis of the mean-field games with common noise.