Path-dependent McKean PDEs with reaction: a discussion on probabilistic interpretations and particle approximations
Daniela Morale, Leonardo Tarquini, Stefania Ugolini
TL;DR
The paper analyzes non-conservative McKean–Vlasov PDEs with reaction and path-dependent coefficients by contrasting two probabilistic interpretations: a measure-valued Feynman–Kac representation and a sub-probability framework arising from a killing mechanism. Each viewpoint yields a distinct stochastic model at the microscale (MKFK-SDE vs. killed MKV SDE) and, correspondingly, two kernel-based particle systems used to estimate the PDE solution. The authors establish well-posedness results for the two SDE models, connect the stochastic dynamics to the macroscopic PDE through measure-valued equations, and show how the empirical densities from the particle systems provide two different kernel estimators of the same PDE solution. The work offers a conceptual bridge between microscopic randomness and macroscopic, non-conservative dynamics, with implications for numerical methods and applications such as marble degradation under sulfur dioxide.
Abstract
In this paper, we discuss and compare two probabilistic approaches for associating a stochastic differential equation with a McKean-type partial differential equation featuring a reaction term and path-dependent coefficients. The non-conservative nature of the macroscopic dynamics leads to two possible interpretations of the sub-probability measure and of the associated SDE equation at the microscale: on the one hand, as a measure-valued solution of a Feynman-Kac-type equation; on the other hand, as the sub-probability associated with an SDE defined up to a survival time with a reaction-dependent rate. These different interpretations give rise to two different microscopic stochastic models and therefore to two different techniques of probabilistic analysis. Finally, by considering the interacting particle systems associated with both models, we discuss how their empirical densities provide two different kernel estimators for the PDE solution.