Path-dependent McKean PDEs with reaction: a discussion on probabilistic interpretations and particle approximations

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.

💡 Research Summary

The paper investigates two distinct probabilistic frameworks for interpreting and numerically approximating a class of non‑conservative McKean‑Vlasov–Fokker‑Planck partial differential equations (PDEs) that contain a reaction term and path‑dependent coefficients. The authors start from a deterministic macroscopic model describing the diffusion of sulfur dioxide (SO₂) in marble and its reaction with calcite, which leads to a coupled PDE‑ODE system. By eliminating the ODE for the calcite concentration, they obtain a single non‑conservative, path‑dependent PDE for the SO₂ concentration ρ(t,x) with a drift term that depends on the convolution of ρ with a smooth kernel K and on an exponential functional of the accumulated mass. This PDE can be written as

∂ₜρ = Δρ – ∇·