We address two probabilistic approaches for associating a specific stochastic dynamics 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 both the sub-probability measure and 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 with memory 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. Then, by considering the interacting particle systems associated with each of the microscopic models, we discuss how their empirical densities provide two different kernel estimators for the PDE solution.

Finally, we discuss how the convergence to a standard advection-diffusion-reaction McKean-type PDE is achieved by rescaling the interaction kernel at an intermediate scale and using a semigroup approach.

The PDE model under consideration arises in applications in materials science: it describes the sulphation phenomenon, a degradation process affecting marble surfaces exposed to atmospheric pollutants.