A Feynman--Kac representation of a non-conservative and path-dependent nonlinear reaction-diffusion-advection system

We provide a probabilistic interpretation of a weakly parabolic PDE--ODE system with a reaction term, which makes the dynamics non-conservative. As a consequence, the solution is represented as the density of a sub-probability measure solving a Feynman--Kac-type equation, where the time-marginal law of the underlying process is weighted by a survival probability induced by the reaction. This leads to a coupled stochastic formulation consisting of a non-Markovian stochastic differential equation with path-dependent coefficients and the associated Feynman--Kac-type equation. We prove well-posedness of the resulting stochastic system. Finally, we introduce the corresponding interacting particle system and show that its empirical measure, suitably weighted by the survival probability associated with the reaction rate, converges to the limiting sub-probability.