We prove existence and pathwise uniqueness of a strong solution for a system of N interacting stochastic particles driven by independent Brownian motions. The particles are subject to pairwise interactions generated by a strongly singular Lennard–Jones-type potential, producing repulsive-attractive forces with singular behavior at the origin, and to a nonlocal interaction with an underlying field c. Each particle evolves until a random reaction time, at which it switches from an active to an inactive state with a rate depending on the value of the field at its position.
The coupled process describes the particle positions, their activity states, and the underlying field. The field influences the dynamics in two ways: it biases the motion of active particles through a nonlocal drift and modulates their reaction rates. In turn, the field evolves according to the instantaneous particle configuration, yielding a fully coupled stochastic system.
The proof of well-posedness proceeds in two steps. We first establish existence and pathwise uniqueness for the diffusion system without reactions. We then incorporate the jump mechanism through an interlacing argument, obtaining well-posedness for the full coupled dynamics. The model is motivated by microscopic stochastic descriptions of sulphation processes in cultural-heritage materials.