A stochastic McKean--Vlasov equation for absorbing diffusions on the half-line

Understand

We study a finite system of diffusions on the half-line, absorbed when they hit zero, with a correlation effect that is controlled by the proportion of the processes that have been absorbed.

• As the number of processes in the system becomes large, the empirical measure of the population converges to the solution of a non-linear stochastic heat equation with Dirichlet boundary condition.
• The diffusion coefficients are allowed to have finitely many discontinuities (piecewise Lipschitz) and we prove pathwise uniqueness of solutions to the limiting stochastic PDE.
• As a corollary we obtain a representation of the limit as the unique solution to a stochastic McKean--Vlasov problem.