Solving semilinear partial differential equations with probabilistic potential theory

Abstract

Techniques of probabilistic potential theory are applied to solve − L u + f ( u ) = μ - Lu + f(u) = \mu , where μ \mu is a signed measure, f f a (possibly discontinuous) function and L L a second order elliptic or parabolic operator on R d {R^d} or, more generally, the infinitesimal generator of a Markov process. Also formulated are sufficient conditions guaranteeing existence of a solution to a countably infinite system of such equations.