ℕ-measures for branching exit Markov systems and their applications to differential equations

Abstract

Semilinear equations Lu = ψ(u) where L is an elliptic differential operator and ψ is a positive function can be investigated by using (L, ψ)-superdiffusions. In a special case Δu = u2 a powerful probabilistic tool - the Brownian snake - introduced by Le Gall was successfully applied by him and his school to get deep results on solutions of this equation. Some of these results (but not all of them) were extended by Dynkin and Kuznetsov to general equations by applying superprocesses. An important role in the theory of the Brownian snake and its applications is played by measures ℕx on the space of continuous paths. Our goal is to introduce analogous measures related to superprocesses (and to general branching exit Markov systems). They are defined on the space of measures and we call them ℕ-measures. Using ℕ-measures allows to combine some advantages of Brownian snakes and of superprocesses as tools for a study of semilinear PDEs.