A Probabilistic Approach to the Equation Lu=−u2

Let L be a second order elliptic differential operator and let D be an arbitrary open subset of Rd. In [1] we introduced a class U1(D) of positive solutions of the equation Lu=−u2 which is in 1–1 correspondence with a convex class H1(D) of positive solutions of the equation Lu=0. In the present paper, we give a probabilistic characterization of U1(D) and a probabilistic representation of u∈U1(D) in terms of a superdiffusion. Similar results are obtained also for a parabolic equation u+Lu=−u2.