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 R-d. In [1] we introduced a class H-1(D) of positive solutions of the equation Lu = -u(2) which is in 1-1 correspondence with a convex class H-1(D) of positive solutions of the equation Lu = 0. In the present paper, we give a probabilistic characterization of H-1(D) and a probabilistic representation of u is an element of H-1, (D) in terms of a superdiffusion. Similar results are obtained also for a parabolic equation (u) over dot + Lu = -u(2). (C) 2000 Academic Press.