A Probabilistic Approach to the Equation Lu=−u2
Eugene B. Dynkin · Journal of Functional Analysis · 2000
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 U 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 U 1 ( D ) and a probabilistic representation of u ∈ U 1 ( D ) in terms of a superdiffusion. Similar results are obtained also for a parabolic equation u + Lu =− u 2 .