A probabilistic Poisson representation for positive solutions of Δu = u2 in a planar domain

We use the stochastic process called the Brownian snake to investigate solutions of the partial differential equation Δu = u² in a domain D of class C² of the plane. We prove that nonnegative solutions are in one-to-one correspondence with pairs (K, v) where K is a closed subset of ?D and v is a Radon measure on ?D\K. Both Kand v are determined from the boundary behavior of the solution u. On the other hand, u can be expressed in terms of the pair (K, v) by an explicit probabilistic representation formula involving the Brownian snake. © 1997 John Wiley & Sons, Inc.