The Brownian snake and solutions of Δu=u 2 in a domain

Abstract

We investigate the connections between the path-valued process called the Brownian snake and nonnegative solutions of the partial differential equation Δu=u2 in a domain of ℝ d . In particular, we prove two conjectures recently formulated by Dynkin. The first one gives a complete characterization of the boundary polar sets, which correspond to boundary removable singularities for the equation Δu=u2. The second one establishes a one-to-one correspondence between nonnegative solutions that are bounded above by a harmonic function, and finite measures on the boundary that do not charge polar sets. This correspondence can be made explicit by a probabilistic formula involving a special class of additive functionals of the Brownian snake. Our proofs combine probabilistic and analytic arguments. An important role is played by a new version of the special Markov property, which is of independent interest.