Polar Sets and Solutions with Boundary Blow-up

Abstract

In this chapter, we consider the case when the spatial motion ξ is Brownian motion in ℝ^d and we continue our investigation of the connections between the Brownian snake and the partial differential equation Δu = 4u ². In partic-ular, we show that the maximal nonnegative solution in a domain D can be interpreted as the hitting probability of D ^c for the Brownian snake. We then combine analytic and probabilistic techniques to give a characterization of po-lar sets for the Brownian snake or equivalently for super-Brownian motion. In the last two sections, we investigate two problems concerning solutions with boundary blow-up. We first give a complete characterization of those domains in ℝ^d in which there exists a (nonnegative) solution which blows up every-where at the boundary. This analytic result is equivalent to a Wiener test for the Brownian snake or for super-Brownian motion. Finally, in the case of a regular domain, we give sufficient conditions that ensure the uniqueness of the solution with boundary blow-up.