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 ℝd 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 ℝd 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.KeywordsBrownian MotionExit TimeNonnegative SolutionRegular DomainStrong Markov PropertyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.