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

✍ Scribed by Jean-François Le Gall

• Publisher: John Wiley and Sons
• Year: 1997
• Tongue: English
• Weight: 255 KB
• Volume: 50
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/(sici)1097-0312(199701)50:1<69::aid-cpa2>3.0.co;2-e

✦ Synopsis

We use the stochastic process called the Brownian snake to investigate solutions of the partial differential equation ∆u = u 2 in a domain D of class C 2 of the plane. We prove that nonnegative solutions are in one-to-one correspondence with pairs (K, ν) where K is a closed subset of ∂D and ν is a Radon measure on ∂D\K. Both K and ν are determined from the boundary behavior of the solution u. On the other hand, u can be expressed in terms of the pair (K, ν) by an explicit probabilistic representation formula involving the Brownian snake.