Solutions of Δu=4u2 with Neumann's conditions using the Brownian snake

• 成果类型: Article
• 署名作者: Abraham, R; Delmas, JF
• 署名单位: Universite Paris Cite; Institut Polytechnique de Paris; Ecole Nationale des Ponts et Chaussees
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-003-0302-2
• 发表日期: 2004
• 页码: 475-516
• 关键词: probabilistic approach; potential-theory; delta-u=u(2); motion; domain

摘要

We consider a Brownian snake (W-s, s greater than or equal to 0) with underlying process a reflected Brownian motion in a bounded domain D. We construct a continuous additive functional (L-s, s greater than or equal to0) of the Brownian snake which counts the time spent by the end points (W) over cap (s) of the Brownian snake paths on partial derivativeD. The random measure Z = integraldelta(delta) over cap (s) dL(s) is supported by partial derivativeD. Then we represent the solution v of Deltau = 4u(2) in D with weak Neumann boundary condition phi greater than or equal to 0 by using exponential moment of (Z, phi) under the excursion measure of the Brownian snake. We then derive an integral equation for v. For small phi it is then possible to describe negative solution of v of Deltau = 4u(2) in D with weak Neumann boundary condition phi. In contrast to the exit measure of the Brownian snake out of D, the measure Z is more regular. In particular we show it is absolutely continuous with respect to the surface measure on partial derivativeD for dimension 2 and 3.