Non-linear Neumann's condition for the heat equation: a probabilistic representation using catalytic super-Brownian motion

详细信息    查看全文

• 作者：Jean-Franç ; ois Delmas ; Pascal Vogt
• 关键词：Non-linear boundary value problem ; Collision local time ; Catalytic super-Brownian motion ; Exit-measure
• 刊名：Annales de l'Institut Henri Poincare (B) Probability and Statistics
• 出版年：2005
• 出版时间：September-October 2005
• 年：2005
• 卷：41
• 期：5
• 页码：817-849
• 全文大小：322 K

文摘

Let D be a bounded domain in version=1&_userid=6230853&md5=60ac7f00b6b58f2ea85ca7f8675953c1""> with smooth boundary ∂D. We give a probabilistic representation formula for the non-negative solution of the mixed Dirichlet non-linear Neumann boundary value problem (DNP)

where version=1&_userid=6230853&md5=bf44f5a95bc6360175370f51d36e0bce"" title=""Click to view the MathML source"">(F₁,F₂) is a non-trivial partition of ∂D, φ is a non-negative, bounded and continuous function defined on version=1&_userid=6230853&md5=c1fbba02bbc17fcf1ecb7f1612766f44"" title=""Click to view the MathML source"">F₂, and version=1&_userid=6230853&md5=ffbdcd53503906f341a4aa5045ad3752"" title=""Click to view the MathML source"">∂_n denotes the outward normal derivative on the boundary of D.

To solve the DNP, we consider a catalytic super-Brownian motion with underlying motion a Brownian motion reflected on ∂D, killed when it reaches version=1&_userid=6230853&md5=9050a50712c031c65fb7095781b55aca"" title=""Click to view the MathML source"">F₂ and catalysed by the set version=1&_userid=6230853&md5=928d8f81857a2e9b804362c887014322"" title=""Click to view the MathML source"">F₁, i.e. the branching rate is given by the local time of the paths on version=1&_userid=6230853&md5=7dee9da143ce35d28330a50e2b4ad88d"" title=""Click to view the MathML source"">F₁. Then we prove that the log-Laplace transform of φ integrated with respect to the exit measure of the catalytic process on version=1&_userid=6230853&md5=f5f6b1ff9ce3f95b4491aa8f2e4a35c6"" title=""Click to view the MathML source"">F₂, is a non-negative weak solution of the DNP.

In a second part we show that we still have a probabilistic representation formula if the Dirichlet condition on version=1&_userid=6230853&md5=bf5431bcb197de96d656ed25b85c7aff"" title=""Click to view the MathML source"">F₂ is replaced by a Neumann condition.