A PROBABILISTIC SOLUTION TO THE STROOCK WILLIAMS EQUATION

• 成果类型: Article
• 署名作者: Peskir, Goran
• 署名单位: University of Manchester
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP865
• 发表日期: 2014
• 页码: 2197-2206
• 关键词: boundary-value problem; simple pde

摘要

We consider the initial boundary value problem ut = mu u(x) + 1/2u(xx) (t > 0, x >= 0) u(0, x) = f(x) (x >= 0), u(t)(t, 0) = vu(x) (t, 0) (t > 0) of Stroock and Williams [Comm. Pure Appl. Math. 58 (2005) 1116-1148] where, mu, v is an element of R and the boundary condition is not of Feller's type when v <0. We show that when f belongs to C-b(1) with f (infinity) = 0 then the following probabilistic representation of the solution is valid: u(t, x) = Ex[f'(Xt)integral(0) (lt0(x)) (e-2(v - mu)s) ds] where X is a reflecting Brownian motion with drift, mu and l0(X) is the local time of X at 0. The solution can be interpreted in terms of X and its creation in 0 at rate proportional to f0(X). Invoking the law of (Xt, 4(X)), this also yields a closed integral formula for u expressed in terms of mu, v and f.