A PROBABILISTIC SOLUTION TO THE STROOCK WILLIAMS EQUATION

作者：Peskir Goran^*

来源：Annals of Probability, 2014, 42(5): 2197-2206.

DOI：10.1214/13-AOP865

• Stroock-Williams equation
• Feller boundary condition
• sticky Brownian motion
• elastic Brownian motion
• reflecting Brownian motion
• local time
• creation
• killing

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.

• 出版日期2014-9