A probabilistic solution to the Stroock.Williams equation

We consider the initial boundary value problem ut=.ux+12uxx(t>0,x.0),u(0,x)=f(x)(x.0),ut(t,0)=.ux(t,0)(t>0) of Stroock and Williams [Comm. Pure Appl. Math. 58 (2005) 1116.1148] where .,..R and the boundary condition is not of Feller.s type when .<0. We show that when f belongs to C1b with f(.)=0 then the following probabilistic representation of the solution is valid: u(t,x)=Ex[f(Xt)].Ex[f.(Xt)..0t(X)0e.2(...)sds], where X is a reflecting Brownian motion with drift . and .0(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 .0(X). Invoking the law of (Xt,.0t(X)), this also yields a closed integral formula for u expressed in terms of ., . and f.