A probabilistic solution to the Stroock-Williams equation

Abstract

We consider the initial boundary value problem \begin{eqnarray*}u_t=\mu u_x+\tfrac{1}{2}u_{xx}\qquad (t>0,x\ge0),\\u(0,x)=f(x)\qquad (x\ge0),\\u_t(t,0)=\nu u_x(t,0)\qquad (t>0)\end{eqnarray*} of Stroock and Williams [Comm. Pure Appl. Math. 58 (2005) 1116-1148] where $\mu,\nu\in \mathbb{R}$ and the boundary condition is not of Feller's type when $\nu<0$. We show that when $f$ belongs to $C_b^1$ with $f(\infty)=0$ then the following probabilistic representation of the solution is valid: \[u(t,x)=\mathsf{E}_x\bigl[f(X_t)\bigr]-\mathsf{E}_x\biggl[f'(X_t)\int_0^{\ell_t^0(X)}e^{-2(\nu-\mu)s}\,ds\biggr],\] where $X$ is a reflecting Brownian motion with drift $\mu$ and $\ell^0(X)$ 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 $\ell^0(X)$. Invoking the law of $(X_t,\ell_t^0(X))$, this also yields a closed integral formula for $u$ expressed in terms of $\mu$, $\nu$ and $f$.