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 $f$.