A probabilistic solution to the Stroock-Williams equation

TL;DR

This paper provides a probabilistic representation for solutions to a boundary value problem related to the Stroock-Williams equation, involving reflecting Brownian motion with drift and local time, even when boundary conditions are not of Feller's type.

Contribution

It introduces a novel probabilistic formula for the solution of the Stroock-Williams equation with non-Feller boundary conditions, using reflecting Brownian motion and local time.

Findings

Valid probabilistic representation of the solution.

Closed integral formula involving drift and boundary parameters.

Applicable to boundary conditions not of Feller's type.

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 $f$.