Coupling of Multidimensional Diffusions by Reflection

Abstract

If $x \neq x'$ are two points of $\mathbb{R}^d, d \geq 2$, and if $X$ is a Brownian motion in $\mathbb{R}^d$ started at $x$, then by reflecting $X$ in the hyperplane $L \equiv \{y: |y - x| = |y - x'|\}$ we obtain a Brownian motion $X'$ started at $x'$, which couples with $X$ when $X$ first hits $L$. This paper deduces a number of well-known results from this observation, and goes on to consider the analogous construction for a diffusion $X$ in $\mathbb{R}^d$ which is the solution of an s.d.e. driven by a Brownian motion $B$; the essential idea is the reflection of the increments of $B$ in a suitable (time-varying) hyperplane. A completely different coupling construction is given for diffusions with radial symmetry.