The Annals of Probability

Coupling of Multidimensional Diffusions by Reflection

Torgny Lindvall and L. C. G. Rogers

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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.

Article information

Ann. Probab., Volume 14, Number 3 (1986), 860-872.

First available in Project Euclid: 19 April 2007

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Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60J65: Brownian motion [See also 58J65] 60H10: Stochastic ordinary differential equations [See also 34F05] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Coupling Brownian motion multidimensional diffusion stochastic differential equation reflection skew product stationary distribution radial process tail $\sigma$-field of a one-dimensional diffusion


Lindvall, Torgny; Rogers, L. C. G. Coupling of Multidimensional Diffusions by Reflection. Ann. Probab. 14 (1986), no. 3, 860--872. doi:10.1214/aop/1176992442.

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