## The Annals of Probability

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

### Coupling of Multidimensional Diffusions by Reflection

Torgny Lindvall and L. C. G. Rogers

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992442

**Digital Object Identifier**

doi:10.1214/aop/1176992442

**Mathematical Reviews number (MathSciNet)**

MR841588

**Zentralblatt MATH identifier**

0593.60076

**JSTOR**

links.jstor.org

**Subjects**

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]

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aop/1176992442