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July, 1986 Coupling of Multidimensional Diffusions by Reflection
Torgny Lindvall, L. C. G. Rogers
Ann. Probab. 14(3): 860-872 (July, 1986). DOI: 10.1214/aop/1176992442


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.


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Torgny Lindvall. L. C. G. Rogers. "Coupling of Multidimensional Diffusions by Reflection." Ann. Probab. 14 (3) 860 - 872, July, 1986.


Published: July, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0593.60076
MathSciNet: MR841588
Digital Object Identifier: 10.1214/aop/1176992442

Primary: 60J60
Secondary: 60H10 , 60J45 , 60J65 , 60J70

Keywords: Brownian motion , coupling , multidimensional diffusion , radial process , reflection , Skew product , stationary distribution , Stochastic differential equation , tail $\sigma$-field of a one-dimensional diffusion

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 3 • July, 1986
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