## Coupling of multidimensional diffusions by reflection.(English)Zbl 0593.60076

Summary: 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 $$L\equiv \{y:\quad | y-x| =| y-x'| \}$$ 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.

### MSC:

 60J60 Diffusion processes 60J65 Brownian motion 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J45 Probabilistic potential theory 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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