If x ≠ x' are two points of Rᵈ, d ≥ 2, and if X is a Brownian motion in Rᵈ started at x, then by reflecting X in the hyperplane L ≡ : |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 Rᵈ 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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Lindvall et al. (1986) studied this question.