Reflection and coalescence between independent one-dimensional Brownian paths

Take two independent one-dimensional processes as follows: (B-r, t is an element of [0, 1]) is a Brownian motion with B-0 = 0, and (B-t, t is an element of [0, 1]) has the same law as (B1-t, t is an element of [0, 1]); in other words, beta(1) = 0 and beta can be seen as Brownian motion running backwards in time. Define (gamma(t), t is an element of [0, 1]) as being the function that is obtained by reflecting B on beta. Then gamma is still a Brownian motion. Similar and more general results (with families of coalescing Brownian motions) are also derived. They enable us to give a precise definition (in terms of reflection) of the joint realization of finite families of coalescing/reflecting Brownian motions. (C) 2000 Editions scientifiques et medicales Elsevier SAS.