Bouncing skew Brownian motions.

Summary We consider two skew Brownian motions, driven by the same Brownian motion, with different starting points and different skewness coefficients. In [13], the evolution of the distance between the two processes, in local time scale and up to their first hitting time is shown to satisfy a stochastic differential equation with jumps. The jumps of this S.D.E. are naturally driven by the excursion process of one of the two skew Brownian motions. In this article, we show that the description of the distance of the two processes after this first hitting time may be studied using the self similarity induced by the previous S.D.E. More precisely, we show that the distance between the two processes in local time scale may be viewed as the unique continuous markovian self-similar extension of the process described in [13]. This permits us to compute the law of the distance of the two skew Brownian motions at any time in the local time scale, when both original skew Brownian motions start from zero. As a by product, we manage to study the markovian dependence on the skewness parameter and answer an open question formulated initially by C. Burdzy and Z.Q. Chen in [6].