Iterating Brownian Motions, Ad Libitum

Iterating Brownian motions, ad libitum

Let B1, B2, ... be independent one-dimensional Brownian motions defined over the whole real line such that Bi(0) = 0 for every i ≥ 1. We consider the nth iterated Brownian motion Wn(t) = Bn(Bn−1(...(B2(B1(t)))...)). Although the sequences of processes (Wn)n≥1 do not converge in a functional sense, we prove that the finite-dimensional marginals converge. As a consequence, we deduce that the rand...

Bouncing skew Brownian motions

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 pro...

Nonintersecting Planar Brownian Motions

In this paper we construct a measure on pairs of Brownian motions starting at the same point conditioned so their paths do not intersect. The construction of this measure is a start towards the rigorous understanding of nonintersecting Brownian motions as a conformal eld. Let B 1 ; B 2 be independent Brownian motions in R 2 starting at distinct points on the unit circle. Let T j r be the rst ti...

Perturbed Brownian Motions

Introduction to Brownian Motions

This paper aims to present some basic facts about Brownian Motions. It will assume a basic familiarity with probability and random variables. It will begin by defining Brownian Motion in one dimension on the dyadic rationals using a countable number of random variables and then proceed to generalize this to the real line using a continuity argument. Some other consequences of continuity will be...