Dirichlet Brownian Motions

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...

Dirichlet Heat Kernel Estimates for Subordinate Brownian Motions with Gaussian Components

In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels, in C open sets D in R, of a large class of subordinate Brownian motions with Gaussian components. When D is bounded, our sharp two-sided Dirichlet heat kernel estimates hold for all t > 0. Integrating the heat kernel estimates with respect to the time variable t, we obtain sharp two-sided estimates for t...