Dimension-free local convergence and perturbations for reflected Brownian motions

Weak Convergence of Reflecting Brownian Motions

1. Introduction. We will show that if a sequence of domains D k increases to a domain D then the reflected Brownian motions in D k 's converge to the reflected Brownian motion in D, under mild technical assumptions. Our theorem follows easily from known results and is perhaps known as a " folk law " among the specialists but it does not seem to be recorded anywhere in an explicit form. The purp...

Differentiability of Stochastic Flow of Reflected Brownian Motions

We prove that a stochastic flow of reflected Brownian motions in a smooth multidimensional domain is differentiable with respect to its initial position. The derivative is a linear map represented by a multiplicative functional for reflected Brownian motion. The method of proof is based on excursion theory and analysis of the deterministic Skorokhod equation.

Local Fields, Gaussian Measures, and Brownian Motions

Synchronous Couplings of Reflected Brownian Motions in Smooth Domains

For every bounded planar domain D with a smooth boundary, we define a “Lyapunov exponent” Λ(D) using a fairly explicit formula. We consider two reflected Brownian motions in D, driven by the same Brownian motion (i.e., a “synchronous coupling”). If Λ(D) > 0 then the distance between the two Brownian particles goes to 0 exponentially fast with rate Λ(D)/(2|D|) as time goes to infinity. The expon...

Reflected Backward Stochastic Differential Equations Driven by Countable Brownian Motions

In this note, we study one-dimensional reflected backward stochastic differential equations (RBSDEs) driven by Countable Brownian Motions with one continuous barrier and continuous generators. Via a comparison theorem, we provide the existence of minimal and maximal solutions to this kind of equations.