Mirror Coupling of Reflecting Brownian Motion and an Application to Chavel's Conjecture

Mirror coupling of reflecting Brownian motion and an application to Chavel’s conjecture∗

In a series of papers, Burdzy et al. introduced the mirror coupling of reflecting Brownian motions in a smooth bounded domain D ⊂ Rd , and used it to prove certain properties of eigenvalues and eigenfunctions of the Neumann Laplaceian on D. In the present paper we show that the construction of the mirror coupling can be extended to the case when the two Brownian motions live in different domain...

The Semimartingale Structure of Reflecting Brownian Motion

We prove that reflecting Brownian motion on a bounded Lipschitz domain is a semimartingale. We also extend the well-known Skorokhod equation to this case. In this note we study the semimartingale property and the Skorokhod equation of reflecting Brownian motion on a bounded Euclidean domain. A R d_ valued continuous stochastic process X = {Xt ; t > 0} is said to be a semimartingale if it can be...

On Reflecting Brownian Motion with Drift

Let B = (Bt)t≥0 be a standard Brownian motion started at zero, and let μ ∈ R be a given and fixed constant. Set B t = Bt+μt and S t = max 0≤s≤t B s for t ≥ 0 . Then the process: (x ∨ Sμ)−Bμ = ((x ∨ S t )−B t )t≥0 realises an explicit construction of the reflecting Brownian motion with drift −μ started at x in R+ . Moreover, if the latter process is denoted by Z = (Z t )t≥0 , then the classic Lé...

Super-brownian Motion with Reflecting Historical Paths

We consider super-Brownian motion whose historical paths reflect from each other, unlike those of the usual historical super-Brownian motion. We prove tightness for the family of distributions corresponding to a sequence of discrete approximations but we leave the problem of uniqueness of the limit open. We prove a few results about path behavior for processes under any limit distribution. In p...

Super-Brownian motion with reflecting historical paths