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Suppose $D$ is a bounded Lipschitz domain in $R^n$ and $X_1$, $X_2$ are two independent reflected Brownian motions (RBMs) in $D$. Is it true that

$$P[X_1(t)\neq X_2(t) \text{ for all }t>0]=1$$

It seems to be true for dimension $n\geq 2$.

But how to prove it? Does anyone know any reference? Thanks!

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I think we need a bit more details: reflected where? And are they indepent?it would be great if you could give a little more specifications of your problem. –  Stephan Sturm Nov 1 '12 at 21:03
    
Thanks! I've added more detail. –  Fantastic Nov 2 '12 at 3:24
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1 Answer 1

The pair $(X_1(t),X_2(t))$ is a reflected Brownian motion in $D \times D \subset {\mathbb R}^{2n}$. Its sample paths have Hausdorff dimension 2 but 2-dimensional Hausdorff measure 0. The diagonal $\{(x,x): x \in D\}$ has dimension $n$, so if $n \ge 2$ the probability of the sample path intersecting the diagonal should be $0$.

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