MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
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

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 $0$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.