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In other words, the probability that Brownian motion stays within $A^{c}$.

What about for connected and fixed compact sets ? Would that involve solving a heat equation? How can I condition it, so that it's solutions will avoid a set?

I am asking about the "specific" probability of not hitting non-polar sets (positive hitting probability). For example for d-sphere we know the hitting probability to be $(\frac{r}{x})^{d-2}$. So avoiding it is $1-(\frac{r}{x})^{d-2}$.

What would be the probability of avoiding two spheres?

Feel free to reference any book or paper.

Thanks

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  • $\begingroup$ How can the probability of avoiding an open set (or set with interior) be one? Suppose an open set $A$ is almost surely avoided. We can take rotated copies of it to to cover a spherical layer $L=B(0,R)\setminus B(0,r)$ for some $0<r<R$. Since the BM is rotationally invariant, it almost surely avoids $L$. By continuity, the BM stays almost surely in a bounded set. This is false. (Or did I miss something?) $\endgroup$
    – Joonas Ilmavirta
    Jul 28, 2014 at 17:36
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    $\begingroup$ This problem was completely solved by Kakutani in the 50s, at least when $A^c$ is closed. In this case, a Brownian motion started in $A$ has zero probability of hitting $A^c$ if and only if $A^c$ has zero capacity for the potential $|x|^{2-d}$. In fact much more is known, see Chapter 8.3 of the Brownian motion book of Mörters and Peres. $\endgroup$
    – Pablo Shmerkin
    Jul 28, 2014 at 17:49
  • $\begingroup$ @PabloShmerkin Thanks. But I was asking about the specific probability. For example for d-sphere we know it to be $1-(\frac{r}{x})^{d-2}$. $\endgroup$
    – TKM
    Jul 30, 2014 at 15:05
  • $\begingroup$ @JoonasIlmavirta here is a paper on the cylinders math.upenn.edu/~pemantle/papers/burdzy.pdf $\endgroup$
    – TKM
    Jul 30, 2014 at 15:09
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    $\begingroup$ @JoonasIlmavirta: The statement is, with probability 1, there exists a cylinder which is not hit. The cylinder depends on the outcome $\omega$, hence the phrase "random cylinder". This is actually quite a different question than computing the probability of avoiding a fixed set, so I think perhaps the question should be clarified. $\endgroup$
    – Nate Eldredge
    Jul 30, 2014 at 17:09

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