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The hitting probability for spheres centered at origin is $P_{x}(T_{B_{r}(0)}<\infty)=\frac{r^{d-2}}{|x|^{d-2}}>0$, where $|x|>r$.

1)So I was wondering how can one compute $P_{x}(T_{B_{r}(0)}<\infty \cap T_{B_{r}(y)}<\infty)$ where the spheres are disjoint and x is "outside" (eg. for $|y|>2r$ and $|x|>|y|+2r$)?

The Strong Markov property doesn't apply because the probability of hitting one sphere increases the probability of hitting the other and so one can't separate their laws.

To make the problem more symmetric, let the starting point x be equidistant to 0 and y.

How about the average hitting probability i.e. $\int_{\mathbb{R}^{n}}P_{x}(T_{B_{r}(0)}<\infty \cap T_{B_{r}(y)}<\infty)dx$? Is that any easier to compute?

2)Is there a PDE whose solution are those paths?

3)How about $P_{x}(T_{B_{r}(0)}<\infty \cap T_{B_{r}(y)}<t)$ (*), where I replaced $\infty$ by time $t>0?$

My research problem is about getting estimates of probabilities like (*) as $t\to \infty$; thus any references and interesting papers are welcomed.

Thanks

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  • $\begingroup$ Any news? There are ways to relate this to capacity calculation, as capacity of the two spheres is related to the probability of a Brownian motion started at a uniformly on the surface of a bounding sphere of ever hitting one of the spheres. Check Sidney and Port on potential theory. $\endgroup$
    – hHhh
    Jun 23, 2015 at 18:41

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