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