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I am looking for references/progress made in estimating the hitting probability for Borel sets.

For spheres we have $P_{x}(T_{B_{r}(0)}<\infty)=(\frac{|r|}{|x|})^{d-2}$, where $x=B_{0}$ for Brownian motion $\{B_{t}:0\leq t\}$ with $d\geq 3$ and $T_{B_{r}(0)}$ is the stopping time.

One estimate for compact sets I know is with Martin Capacities: $\frac{1}{2}Cap_{M}(A)\leq P_{x}(T_{A}<\infty)\leq Cap_{M}(A)$ (8.24 B.M. by Peres and Morters).

I am also curious for what types of sets we've calculated the probabilities exactly. It would be nice to have all of them in one place. Can you suggest me some experts in that field so I can ask them about the up-to-date progress?

1)On the First Hitting Time and the Last Exit Time for a Brownian Motion to/from a Moving Boundary, 1988

2)Boundary Crossing Probability for Brownian Motion, 2000

3)Some conditional crossing results of Brownian motion over a piecewise-linear boundary, 2002

Thank you

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  • $\begingroup$ It would be helpful if you could give full citations for the papers you cite (including author, journal, year, etc.) $\endgroup$
    – Nate Eldredge
    Commented Jul 12, 2014 at 0:17

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