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The problem is: Given compact set A with positive finite volume (eg. ball,cube), what happens to $P_{x}(T_{A}>t)$ as $t\to \infty$, where $T_{A}=inf_{t>0}(B_{t}\in A)$ and x is in the "exterior" of A (consider A to be a solid object, then x is in the complement).

More specifically, do we have for all finite t: $P_{x}(\infty>T_{A}>t)>0$ for $d\geq 3$?

Also, do we have decay estimates of $P_{x}(\infty>T_{A}>t)$?

In d=1,2, Brownian motion is recurrent and so $P_{x}(T_{A}>t)=0$ as $t\to \infty$. I want to know the behavior of $P_{x}(T_{A}>t)$ for $d\geq 3$ as $t\to \infty$. For example does it decay exponentially? Is it always positive?

What are the methods I can use to study this problem (eg. Wiener Sausage asymptotics, Feyman-Kac etc)?

I am also curious, how far we've gone in finding densities of stopping times for various shapes. For $T_{a}=inf_{t>0}(B_{t}=a)$, we have $P(T_{a}>t)=\int_{t}^{\infty}\frac{a}{\sqrt{2\pi(s-t)^{3}}}e^{\frac{-a^{2}}{2(s-t)}}ds$ and for the sphere $P_{x}[t<T_{B_{0,r}}<\infty]=\int_{t}^{\infty}(\frac{1}{4\pi s^{3}})^{1/2}\frac{r(|x|-r)}{|x|}e^{-\frac{(|x|-r)^{2}}{4s}}ds$

Any help will be appreciated.

Thanks

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  • $\begingroup$ In dimensions 1 and 2, I think you mean to say $\lim_{t \to \infty} P_x(T_A > t) = 0$. In dimensions 3 and higher, we certainly have $\lim_{t \to \infty} P_x(T_A > t) = P_x(T_A = \infty) < 1$. Depending on exactly what you mean by "exterior", you may even get $P_x(T_A = \infty) = 0$. But it is clearly true that $P_x(T_A > t) > 0$ for all finite $t$: consider a ball around $x$ disjoint from $A$, and there is a positive probability that the Brownian motion remains in this ball up to time $t$. $\endgroup$ – Nate Eldredge Oct 16 '14 at 20:53
  • $\begingroup$ For $d\geq 3$, is there any property of BM, that ensures $P_{x}(\infty >T_{A}>t)>0$ as t grows ? So I want the probability that BM hits A but after time t. The $P_{x}(T_{A}>t)$ follows from the ball example you gave. But I wonder about the finiteness of $T_{A}$ as $t\to \infty$. $\endgroup$ – TKM Oct 16 '14 at 23:51
  • $\begingroup$ consider A to be a solid object and x a point in the complement. $\endgroup$ – TKM Oct 16 '14 at 23:52
  • $\begingroup$ Just out of curiosity, is the probability of staying in the ball centered at x, decrease as $t\to \infty$? $\endgroup$ – TKM Oct 16 '14 at 23:53
  • $\begingroup$ What is a "solid object"? What if $A$ is the set $\{p : 1 \le |p| \le 2\}$ (a spherical shell) and $x=0$? $\endgroup$ – Nate Eldredge Oct 17 '14 at 15:56

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