I have been trying to prove the following conjecture for a while, but so far to no avail. Would be very grateful for some tips!
The conjecture is the following;
Think of an $n$ dimensional Brownian Motion (BM) that starts at $(x_0,t_0)=(0,0)\in \mathbb{R}^{n+1}$. Define the set $A$ by $A=\{x\in \mathbb{R}^2|\; ||x|| >= \eta\}$ for some $\eta>0$, and define a continuous function $f$ on the boundary of $A$. Obviously the BM induces a hitting-time distribution on the boundary of $A$. Define $F(x,t)$ as the expectation of $f$ with respect to this distribution given that the BM starts at some $(x,t)$ in the interior of the complement of $A$. What I want to show is that $F$ is a differentiable function of $(x,t)$ on this set.
I know from the literature on harmonic functions that an analogous theorem is true if the set $A$ does not change in time, but I have not found how to use the techniques from that literature in the present case.
It would be awesome if someone could give me a hint as to how to go about this!
Or, if someone knows of another stochastic process that has this property that would also be great!