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!
I'm still looking for an answer to this. I managed to find a way to prove continuity of $F$, but not yet differentiability. So I'd still be extremely glad if someone could give me a helpful hint!