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!

  • $\begingroup$ What is $IR$? Do you perhaps mean $\mathbb{R}$? $\endgroup$ – Nate Eldredge Nov 1 '12 at 2:25
  • $\begingroup$ Also, I don't understand what role $t$ is playing here. What does it mean for Brownian motion to start at $(x_0, t_0)$? Do you just mean it starts at location $x_0$ at time $t_0$? The starting time doesn't have any effect on the hitting distribution, because Brownian motion is time homogeneous. I'm also confused by the dimenions: is the state space of your Brownian motion supposed to be $\mathbb{R}^n$, $\mathbb{R}^{n+1}$, or $\mathbb{R}^2$? $\endgroup$ – Nate Eldredge Nov 1 '12 at 2:28
  • $\begingroup$ Sorry for the confusion! $\endgroup$ – Sandro Nov 1 '12 at 2:52
  • $\begingroup$ Yes to both of your questions: $IR$ is $\mathbb{R}$, and the n-dimensional BM starts at location $x_0\in IR^n$ at time $t_0\in IR$. I want to fix the ball around $(x_0,t_0)$, but let the BM start at points (x,t) in a neighborhood of $(x_0,t_0)$, and take the integral of the induced distribution on the ball centered at $(x_0,t_0)$ with respect to the measure induced by the BM that starts at $(x,t)$. And I want to show that this integral is a differentiable function of the starting point. $\endgroup$ – Sandro Nov 1 '12 at 3:01
  • $\begingroup$ You can also think of the problem like this: suppose the BM is $n$-dimensional. In the classic setting of harmonic measures there would be a subset of $\mathbb{R}^n$, and we'd be interested in the hitting distribution on the boundary of this set. In my case, this set changes over time. So it is a set in $\mathbb{R}^{n+1}$. And I want to integrate a function on the boundary of that set with respect to the distribution induced by the first hitting time of the BM on that set, and show that this integral is a differentiable function of the starting point of the BM. $\endgroup$ – Sandro Nov 1 '12 at 3:09

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!


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