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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!

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    What is $IR$? Do you perhaps mean $\mathbb{R}$?2012-11-01
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    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 $\mathbb{R}^2$?2012-11-01
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    Sorry for the confusion!2012-11-01
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    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.2012-11-01
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    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.2012-11-01
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    So to rephrase: if we set $B_{t_0} = x_0$ and let $\tau = \inf\{t : \sqrt{|B_t|^2 + (t-t_0)^2} \ge \eta\}$, you are interested in the distribution of $(B_\tau, \tau)$?2012-11-01
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    By the way, if you register your account, you can edit your question to make corrections.2012-11-01
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    Yes - almost. Let $\tau(x,t):=\inf\{t':\; |B_{t'}-x_0|^2+(t'-t_0)^2 \geq \eta^2\}$, where $B_{t}=x$ is the starting condition, i.e. the BM starts at time $t$ at location $x$. Then, for a continuous function $f$ defined on the boundary of the ball $A$, I am interested in $F(x,t)=\int f(B_\tau(x,t)(\omega))dW(\omega)$. What I want to show about this is that it is differentiable in $(x,t)$.2012-11-01
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    sorry, the integrand is $f(B_{\tau(x,t)(\omega)})$. I'm logged in, but still can edit only for 5 min...2012-11-01

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