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I would like to find a function $f(s)$, which solves the following equation:

$ \int_0^t \int_0^L f(s,x) p(t-s,x,y) dy ds = 1 $

The function $p(\tau,x,y)$ is

$p(\tau,x,y) = \sum_n e^{-\lambda_n \tau} \phi_n(x) \phi_n(y)$


$\phi_n(x) = \sqrt{\frac{2}{L}} \sin \left( n \pi x / L \right)$


$\lambda_n = \frac{n^2\pi^2}{L^2}$.

i.e. $\psi_n$ and $\lambda_n$ are the eigenfunctions and eigenvalues corresponding to $(-\tfrac{1}{2}\partial^2_{xx})$. Physically, the above equations correspond to the following situation. At time $t=0$ start a Brownian particle at $x \in (0,L)$. Whenever the BM touches a boundary (either $0$ or $L$) immediately send the particle back to $x$, where it begins a new Brownian path. The function $f(s,x)$ represents the probability that a particle found in infinitesimal element $dy$ at time $t$ was started at $x$ at time $s$. The function $p(t,x,y)$ is the transition density of a Brownian particle with a killing boundary condition at $0$ and $L$.

There seems to be a good deal of literature that analyzes the spectrum of BM with a jump boundary. But, as of yet, I have found no papers that specifically say what the transition density of such a process would be. And, that is my interest (i.e. find the transition density of a diffusion in a bounded domain with a jump boundary condition). Any help in solving the top equation or any suggestions for papers to look at would be greatly appreciated.

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In some of the places where you write $f(s,x)$ do you mean $f(s,y)$? For example, in your first equation can't you just bring the $f(s,x)$ out of the integral? – Paul Tupper Oct 25 '11 at 4:03

the transition for the jump process started from x satisfies a renewal equation where the lifetime distribution is the hitting time for the boundary. You can write a formal solution in the usual manner of solving renewal equations. I have seen this as problem somewhere, but a quick search of Karlin & Taylor did not turn it up.

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Thanks Mike. To be honest, I have really looked into this question much since I posted it a few months ago, since this question isn't really a major area of research for me. But, I did find the problem interesting enough to look for an answer. Now that I know where to look, I think I will revisit this problem. Thanks. – psyduck Apr 5 '12 at 3:09

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