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

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

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

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

where

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

and

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

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 $f(s)$ 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. 1 # Brownian particle with jump boundary condition I would like to find a function$f(s)$, which solves the following equation:$ \int_0^t \int_0^L f(s) 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)$where$\phi_n(x) = \sqrt{\frac{2}{L}} \sin \left( n \pi x / L \right)$and$\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$f(s)$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.