Stochastic flows and an interface SDE on metric graphs
Résumé
This paper consists in the study of a stochastic differential equation on a metric graph, called an interface SDE $(\hbox{ISDE})$.
To each edge of the graph is associated an independent white noise, which drives $(\hbox{ISDE})$ on this edge.
This produces an interface at each vertex of the graph. We first do our study on star graphs with $N\ge 2$ rays. The case $N=2$ corresponds to the perturbed Tanaka's equation recently
studied by Prokaj \cite{MR18} and Le Jan-Raimond \cite{MR000} among others.
It is proved that $(\hbox{ISDE})$ has a unique in law solution, which is a Walsh's Brownian motion. This solution is strong if and only if $N=2$.
Solution flows are also considered. There is a (unique in law) coalescing stochastic flow of
mappings $\p$ solving $(\hbox{ISDE})$. For $N=2$, it is the only solution flow. For $N\ge 3$, $\p$ is not a strong solution and by filtering $\p$ with respect to the
family of white noises, we obtain a (Wiener) stochastic flow of kernels solution of $(\hbox{ISDE})$.
There are no other Wiener solutions.
Our previous results \cite{MR501011} in hand, these results are extended to more general metric graphs.
The proofs involve the study of $(X,Y)$ a Brownian motion in a two dimensional quadrant obliquely reflected at the boundary, with time dependent
angle of reflection. We prove in particular that, when $(X_0,Y_0)=(1,0)$ and if $S$ is the first time $X$ hits $0$, then $Y_S^2$ is a beta random variable of the second kind. We also calculate $\EE[L_{\sigma_0}]$, where $L$ is the local time accumulated at the boundary, and $\sigma_0$ is the first time $(X,Y)$ hits $(0,0)$.
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