Stochastic flows and an interface SDE on metric graphs

Abstract

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 $(0,0)$.