# Stochastic flows and an interface SDE on metric graphs

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1 Uni.lu - Université du Luxembourg 2 MODAL-X - Modélisation aléatoire de Paris X

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 recentlystudied 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 ofmappings $\p$ 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 thefamily 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 dependentangle 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 $\EEL {\sigma 0}$, where $L$ is the local time accumulated at the boundary, and $\sigma 0$ is the first time $X,Y$ hits $0,0$.

Keywords : Stochastic flows Tsirelson theorem Perturbed Tanaka-s SDE metric graphs Walsh-s Brownian motion Tsirelson theorem.

Author: ** Hatem Hajri - Olivier Raimond - **

Source: https://hal.archives-ouvertes.fr/