Stochastic flows and an interface SDE on metric graphs

Hatem Hajri (1) , Olivier Raimond (2)

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

We study a stochastic differential equation (SDE) driven by a finite family of independent white noises on a star graph, each of these white noises driving the SDE on a ray of the graph. This equation extends the perturbed Tanaka's equation recently studied by Prokaj and Le Jan-Raimond among others. We prove that there exists a (unique in law) coalescing stochastic flow of mappings solution of this equation. Our proofs involve the study of a Brownian motion in the two dimensional quadrant obliquely reflected at the boundary, with time dependent angle of reflection. Filtering this coalescing flow with respect to the family of white noises yields a Wiener stochastic flow of kernels also solution of this SDE. This Wiener solution is also unique. Moreover, if $N$ denotes the number of rays constituting the star graph, the Wiener solution and the coalescing solution coincide if and only if $N=2$. When $N\ge 3$, the problem of classifying all solutions is left open. Finally, we define an extension of this equation on more general metric graphs to which we apply some of our previous results. As a consequence, we deduce the existence and uniqueness in law of a flow of mappings and a Wiener flow solutions of this SDE.

Mots clés: Stochastic flows; Perturbed Tanaka's SDE; metric graphs; Walsh's Brownian motion; Tsirelson theorem.; Tsirelson theorem

Domaines: Probabilités [math.PR]

Hatem Hajri, Olivier Raimond. Stochastic flows and an interface SDE on metric graphs. 2013. ⟨hal-00872458v4⟩