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 coalescing stochastic flow of mappings solution of this equation. This flow is unique in law and is coalescing. Our proofs involve the study of a reflected Brownian motion in the two dimensional quadrant obliquely reflected at the boundary, with time dependent reflections. Filtering this flow solution of the SDE with respect to the family of white noises yields a Wiener stochastic flow of kernels also solution of this SDE. This Wiener soltution is also unique. Moreover, if $N$ denotes the number of rays in 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 a general metric graph with finite sets of vertices and edges to which we apply our previous results in the paper stochastic flows on metric graphs. As a consequence, we get a flow of mappings and a Wiener flow solutions for this SDE.

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

Domaines: Probabilités [math.PR]

Hatem Hajri, Olivier Raimond. Stochastic flows and an interface SDE on metric graphs. 2013. ⟨hal-00872458v3⟩