Stochastic flows and an interface SDE on metric graphs

This paper consists in the study of a stochastic differential equation on a metric graph, called an interface SDE (ISDE)(ISDE). To each edge of the graph is associated an independent white noise, which drives (ISDE)(ISDE) on this edge. This produces an interface at each vertex of the graph. This study is first done on star graphs with N≥2N≥2 rays. The case N=2N=2 corresponds to the perturbed Tanaka’s equation recently studied by Prokaj (2013) and Le Jan and Raimond (2014) among others. It is proved that (ISDE)(ISDE) has a unique in law solution, which is a Walsh’s Brownian motion. This solution is strong if and only if N=2N=2.Solution flows are also considered. There is a (unique in law) coalescing stochastic flow of mappings φφ solving (ISDE)(ISDE). For N=2N=2, it is the only solution flow. For N≥3N≥3, φφ is not a strong solution and by filtering φφ with respect to the family of white noises, we obtain a (Wiener) stochastic flow of kernels solution of (ISDE)(ISDE). There are no other Wiener solutions. By applying our previous work Hajri and Raimond (2014), these results are extended to more general metric graphs.The proofs involve the study of (X,Y)(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 (X0,Y0)=(1,0)(X0,Y0)=(1,0) and if SS is the first time XX hits 00, then YS2 is a beta random variable of the second kind. We also calculate E[Lσ0]E[Lσ0], where LL is the local time accumulated at the boundary, and σ0σ0 is the first time (X,Y)(X,Y) hits (0,0)(0,0).