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 (ISDE). To each edge of the graph is associated an independent white noise, which drives (ISDE) on this edge. This produces an interface at each vertex of the graph. We first do our study on star graphs with N ≥ 2 rays. The case N = 2 corresponds to the perturbed Tanaka's equation recently studied by Prokaj [18] and Le Jan-Raimond [12] among others. It is proved that (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 ϕ solving (ISDE). For N = 2, it is the only solution flow. For N ≥ 3, ϕ is not a strong solution and by filtering ϕ with respect to the family of white noises, we obtain a (Wiener) stochastic flow of kernels solution of (ISDE). There are no other Wiener solutions. Our previous results [8] 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 (X0, Y0) = (1, 0) and if S is the first time X hits 0, then Y 2 S is a beta random variable of the second kind. We also calculate E[Lσ 0 ], where L is the local time accumulated at the boundary, and σ0 is the first time (X, Y ) hits (0, 0).

On a Coupling of Solutions to the Interface Stochastic Differential Equation on a Star Graph

“…The case N = 2. Point (i) in Theorem 1.5 is also true for N = 2 since X = Y in this case [6]. This can also be deduced from the proofs below.…”

“…The main result proved in [6] was the existence of a stochastic flow of mappings, unique in law and a Wiener stochastic flow [8] which solve the interface SDE. The problem of finding the flows of kernels which "interpolate" between these two particular flows was left open in [6]. The answer to this question needs a complete understanding of weak solutions of this equation.…”

“…Section 3 is a complement based on stochastic flows to the previous results. We consider the Wiener stochastic flow of kernels K constructed in [6] which is a strong solution to the flows of kernels version of (2) driven by a real white noise (W s,t ) s≤t . The Wiener coupling (X, Y, W ) is shown to be the strong Markov process associated to a Feller semigroup Q obtained from K. The subject of this paragraph is to prove the following result which, in the case N = 2, is proved in [6].…”

“…Tsirelson theorem 1.1 combined with Theorem 2.3 in [6] show that X is σ(W )measurable if and only if N ≤ 2.…”

“…To emphasize on the filtration (F t ) t , we will sometimes say (X, W ) is an (F t ) tsolution. It has been proved in [6] (Theorem 2.3) that for all x ∈ G, (I) admits a solution (X, W ) with X 0 = x, the law of (X, W ) is unique and X is an (F t )-WBM on G. We will denote by Q x the law of a solution (X, W ) with X 0 = x.…”