Martingale problem for a Walsh spider process with spinning measure selected from its own local time

TL;DR

This paper establishes existence and uniqueness of a Walsh spider diffusion process with local time-dependent spinning measure, using martingale problems and PDE techniques, and computes explicit laws for certain cases.

Contribution

It introduces a novel martingale problem framework for Walsh spider processes with local time-dependent parameters, extending previous methods and solving a new class of transmission problems.

Findings

Proved existence and weak uniqueness of the Walsh spider diffusion process.

Derived explicit law of the diffusion as a standard Brownian motion on each branch.

Established existence and uniqueness for generalized SDEs involving local time in coefficients.

Abstract

The objective of this article is to prove existence and weak uniqueness of a Walsh spider diffusion process, whose spinning measure and coefficients are allowed to depend on the local time spent at the junction vertex. The methodology is to show carefully that an effectively designed martingale problem is well-posed. Exploiting fully the results coming from the pioneering work of [16], the construction of the solution is performed using a concatenation procedure, as introduced in the seminal reference [26]. Uniqueness is shown by making use of the recent results obtained in [25] for the solution of the corresponding parabolic PDE that involves a new class of transmission condition called local time Kirchhoff 's transmission condition. As a byproduct of our main result, we manage to compute the explicit law of the diffusion when it behaves as a standard Brownian motion on each branch. The case I = 2 permits us to derive also that there is existence and uniqueness for solutions of generalized SDE on the real line that involve the local time of the unknown process in all its coefficients.