Non Homogeneous Stochastic Diffusion on a Junction

TL;DR

This paper provides a new proof for the existence of a non-homogeneous stochastic diffusion process on a junction, extending previous results to include time-dependent coefficients and offering an Itô's formula and local time estimates.

Contribution

It generalizes existing diffusion on junction results to time-dependent coefficients and derives an Itô's formula and local time estimates for such processes.

Findings

Existence proof for non-homogeneous diffusion on a junction.

Extension to time-dependent and Borel coefficients.

Itô's formula and local time estimates at the junction.

Abstract

The purpose of this article is to give another proof on the existence of a diffusion on a junction, which has been already done by M.Freidlin and S-J.Sheu, in Diffusion processes on graphs, (2000). We generalize the result to time dependent and borel coefficients. Such a process can be seen as a couple (x, i) with x a one dimensional continuous diffusion whose coefficients depends on the edge i where it is located. We then provide an It{\^o}'s formula for this process. Finally, we give an estimate of the local time of the process at the junction point.