On spider diffusions having a spinning measure selected from their own local time

TL;DR

This paper studies Walsh's spider diffusions on star-shaped networks with a focus on local time-based spinning measures, providing new formulas, properties, and characterizations of process behavior at the junction point.

Contribution

It introduces new Itô's formula, trajectory properties, and a Feynman-Kac representation for star network diffusions with local-time-based boundary conditions.

Findings

The process distribution is non-atomic at the junction.

Provides an $L^1$-approximation of local time.

Characterizes the scattering distribution using diffraction coefficients.

Abstract

The aim of this article is to give several results related to Walsh's spider diffusions living on a star-shaped network that have a spinning measure selected from the own local time of the motion at the vertex (cf.[17]). We prove the corresponding It\^o's formula and give some global trajectory properties such as $L^1$-approximation of the local time and the Markov property. Regarding the behavior of the process at the vertex, we show that that the distribution of the process is non atomic at the junction point and we characterize the instantaneous scattering distribution along some ray with the aid of the probability coefficients of diffraction. We obtain also a Feynmann-Kac representation for linear parabolic systems posed on star-shaped networks that where introduced in [18] possessing a so-called local-time Kirchhoff's boundary condition.