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

Isaac Ohavi, Miguel Martinez

TL;DR

The paper advances Walsh's spider diffusion theory by allowing the spinning measure to depend on the process's own local time at the junction, and develops a comprehensive framework encompassing an Itô formula, absolute continuity, Feynman-Kac representations for local-time Kirchhoff problems, quadratic local-time approximations, strong Markov properties, and diffusion scattering at the vertex. The approach unifies stochastic martingale problems with PDEs on networks, yielding probabilistic representations for backward parabolic systems and rigorous analysis of density and non-atomic behavior at the junction. The results collectively extend previous constant-spinning-measure theories to non-constant, time- and local-time-dependent regimes, with potential implications for stochastic scattering control and boundary conditions on graph-like domains. The work provides a robust toolkit for analyzing star-shaped network diffusions with locally time-driven boundary behavior and offers precise asymptotics for scattering distributions in terms of the dynamic spinning coefficients $\alpha_i(t,l)$.

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 Itô'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.

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

TL;DR

The paper advances Walsh's spider diffusion theory by allowing the spinning measure to depend on the process's own local time at the junction, and develops a comprehensive framework encompassing an Itô formula, absolute continuity, Feynman-Kac representations for local-time Kirchhoff problems, quadratic local-time approximations, strong Markov properties, and diffusion scattering at the vertex. The approach unifies stochastic martingale problems with PDEs on networks, yielding probabilistic representations for backward parabolic systems and rigorous analysis of density and non-atomic behavior at the junction. The results collectively extend previous constant-spinning-measure theories to non-constant, time- and local-time-dependent regimes, with potential implications for stochastic scattering control and boundary conditions on graph-like domains. The work provides a robust toolkit for analyzing star-shaped network diffusions with locally time-driven boundary behavior and offers precise asymptotics for scattering distributions in terms of the dynamic spinning 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 Itô's formula and give some global trajectory properties such as -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.

Paper Structure

This paper contains 17 sections, 11 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Problem 1 - Itô's formula
  3. Problem 2 - Absolute continuity
  4. Problem 3 - Feynman-Kac representation
  5. Problem 4 - Approximations of the local time at the junction vertex
  6. Problem 5 - Strong Markov property
  7. Problem 6 - Diffusion scattering at the junction vertex
  8. Notations and a remainder of the main result of Martinez-Ohavi Walsh
  9. Problem 1 - Itô's formula
  10. Problem 2 - Absolute continuity
  11. Preliminaries
  12. No atom at zero
  13. Existence of a density
  14. Problem 3 - Feynman-Kac representation
  15. Problem 4 - Quadratic approximations of the local time
  16. ...and 2 more sections

Key Result

Theorem 2.1

Assume assumption $(\mathcal{H})$. Then, the martingale problem $(\mathcal{S}_{pi}-\mathcal{M}_{ar})$ is well-posed.

Theorems & Definitions (23)

  • Theorem 2.1
  • Corollary 2.2
  • Theorem 3.1
  • proof
  • Definition 5.1
  • Theorem 5.2
  • proof
  • Remark 5.3
  • Proposition 6.1
  • proof
  • ...and 13 more