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Stochastic dynamics on evolving geometric graphs

Alexei Daletskii, Dmitri Finkelshtein

TL;DR

We address stochastic dynamics on evolving geometric graphs by coupling a Birth-and-Death evolution of particle positions with a coupled infinite-dimensional diffusion of marks on the particles. The authors develop a rigorous framework on marked configuration spaces $\\Gamma(X,S)$, employing a phantom-space construction and a fibre-bundle viewpoint to define and analyze the joint dynamics. They prove existence and uniqueness of a càdlàg $\\Gamma(X,S)$-valued process and establish detailed regularity and growth estimates, both for the spin system along a fixed path and for the coupled evolution with the underlying configuration. The results lay a solid mathematical foundation for stochastic dynamics on random evolving graphs and have potential implications for modeling complex systems with moving constituents and interacting internal states.

Abstract

We consider an infinite locally finite system (configuration) $γ$ of particles distributed over a Euclidean space $X$. Each particle located at $x\in X$ carries an internal parameter (mark, or ``spin'') $σ_{x}\in S=\mathbb{R}.$ Such collections of particles form the space of marked configurations $Γ(X,S)$. We construct the following stochastic dynamics in $Γ(X,S)$: while the configuration $γ$ of particle positions performs a random evolution, the corresponding marks interact with each other and perform a coupled infinite-dimensional diffusion. The study of a spin dynamics on a fixed configuration $γ$ was initiated by Daletskii and Finkelshtein, J. Stat. Phys. 122 ( 2018), and continued by Chargaziya and Daletskii, J. Math. Phys. 66 (2025), and is based on the generalisation of the Ovsjannikov method. In the present paper, the underlying configuration evolves according to a Birth-and-Death process. We prove the existence and uniqueness of such dynamics and show that it forms a càdlàg process in $Γ(X,S)$.

Stochastic dynamics on evolving geometric graphs

TL;DR

We address stochastic dynamics on evolving geometric graphs by coupling a Birth-and-Death evolution of particle positions with a coupled infinite-dimensional diffusion of marks on the particles. The authors develop a rigorous framework on marked configuration spaces , employing a phantom-space construction and a fibre-bundle viewpoint to define and analyze the joint dynamics. They prove existence and uniqueness of a càdlàg -valued process and establish detailed regularity and growth estimates, both for the spin system along a fixed path and for the coupled evolution with the underlying configuration. The results lay a solid mathematical foundation for stochastic dynamics on random evolving graphs and have potential implications for modeling complex systems with moving constituents and interacting internal states.

Abstract

We consider an infinite locally finite system (configuration) of particles distributed over a Euclidean space . Each particle located at carries an internal parameter (mark, or ``spin'') Such collections of particles form the space of marked configurations . We construct the following stochastic dynamics in : while the configuration of particle positions performs a random evolution, the corresponding marks interact with each other and perform a coupled infinite-dimensional diffusion. The study of a spin dynamics on a fixed configuration was initiated by Daletskii and Finkelshtein, J. Stat. Phys. 122 ( 2018), and continued by Chargaziya and Daletskii, J. Math. Phys. 66 (2025), and is based on the generalisation of the Ovsjannikov method. In the present paper, the underlying configuration evolves according to a Birth-and-Death process. We prove the existence and uniqueness of such dynamics and show that it forms a càdlàg process in .

Paper Structure

This paper contains 10 sections, 17 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Marked configuration spaces $\Gamma(X,S)$
  3. A class of jump processes on $\Gamma(X)$ and their phantoms
  4. Dynamics in the Phantom Space
  5. Dynamics in $\Gamma(X,S)$.
  6. Dynamics of underlying configurations
  7. Appendix
  8. Finite volume cut-offs
  9. Gronwall inequality in the scale of spaces of sequences
  10. Estimates of the solution

Key Result

Theorem 1.1

For any $(\gamma_t)_{t\in [0,T]}\in\mathcal{R}(\mathrm{L}\Gamma(X))$ and $\widehat{\gamma}_0\in \Gamma(X,S)$, there exists a unique $\Gamma(X,S)$-valued process $(\widehat{\gamma}_{t})_{t\in[0,T]}$ with initial value $\widehat{\gamma}_0$ and such that, under identification eq:fibre, where $\Xi_t=(\xi_{x,t})_{x\in\gamma}$ is a solution of system eq:SDE along $\gamma_{t}$. The process $(\widehat{\g

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark 2.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.5
  • Lemma 4.2
  • proof
  • Definition 4.3
  • Theorem 4.4
  • Remark 4.5
  • ...and 43 more