Stochastic dynamics on evolving geometric graphs

We consider an infinite locally finite system (configuration) $\gamma$ of particles distributed over a Euclidean space $X$. Each particle located at $x\in X$ carries an internal parameter (mark, or ``spin'') $\sigma_{x}\in S=\mathbb{R}.$ Such collections of particles form the space of marked configurations $\Gamma(X,S)$. We construct the following stochastic dynamics in $\Gamma(X,S)$: while the configuration $\gamma$ 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 $\gamma$ 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 c\`adl\`ag process in $\Gamma(X,S)$.