Finite system scheme for infinite rate mutually catalytic branching with infinite branching rate

TL;DR

This paper proves the convergence of finite system schemes for mutually catalytic branching processes with infinite rate, identifying the limit as a finite rate diffusion and analyzing different time scales for convergence.

Contribution

It extends the finite system scheme convergence results to infinite rate mutually catalytic branching processes, accounting for the absence of second moments.

Findings

Rescaled total mass processes converge to a finite rate diffusion.

Convergence occurs jointly with coordinate processes at different time scales.

The rescaling of time differs from the finite rate case due to lack of second moments.

Abstract

For many stochastic diffusion processes with mean field interaction, convergence of the rescaled total mass processes towards a diffusion process is known. Here we show convergence of the so-called finite system scheme for interacting jump-type processes known as mutually catalytic branching processes with infinite branching rate. Due to the lack of second moments the rescaling of time is different from the finite rate mutually catalytic case. The limit of rescaled total mass processes is identified as the finite rate mutually catalytic branching diffusion. The convergence of rescaled processes holds jointly with convergence of coordinate processes, whereas the latter converge at a different time scale.