## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 5 (2017), 3113-3152.

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

Leif Döring, Achim Klenke, and Leonid Mytnik

#### 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, where the latter converge at a different time scale.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 5 (2017), 3113-3152.

**Dates**

Received: October 2015

Revised: September 2016

First available in Project Euclid: 3 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1509696042

**Digital Object Identifier**

doi:10.1214/17-AAP1277

**Mathematical Reviews number (MathSciNet)**

MR3719954

**Zentralblatt MATH identifier**

1379.60097

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J60: Diffusion processes [See also 58J65] 60J75: Jump processes 60F05: Central limit and other weak theorems 60H20: Stochastic integral equations

**Keywords**

Finite systems scheme interacting diffusions meanfield limit mutually catalytic branching

#### Citation

Döring, Leif; Klenke, Achim; Mytnik, Leonid. Finite system scheme for mutually catalytic branching with infinite branching rate. Ann. Appl. Probab. 27 (2017), no. 5, 3113--3152. doi:10.1214/17-AAP1277. https://projecteuclid.org/euclid.aoap/1509696042