Infinite rate symbiotic branching on the real line: The tired frogs model

TL;DR

This paper introduces a new stochastic model of interacting frogs on the real line, capturing complex waking and sleeping dynamics through a limit of approximations and a jump-driven SPDE.

Contribution

It constructs the infinite rate symbiotic branching process as a limit of approximations and characterizes it via a jump-type stochastic PDE.

Findings

The model is constructed as a limit of approximating processes.

The structure of jumps in the process is explicitly computed.

The process is described by a stochastic PDE with jump noise.

Abstract

Consider a population of infinitesimally small frogs on the real line. Initially the frogs on the positive half-line are dormant while those on the negative half-line are awake and move according to the heat flow. At the interface, the incoming wake frogs try to wake up the dormant frogs and succeed with a probability proportional to their amount among the total amount of involved frogs at the specific site. Otherwise, the incoming frogs also fall asleep. This frog model is a special case of the infinite rate symbiotic branching process on the real line with different motion speeds for the two types. We construct this frog model as the limit of approximating processes and compute the structure of jumps. We show that our frog model can be described by a stochastic partial differential equation on the real line with a jump type noise.