On/off super-Brownian motion: Characterization, construction and long-term behaviour

TL;DR

This paper introduces on/off super-Brownian motion, a measure-valued process modeling particles that can switch between active and dormant states, revealing unique long-term behaviors different from classical models.

Contribution

It constructs and characterizes on/off super-Brownian motion as a new measure-valued limit of branching Brownian motions with dormancy features.

Findings

The process does not die out in finite time with probability one.

The active population size hits zero in finite time with positive probability.

Distinct qualitative behaviors from classical super-Brownian motion are observed.

Abstract

We introduce and construct on/off super-Brownian motion (on/off SBM) as a measure-valued scaling limit of critical on/off branching Brownian motions. The distinguishing feature of this process is that its infinitesimal particles can switch individually into and out of a state of dormancy, in which they neither move nor reproduce. Related dormancy traits have received interest in mathematical population biology recently, introducing memory and delay (in the form of a seed bank) into the corresponding processes. It turns out that the properties of on/off SBM differ significantly from those of classical super-Brownian motion. In particular, the process does not die out in finite time with probability one despite criticality of reproduction. However, the size of the active subpopulation does hit 0 in finite time with positive probability, a result which can be shown using methods from polynomial diffusion theory. We expect distinct behaviour in various other qualitative and quantitative respects, for which we provide heuristics, and conclude with a brief discussion of several directions for future research.