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arXiv · 2307.10968

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

Jochen Blath·Dave Jacobi
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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.

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Jochen Blath, Dave Jacobi. 2023-07-20. On/off super-Brownian motion: Characterization, construction and long-term behaviour. https://arxiv.org/abs/2307.10968

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