Abstract
A fundamental question concerning general spatial branching processes, both superprocesses and branching Markov processes, pertains to their moments. Whilst the setting of first and second moments has received quite some attention, limited information seems to be known about higher moments, in particular, their asymptotic behaviour with time. In this work, we provide general results that pertains to both superprocesses and spatial branching Markov processes and which provides a very precise result for moment growth.We show that, under the assumption that the first moment semigroup of the process exhibits a natural Perron Frobenious type behaviour, the k-th moment functional of either a superprocess or branching Markov process, when appropriately normalised, limits to a precise constant. The setting in which we work is remarkably general, even allowing for the setting of nonlocal branching; that is, where mass is created at a different point in space to the position of the parent. Moreover, the methodology we use appears to be extremely robust and we show that the asymptotic k-th moments of the running occupation measure are equally accessible using essentially the same approach. Our results will thus expand on what is known for branching diffusions and superdiffusions, as well as giving precise growth rates for the moments of occupations.
| Date of Award | 14 Sept 2022 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Andreas Kyprianou (Supervisor) & Matthew Roberts (Supervisor) |