Law of large numbers for supercritical superprocesses with non-local branching

Palau, Sandra; Yang, Ting

doi:10.1016/j.spa.2019.04.007

Cited by 4 publications

(4 citation statements)

References 29 publications

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“…Note that assumption (G2) entails that the MBP X is what we would call a "critical" setting. Indeed, for a lot of literature surrounding spatial branching processes, there has been emphasis on results for which an underlying assumption of exponential ergodic growth in the first moment is present; see, for example, [7,18,20,26,35,38]. In other words, this means that the mean semigroup exhibits a Perron-Frobenius type asymptotic of the form…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Yaglom limit for critical nonlocal branching Markov processes

et al. 2022

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We consider the classical Yaglom limit theorem for a branching Markov process X = (X t , t ≥ 0), with nonlocal branching mechanism in the setting that the mean semigroup is critical, that is, its leading eigenvalue is zero. In particular, we show that there exists a constant c(fwhere e c(f ) is an exponential random variable with rate c(f ) and the convergence is in distribution. As part of the proof, we also show that the probability of survival decays inversely proportionally to time. Although Yaglom limit theorems have recently been handled in the setting of branching Brownian motion in a bounded domain and superprocesses, (Probab. Theory Related Fields 173 (2019) 999-1062; Electron. Commun. Probab. 23 (2018) 42), these results do not allow for nonlocal branching which complicates the analysis. Our approach and the main novelty of this work is based around a precise result for the scaled asymptotics for the kth martingale moments of X (rather than the Yaglom limit itself). We then illustrate our results in the setting of neutron transport for which the nonlocality is essential, complementing recent developments in this domain (Ann.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Yaglom limit for critical nonlocal branching Markov processes

et al. 2022

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“…For a lot of literature surrounding spatial branching processes, there has been emphasis on results for which an underlying assumption of exponential ergodic growth in the first moment is present as in (H1); see e.g. [1,16,19,22,27,29]. Due to this, we may characterise the process as supercritical if λ > 0, critical if λ = 0 and subcritical if λ < 0.…”

Section: Main Results: K-th Momentsmentioning

confidence: 99%

“…In a number of settings, we would expect X to obey a strong law of large numbers (cf. [1,16,19,27]) in the sense that…”

Section: Main Results: K-th Momentsmentioning

confidence: 99%

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Asymptotic moments of spatial branching processes

Gonzàlez

Horton

Kyprianou

2022

Probab. Theory Relat. Fields

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Suppose that $$X =(X_t, t\ge 0)$$ X = ( X t , t ≥ 0 ) is either a superprocess or a branching Markov process on a general space E, with non-local branching mechanism and probabilities $${\mathbb {P}}_{\delta _x}$$ P δ x , when issued from a unit mass at $$x\in E$$ x ∈ E . For a general setting in which the first moment semigroup of X displays a Perron–Frobenius type behaviour, we show that, for $$k\ge 2$$ k ≥ 2 and any positive bounded measurable function f on E, $$\begin{aligned} \lim _{t\rightarrow \infty } g_k(t){\mathbb {E}}_{\delta _x}[\langle f, X_t\rangle ^k] = C_k(x, f), \end{aligned}$$ lim t → ∞ g k ( t ) E δ x [ ⟨ f , X t ⟩ k ] = C k ( x , f ) , where the constant $$C_k(x, f)$$ C k ( x , f ) can be identified in terms of the principal right eigenfunction and left eigenmeasure and $$g_k(t)$$ g k ( t ) is an appropriate deterministic normalisation, which can be identified explicitly as either polynomial in t or exponential in t, depending on whether X is a critical, supercritical or subcritical process. The method we employ is extremely robust and we are able to extract similarly precise results that additionally give us the moment growth with time of $$\int _0^t \langle f, X_t \rangle \mathrm{d}s$$ ∫ 0 t ⟨ f , X t ⟩ d s , for bounded measurable f on E.

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Limiting Distributions for a Class of Super-Brownian Motions with Spatially Dependent Branching Mechanisms

Ren,

Yang

2023

J Theor Probab

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Law of large numbers for supercritical superprocesses with non-local branching

Cited by 4 publications

References 29 publications

Yaglom limit for critical nonlocal branching Markov processes

Yaglom limit for critical nonlocal branching Markov processes

Asymptotic moments of spatial branching processes

Limiting Distributions for a Class of Super-Brownian Motions with Spatially Dependent Branching Mechanisms

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