Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.17929 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909779980451840 |
|---|---|
| author | Yang, Ting |
| author_facet | Yang, Ting |
| contents | Suppose $\{X_{t}:t\ge 0\}$ is a supercritical superprocess on a Luzin space $E$, with a non-local branching mechanism and probabilities $\mathbb{P}_{δ_{x}}$, when initiated from a unit mass at $x\in E$. By ``supercritical", we mean that the first moment semigroup of $X_{t}$ exhibits a Perron-Frobenius type behaviour characterized by an eigentriplet $(λ_{1},φ,\widetildeφ)$, where the principal eigenvalue $λ_{1}$ is greater than $0$. Under a second moment condition, we prove that $X_{t}$ satisfies a law of large numbers. The main purpose of this paper is to further investigate the fluctuations of the linear functional $\mathrm{e}^{-λ_{1}t}\langle f,X_{t}\rangle$ around the limit given by the law of large numbers. To this end, we introduce a parameter $ε(f)$ for a bounded measurable function $f$, which determines the exponent term of the decay rate for the first moment of the fluctuation. Qualitatively, the second-order behaviour of $\langle f,X_{t}\rangle$ depends on the sign of $ε(f)-λ_{1}/2$. We prove that, for a suitable test function $f$, the fluctuation of the associated linear functional exhibits distinct asymptotic behaviours depending on the magnitude of $ε(f)$: If $ε(f)\ge λ_{1}/2$, the fluctuation converges in distribution to a Gaussian limit under appropriate normalization; If $ε(f)<λ_{1}/2$, the fluctuation converges to an $L^{2}$ limit with a larger normalization factor. In particular, when the test function is chosen as the right eigenfunction $φ$, we establish a functional central limit theorem. As an application, we consider a multitype superdiffusion in a bounded domain. For this model, we derive limit theorems for the fluctuations of arbitrary linear functionals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17929 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fluctuations of the linear functionals for supercritical non-local branching superprocesses Yang, Ting Probability 60J68, 60F05, 60G57 Suppose $\{X_{t}:t\ge 0\}$ is a supercritical superprocess on a Luzin space $E$, with a non-local branching mechanism and probabilities $\mathbb{P}_{δ_{x}}$, when initiated from a unit mass at $x\in E$. By ``supercritical", we mean that the first moment semigroup of $X_{t}$ exhibits a Perron-Frobenius type behaviour characterized by an eigentriplet $(λ_{1},φ,\widetildeφ)$, where the principal eigenvalue $λ_{1}$ is greater than $0$. Under a second moment condition, we prove that $X_{t}$ satisfies a law of large numbers. The main purpose of this paper is to further investigate the fluctuations of the linear functional $\mathrm{e}^{-λ_{1}t}\langle f,X_{t}\rangle$ around the limit given by the law of large numbers. To this end, we introduce a parameter $ε(f)$ for a bounded measurable function $f$, which determines the exponent term of the decay rate for the first moment of the fluctuation. Qualitatively, the second-order behaviour of $\langle f,X_{t}\rangle$ depends on the sign of $ε(f)-λ_{1}/2$. We prove that, for a suitable test function $f$, the fluctuation of the associated linear functional exhibits distinct asymptotic behaviours depending on the magnitude of $ε(f)$: If $ε(f)\ge λ_{1}/2$, the fluctuation converges in distribution to a Gaussian limit under appropriate normalization; If $ε(f)<λ_{1}/2$, the fluctuation converges to an $L^{2}$ limit with a larger normalization factor. In particular, when the test function is chosen as the right eigenfunction $φ$, we establish a functional central limit theorem. As an application, we consider a multitype superdiffusion in a bounded domain. For this model, we derive limit theorems for the fluctuations of arbitrary linear functionals. |
| title | Fluctuations of the linear functionals for supercritical non-local branching superprocesses |
| topic | Probability 60J68, 60F05, 60G57 |
| url | https://arxiv.org/abs/2503.17929 |