Occupation time large deviations for critical branching Brownian motion, super-Brownian motion and related processes
type: Item
instance of: scholarly article
title: Occupation time large deviations for critical branching Brownian motion, super-Brownian motion and related processes
zbMATH Open document ID: 0962.60013
author: Jean-Dominique Deuschel
author: Jay Rosen
published in: The Annals of Probability
publication date: 2000-02-01T00:00:00Z
review text: It is known [\textit{I. Iscoe}, Probab. Theory Relat. Fields 71, No. 1, 85-116 (1986; Zbl 0555.60034)] that for 3-dimensional super-Brownian motion \(\mu_t\) the occupation time fluctuations \[ T^{-3/4} \int^T_0(\langle\mu_s, h\rangle- \langle\lambda, h\rangle) ds \] converge in distribution as \(T\to\infty\) towards perfectly correlated centered Gaussian random variables with variance proportional to the Lebesgue integral \(\langle\lambda, h\rangle\) of the test function \(h\). In particular, for functions \(h\) with \(\langle\lambda, h\rangle= 0\), the normed occupation time \(T^{-3/4} \int^T_0\langle\mu_s, h\rangle ds\) converges to zero as \(T\to\infty\). The large deviations of this normed occupation time are investigated. The authors prove the large deviation principle \[ \lim_{T\to\infty} T^{-1/2}\log P\Biggl(T^{- 3/4} \int^T_0 \langle\mu_s, h\rangle ds\geq x\Biggr)= \Lambda^*_h(x), \] where \(\Lambda^*_h(x)\) is expressed in terms of the Legendre transform of the moment generating function of \(L^0_T\), the super-Browian local time up to time one. This settles a conjecture of \textit{T.-Y. Lee} and \textit{B. Rémillard} [Ann. Probab. 23, No. 4, 1755-1771 (1995; Zbl 0852.60004)]. The proof is based on an analysis of the rescaled moment generating functions of the occupation time, using Dynkin's graphical moment formula. As a local version of their result, the authors also prove a large deviation principle for the rescaled difference \(T^{-3/4}(L^x_t- L^0_T)\) of two local times. In addition, they derive from their large deviation principle a central limit theorem which states that \(T^{-1/2} \int^T_0 \langle\mu_s, h\rangle ds\) converges in distribution towards a centered Gaussian random variable with variance \[ -{1\over \pi} \iint|y- z|h(y) h(z) dy dz= \langle\lambda,(Gh)^2\rangle. \] The integral of the squared potential \((Gh)^2\) is finite only due to the assumption \(\langle\lambda, h\rangle= 0\); this property of \(h\) also guarantees that the ``global…
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 1355248
zbMATH Keywords: Legendre transform
zbMATH Keywords: moment generating function
zbMATH Keywords: Dynkin's graphical moment formula
zbMATH Keywords: symmetric \(\beta\)-stable motion
zbMATH Keywords: branching particle systems
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022855645
OpenAlex ID: W1491982597
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DOI: 10.1214/AOP/1022855645
description: scientific article; zbMATH DE number 1355248