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October, 1995 Large Deviations for the Three-Dimensional Super-Brownian Motion
Tzong-Yow Lee, Bruno Remillard
Ann. Probab. 23(4): 1755-1771 (October, 1995). DOI: 10.1214/aop/1176987802


Let $\mu_t(dx)$ denote a three-dimensional super-Brownian motion with deterministic initial state $\mu_0(dx) = dx$, the Lebesgue measure. Let $V: \mathbb{R}^3 \mapsto \mathbb{R}$ be Holder-continuous with compact support, not identically zero and such that $\int_{\mathbb{R}^3}V(x)dx = 0$. We show that $\log P\big\{\int^t_0\int_{\mathbb{R}^3} V(x)\mu_s(dx) ds > bt^{3/4}\big\}$ is of order $t^{1/2}$ as $t \rightarrow \infty$, for $b > 0$. This should be compared with the known result for the case $\int_{\mathbb{R}^3}V(x)dx > 0$. In that case the normalization $bt^{3/4}, b > 0$, must be replaced by $bt, b > \int_{\mathbb{R}^3}V(x)dx$, in order that the same statement hold true. While this result only captures the logarithmic order, the method of proof enables us to obtain complete results for the corresponding moderate deviations and central limit theorems.


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Tzong-Yow Lee. Bruno Remillard. "Large Deviations for the Three-Dimensional Super-Brownian Motion." Ann. Probab. 23 (4) 1755 - 1771, October, 1995.


Published: October, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0852.60004
MathSciNet: MR1379167
Digital Object Identifier: 10.1214/aop/1176987802

Primary: 60B12
Secondary: 60F05 , 60F10 , 60J15

Keywords: large deviations , Super-Brownian motion

Rights: Copyright © 1995 Institute of Mathematical Statistics

Vol.23 • No. 4 • October, 1995
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