Large Deviations for the Three-Dimensional Super-Brownian Motion

Abstract

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.