Moderate deviation for super-Brownian motion with immigration
The author considers \(d\)-dimensional super-Brownian motion with immigration determined by Lebesgue measure \(\lambda\). This is a measure-valued process \(X^\lambda= (X^\lambda_t, \mathbb{P}^\lambda_\mu)\) with transition probabilities given by \[ \mathbb{P}^\lambda_\mu\exp\{-\langle X^\lambda_t, f\rangle\}= \exp\Biggl\{- \langle\mu, V(t,\cdot)\rangle- \int^t_0 \langle\lambda, V(s,\cdot)\rangle\,ds\Biggr\}, \] where \(X^\lambda_0= \mu\), \(f\) belongs to a suitable class of test functions on \(\mathbb{R}^d\), and \(V(\cdot,\cdot)\) is the unique mild solution of \(\partial V(t)/\partial t= \Delta V(t)- V^2(t)\), \(V(0)= f\). A moderate deviation principle is proved for \(X^\lambda_t\) under the normalizations \(t^{1-\delta/4}\) for \(d= 1\), \(t^{1-\delta/2}(\log t)^{\delta/2}\) for \(d= 2\), \(t^{1-\delta/2}\) for \(d\geq 3\), for each \(\delta\in (0,1)\). This principle ``fills the gap'' between the central limit theorem \((\delta= 1)\) and the large deviation principle \((\delta= 0)\) for \(X^\lambda_t\). The speed functions also vary between the corresponding ones for these two results. The proof is based on a series expansion of the cumulant semigroup.
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