Large Deviations for the Three Dimensional Super-brownian Motion

Let t (dx) denote a three dimensional super-Brownian motion with deterministic initial state 0 (dx) = dx, the Lebesgue measure. Let V : R 3 7 ! R be HH older continuous with compact support, not identically zero and such that Z R 3 V (x)dx = 0. We show that log P Z t 0 Z R 3 V (x) s (dx)ds > bt 3=4 is of order t 1=2 as t ! 1, for b > 0. This should be compared with the known result for the case Z R 3 V (x) dx > 0. In that case the normalizing bt 3=4 , b > 0, must be replaced by bt, b > R R 3 V (x) dx, in order that the same statement holds 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.