LARGE DEVIATIONS FOR THE 3-DIMENSIONAL SUPER-BROWNIAN MOTION

Let mu(t)(dx) denote a three-dimensional super-Brownian motion with de terministic initial state mu(0)(dx) = dx, the Lebesgue measure. Let V: R(3) --> R be Holder-continuous with compact support, not identically zero and such that integral(R3)V(x) dx = 0. We show that log P {integ ral(0)(t) integral(R3)V(x)mu(s)(dx)ds > bt(3/4)} is of order t(1/2) as t --> infinity, for b > 0. This should be compared with the known res ult for the case integral(R3)V(x)dx > 0. In that case the normalizatio n bt(3/4), b > 0, must be replaced by bt, b > integral(R3)V(x)dx, in o rder that the same statement hold true. While this result only capture s the logarithmic order, the method of proof enables us to obtain comp lete results for the corresponding moderate deviations and central lim it theorems.