ON THE LARGE TIME GROWTH-RATE OF THE SUPPORT OF SUPERCRITICAL, SUPER-BROWNIAN MOTION

Consider the supercritical super-Brownian motion X(t, .) on R(d) corre sponding to the evolution equation u(t) = D/2 Delta u + u - u(2). We o btain rather tight bounds on P-mu(X(s, B-n(c)(0)) = 0, for all s is an element of [0, t]) and on P-mu(X(t, B-n(c)(0)) = 0), for large n, whe re P-mu denotes the measure corresponding to the supercritical super-B rownian motion starting from the finite measure, mu, B-n(0) subset of R(d) denotes the ball of radius n centered at the origin and B-n(c)(0) denotes its complement. In particular, we show, for example, that if mu is a compactly supported, finite measure on R(d), then [GRAPHICS] a nd [GRAPHICS]