CONDITIONED SUPER-BROWNIAN MOTION

We investigate classes of conditioned super-Brownian motions, namely H-transforms P(H) with non-negative finitely-based space-time harmonic functions H(t, mu). We prove that P(H) is the unique solution of a martingale problem with interaction and is a weak limit of a sequence of rescaled interacting branching Brownian motions. We identify the limit behaviour of H-transforms with functions H(t, mu)=h(t, mu(1)) depending only on the total mass mu(1). Using the Palm measures of the super-Brownian motion we describe for an additive space-time harmonic function H (t, mu) = integral h(t, x) mu(dx) the H-transform P(H) as a conditioned super-Brownian motion in which an immortal particle moves like an h-transform of Brownian motion.