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Conditioned super-Brownian motion

Research paper by L. Overbeck

Indexed on: 01 Dec '93Published on: 01 Dec '93Published in: Probability Theory and Related Fields



Abstract

We investigate classes of conditioned super-Brownian motions, namely H-transformsPH with non-negative finitely-based space-time harmonic functionsH(t, μ). We prove thatHH 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 functionsH(t, μ)=h(t, μ(1)) depending only on the total mass μ(1). Using the Palm measures of the super-Brownian motion we describe for an additive spacetime harmonic functionH(t, μ)=∝h(t, x) μ(dx) theH-transformPH as a conditioned super-Brownian motion in which an immortal particle moves like an h-transform of Brownian motion.