Conditioned super-Brownian motion

Summary

We investigate classes of conditioned super-Brownian motions, namely H-transformsP H with non-negative finitely-based space-time harmonic functionsH(t, μ). We prove thatH 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 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-transformP H as a conditioned super-Brownian motion in which an immortal particle moves like an h-transform of Brownian motion.

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Overbeck, L. Conditioned super-Brownian motion. Probab. Th. Rel. Fields 96, 545–570 (1993). https://doi.org/10.1007/BF01200209

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Mathematics Subject Classification (1991)

  • 60G57
  • 60J50
  • 60K35
  • 60J80