Conditioned super-Brownian motion (Q1326257): Difference between revisions

We investigate classes of conditioned super-Brownian motions, namely \(H\)- transforms \(P^ H\) with nonnegative 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 \(H(t,\mu) = \int h(t,x)\) \(\mu(dx)\) the \(H\)-transform \(P^ H\) as a conditioned super-Brownian motion in which an immortal particle moves like an \(h\)-transform of Brownian motion.