Super-Brownian Motion: L ^p -Convergence of Martingales Through the Pathwise Spine Decomposition

Evans [7] described the semigroup of a superprocess with quadratic branching mechanism under a martingale change of measure in terms of the semigroup of an immortal particle and the semigroup of the superprocess prior to the change of measure. This result, commonly referred to as the spine decomposition, alludes to a pathwise decomposition in which independent copies of the original process “immigrate” along the path of the immortal particle. For branching particle diffusions, the analogue of this decomposition has already been demonstrated in the pathwise sense; see, for example, [11, 10]. The purpose of this short note is to exemplify a new pathwise spine decomposition for supercritical super-Brownian motion with general branching mechanism (cf. [13]) by studying L ^p -convergence of naturally underlying additive martingales in the spirit of analogous arguments for branching particle diffusions due to Harris and Hardy [10]. Amongst other ingredients, the Dynkin–Kuznetsov ℕ-measure plays a pivotal role in the analysis.