Super-Brownian motion: Lp-convergence of martingales through the pathwise spine decomposition

TL;DR

This paper introduces a new pathwise spine decomposition for supercritical super-Brownian motion with general branching, demonstrating $L^p$ convergence of associated martingales using the Dynkin-Kuznetsov measure.

Contribution

It extends the spine decomposition concept to super-Brownian motion with general branching, providing a novel pathwise analysis and convergence results.

Findings

Established $L^p$ convergence of additive martingales

Developed a new pathwise spine decomposition for super-Brownian motion

Utilized Dynkin-Kuznetsov measure in the analysis

Abstract

Evans (1992) described the semi-group of a superprocess with quadratic branching mechanism under a martingale change of measure in terms of the semi-group 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 Hardy and Harris (2009). The purpose of this short note is to exemplify a new {\it pathwise} spine decomposition for supercritical super-Brownian motion with general branching mechanism (cf. Kyprianou et al. (2010)) by studying $L^p$ convergence of naturally underlying additive martingales in the spirit of analogous arguments for branching particle diffusions due to Hardys and Harris (2009). Amongst other ingredients, the Dynkin-Kuznetsov $\mathbb{N}$-measure plays a pivotal role in the analysis.