Path-valued branching processes and nonlocal branching superprocesses

TL;DR

This paper constructs a family of continuous-state branching processes with immigration as solutions to stochastic equations driven by space-time noise, revealing new insights into nonlocal superprocesses and their mechanisms.

Contribution

It explicitly identifies branching and immigration mechanisms of nonlocal superprocesses derived from path-valued branching processes.

Findings

Explicit mechanisms for nonlocal branching superprocesses

Connection to tree-valued Markov processes by Aldous and Pitman

New perspectives on inhomogeneous path-valued processes

Abstract

A family of continuous-state branching processes with immigration are constructed as the solution flow of a stochastic equation system driven by time-space noises. The family can be regarded as an inhomogeneous increasing path-valued branching process with immigration. Two nonlocal branching immigration superprocesses can be defined from the flow. We identify explicitly the branching and immigration mechanisms of those processes. The results provide new perspectives into the tree-valued Markov processes of Aldous and Pitman [Ann. Inst. Henri Poincar\'{e} Probab. Stat. 34 (1998) 637-686] and Abraham and Delmas [Ann. Probab. 40 (2012) 1167-1211].