Path-valued branching processes and nonlocal branching superprocesses

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].