Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions

TL;DR

This paper studies the long-term behavior of conditioned multitype Dawson-Watanabe processes and Feller diffusions, providing new limit theorems and characterizations for these processes under various mutation matrix conditions.

Contribution

It introduces new limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions, including a martingale problem characterization and analysis of different mutation matrix structures.

Findings

Conditioned processes are absolutely continuous w.r.t. unconditioned laws on finite intervals.

Long-term behavior of conditioned mass processes is characterized.

Results cover irreducible and decomposable mutation matrices.

Abstract

A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process|the conditioned multitype Feller branching diffusion are then proved. The general case is first considered, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are analyzed too.