Generalised principal eigenvalues and global survival of branching Markov processes

TL;DR

This paper establishes criteria for the global survival of branching Markov processes by linking them to generalized principal eigenvalues of elliptic operators, extending existing notions, and providing new insights into stationary solutions.

Contribution

It extends the concept of generalized principal eigenvalues to broader semigroups and connects these to the survival and stationary solutions of branching processes.

Findings

Criteria for global survival are established.

Relations between eigenvalues and stationary solutions are clarified.

New results and counterexamples for conjectures are provided.

Abstract

We study necessary and sufficient criteria for global survival of discrete or continuous-time branching Markov processes. We relate these to several definitions of generalised principle eigenvalues for elliptic operators due to Berestycki and Rossi. In doing so, we extend these notions to fairly general semigroups of linear positive operators. We use this relation to prove new results about the generalised principle eigenvalues, as well as about uniqueness and non-uniqueness of stationary solutions of a generalised FKPP equation. The probabilistic approach through branching processes gives rise to relatively simple and transparent proofs under much more general assumptions, as well as constructions of (counter-)examples to certain conjectures.