Description
In this mini-course, I will present an approach to the study of local and global survival of spatial branching processes through generalized principal eigenvalues of linear positive semigroups. The fact that local survival of a branching diffusion process can be characterized via positivity of the generalized principal eigenvalue in the sense of Berestycki-Nirenberg-Varadhan of an associate Schrödinger operator has been established in works by Engländer, Kyprianou and Pinsky in the 2000's. The case of global survival has been studied recently by Oliver Tough and myself. We relate it to several definitions of generalized 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 generalized principle eigenvalues, as well as about uniqueness and non-uniqueness of stationary solutions of a generalized 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.
P. Maillard, O. Tough (2025) Generalised principal eigenvalues and global survival of branching Markov processes, https://arxiv.org/abs/2505.12127