Conference Agenda

We condition a continuous-state branching process with quadratic competition on non-extinction by requiring the total progeny to exceed arbitrarily large exponential random variables. This is related to a Doob’s h-transform with an explicit excessive function h. We show that the h-transformed process has a finite lifetime (it is either killed or it explodes continuously) almost surely. When starting from positive values, we can characterize the conditioned process up to its lifetime as the solution to a certain stochastic equation with jumps. The latter superposes the dynamics of the initial logistic CB process with an additional density-dependent immigration term. Last, we establish that the conditioned process can be started from zero. Key tools employed are a Laplace and Siegmund duality relationships with auxiliary diffusion processes.

Motivated by applications to COVID and other epidemics dynamics, we describe a branching process in random environments model whose characteristics change when crossing upper and lower thresholds. This introduces a cyclical path behavior involving periods of increase and decrease leading to supercritical and subcritical regimes. We establish law of large numbers and central limit theorems for the average length of these regimes and the proportion of time spent in each regime. We also derive the limiting joint distribution of offspring mean estimators in the supercritical and subcritical regimes and thereby show that they are asymptotically independent. We explicitly identify the limiting variances in terms of functionals of the offspring distribution, threshold distribution, and environmental sequences.

References

G. Francisci and A. N. Vidyashankar. Branching processes in random environments with thresholds. Advances in Applied Probability, 56(2):495–544, 2024.