A non-conservative Harris ergodic theorem

Abstract

We consider non-conservative positive semigroups and obtain necessary and sufficient conditions for uniform exponential contraction in weighted total variation norm. This ensures the existence of Perron eigenelements and provides quantitative estimates of the spectral gap, complementing Krein-Rutman theorems and generalizing probabilistic approaches. The proof is based on a non-homogenous $h$-transform of the semigroup and the construction of Lyapunov functions for this latter. It exploits then the classical necessary and sufficient conditions of Harris's theorem for conservative semigroups and recent techniques developed for the study of absorbed Markov processes. We apply these results to population dynamics. We obtain exponential convergence of birth and death processes conditioned on survival to their quasi-stationary distribution, as well as estimates on exponential relaxation to stationary profiles in growth-fragmentation PDEs.

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• Unique Quasi-Stationary Distribution, with a possibly stabilizing extinction
• Analytical cell size distribution: lineage-population bias and parameter inference
• Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation
• On quantitative hypocoercivity estimates based on Harris-type theorems
• Uniform in time propagation of chaos for a Moran model
• Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics
• Exponential ergodicity of a degenerate age-size piecewise deterministic process
• Dynamics of a Fleming-Viot type particle system on the cycle graph
• Strong laws of large numbers for a growth-fragmentation process with bounded cell sizes
• On the asymptotic behaviour of a run and tumble equation for bacterial chemotaxis
• Quasi-stationary distributions in reducible state spaces
• Large deviations for surviving trajectories of general Markov processes