A non?conservative Harris ergodic theorem

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