1 IECL (UMR CNRS 7502), Université de Lorraine, Site de Nancy, B.P. 70239, 54506 Vandœuvre-lès-Nancy Cedex, France.
2 Inria, TOSCA Team, Villers-lès-Nancy 54600, France.
We provide a general criterion ensuring the exponential contraction of Feynman–Kac semi-groups of penalized processes. This criterion applies to time-inhomogeneous Markov processes with absorption and killing through penalization. We also give the asymptotic behavior of the expected penalization and provide results of convergence in total variation of the process penalized up to infinite time. For exponential convergence of penalized semi-groups with bounded penalization, a converse result is obtained, showing that our criterion is sharp in this case. Several cases are studied: we first show how our criterion can be simply checked for processes with bounded penalization, and we then study in detail more delicate examples, including one-dimensional diffusion processes conditioned not to hit 0 and penalized birth and death processes evolving in a quenched random environment.
Key words: Feynman–Kac formula / time-inhomogeneous Markov processes / penalized processes / one-dimensional diffusions with absorption / birth and death processes in random environment with killing / asymptotic stability / uniform exponential mixing / Dobrushin’s ergodic coefficient.