More on the long time stability of Feynman–Kac semigroups

Large deviations of empirical measures of diffusions in weighted topologies

“…This allows to define (1.6) for functions ϕ such that ϕ B ∞ W < +∞, as soon as f is dominated by the function Ψ defined in (1.4). As a result, for any such f , the operator (1.6) can be shown to be compact over the space of functions controlled by W (see [55,47]), and the functional (1.5) is obtained as the largest eigenvalue of the operator (1.6) through a generalized Perron-Frobenius theorem (the Krein-Rutman theorem [27]).…”

“…We mention that the proof of (1.8) relies on the spectral problem associated with the Feynman-Kac operator (1.6), and uses tools from the recent work [47].…”

“…This amounts to showing the well-posedness and regularity of a family of spectral problems. For this, we use several ideas from [47], which shows that under Lyapunov and irreducibility conditions, the eigenvalue problem to which λ is associated is well defined. In order to avoid technical difficulties related to unbounded operators, we study the semigroup (P f t ) t 0 rather than its generator L + f , see Remark 6.11 below for more details.…”

“…We can now bound the right hand side of the above equation with a technique similar to the one used in [47,Section 2.3]. Indeed, for any x ∈ X , Since κ − LW W , there exists a constant c 0 depending on f such that…”

Convergence to quasi-stationarity through Poincaré inequalities and Bakry-Émery criteria

“…Since, other papers showed exponential convergences in total variation under weaker assumptions, allowing convergences in total variation holding non-uniformly in the initial measure. In particular, we refer the reader to [10,36] for the study of absorbed Markov processes, and [4,12,19] for the study of general renormalized Feynman-Kac semi-groups.…”