Convergence of Feynman-Kac semigroups and large deviations of diffusions

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Date(s) - 12/04/2019
11 h 00 min - 12 h 00 min

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Feynman-Kac semigroups appear in various places of mathematics. They can be used in relation with rare events analysis, or in quantum mechanics as a probabilistic representation of the ground state. It is also an everyday tool in nonlinear filtering. Just like Markov chains, their long time behaviour is an important question, be it for theoretical or practical purposes.

However, the stability of such a dynamics is made difficult by its nonlinearity. We prove in [G.F., M. Rousset in G. Stoltz, 2018] that ergodic results can be obtained through assumptions similar to that for Markov chains, by performing a fine spectral analysis of the evolution operator. This shows an interesting competition between the dynamics itself and the weight given to its paths. As a by-product, we show [G.F., G. Stoltz, in prep.] that the technique allows to derive large deviations principles for empirical measures of diffusions in Wasserstein topology, which comes as a generalization of spectral gap conditions for Witten-Laplacian operators in a non-symmetric setting.

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