Generalized $\Gamma$ calculus and application to interacting particles on a graph

Pierre Monmarché

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arXiv · 1510.05936

Generalized $\Gamma$ calculus and application to interacting particles on a graph

Pierre Monmarché

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Abstract

The classical Bakry-\'Emery calculus is extended to study, for degenerated (non-elliptic, non-reversible, or non-diffusive) Markov processes, questions such as hypoellipticity, hypocoercivity, functional inequalities or Wasserstein contraction. In particular we obtain the optimal speed of convergence to equilibrium for any ergodic Ornstein-Uhlenbeck process, which is given by the spectral gap of the drift matrix and the size of the corresponding Jordan blocks. We also study chains of $N$ interacting overdamped particles and establish for their invariant measures log-Sobolev inequalities with constants of order $N^2$, which is optimal.

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Pierre Monmarché. 2015-10-20. Generalized $\Gamma$ calculus and application to interacting particles on a graph. https://arxiv.org/abs/1510.05936

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