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| The dissipation of general convex entropies for continuous time Markov processes can be described in terms of backward martingales. The relative entropy is the expected value of a backward submartingale. In the case of (non necessarily reversible) Markov diffusion processes, we use Girsanov theory to explicit its Doob-Meyer decomposition and provide thereby
a stochastic analogue of the well known entropy dissipation formula, which is valid for general convex entropies, including total
variation distance. Under additional regularity assumptions, and using It\^o's calculus and ideas of
Arnold, Carlen and Ju 2008, we obtain a new Bakry Emery criterion which ensures exponential convergence of the entropy to $0$. This criterion is non-intrisic since it depends on the square root of the diffusion matrix, and cannot be written only in terms of the diffusion matrix itself. |
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