  The dissipation of general convex entropies for continuous time Markov
processes can be described in terms of backward martingales with respect to the
tail filtration. 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 the Doob-Meyer decomposition of
this submartingale. We deduce a stochastic analogue of the well known entropy
dissipation formula, which is valid for general convex entropies, including the
total variation distance. Under additional regularity assumptions, and using
It\^o's calculus and ideas of Arnold, Carlen and Ju \cite{Arnoldcarlenju}, we
obtain moreover 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. We provide examples where the classic
Bakry Emery criterion fails, but our non-intrisic criterion applies without
modifying the law of the diffusion process.
