Entropy, Reversible Diffusion Processes and Markov Uniqueness

        Patrick Cattiaux
        Ecole Polytechnique, CMAP, F-91128 Palaiseau Cedex,
        CNRS 756 ~~ and ~~
        Universit\'e Paris X Nanterre, equipe MODAL'X,
        UFR SEGMI, 200 av. de la r\'epublique, 
        F-91001 Nanterre Cedex.
        e-mail : cattiaux@paris.polytechnique.fr}
        
       Myriam Fradon
        Universit\'e Paris-Sud, D\'ep. de Math\'ematiques,
                Bat. 425, F-91405 Orsay Cedex.
                CNRS 743 . ~~~
                e-mail : fradon@stats.matups.fr

November 30, 1994

Consider a symmetric bilinear form $ \Ep $ defined on
$ \Cic(\R^d) $ by
\[
\Ep(f,g) = \int_{\R^d} \Df . \Dg ~ \p^2 \dx ~~,~~~~ \p \in H^1_{loc}(\R^d)
\]
In this paper we study the stochastic process
associated with the smallest closed markovian extension of
$ ( \Ep , \Cic ) $,
and give a new proof of Markov uniqueness
(~i.e. the uniqueness of a closed markovian extension~)
based on purely probabilistic arguments.
We also give another purely analytic one.
As a consequence, we show that all invariant measures are reversible,
provided they are of finite energy.
The problem of uniqueness of such measures is also partially solved.

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