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P. Cattiaux (1)
M. Fradon (2)
| (1) | Ecole Polytechnique, CMAP, F-91128 Palaiseau Cedex, CNRS 756 Université Paris X Nanterre, equipe MODAL'X, UFR SEGMI, 200 av. de la république, F-91001 Nanterre Cedex. |
| (2) | Université Paris-Sud, Dép. de Mathématiques, Bat. 425, F-91405 Orsay Cedex. CNRS 743 . |
Abstract : 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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