By Jonathan Warren (auth.), Jacques Azéma, Marc Yor, Michel Emery (eds.)

The 31 papers gathered right here current unique learn effects got in 1995-96, on Brownian movement and, extra commonly, diffusion tactics, martingales, Wiener areas, polymer measures.

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Additional resources for Séminaire de Probabilités XXXI

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On s'int~resse aux semi-groupes de Markov, sym~triques dens L2(E), qui admettent la suite (Q,) comme d~composition spectrale. On d~finit un semi-groupe de Markov admettant # comme mesure stationneire per une famille (Pt)t>o d'op~reteurs egissant sur les fonctions bor~liennes born~es, et v~rifiant : 9 PtP,=Pt+,; Po=Id, 9 f>_O=~Ptf>O; Ptl=l, 42 9 Vf C LI(E) rjt Ptfd# = f f d p (hypoth~se d'invariunce). On voit alors que (Pt) est une contraction de LP(E) duns lui-m6me, et ce Vp E [1, +oo[. On exige alors de plus, que 9 Y f E L2(E) 9 Vf, g e L~(E) limPtf = f dans L2(E), t--+O fl(Ptf)gdp = f f ( P t g ) d# (hypoth~se de sym&trie).

Bakry and M. Emery. Hypercontractivit6 de semi-groupes de diffusion. Acad. Paris, 299, S~rie I(15):775-778, 1984. [6] S. Bochner. Sturm-Liouville and heat equations whose eigenfunctions are ultraspherical polynomials or associated Bessel functions. Proc. Conf. Differential Equations, pages 23-48, 1955. [7] W. Feller. The parabolic differential equations and the associated semi-groups of transformations. Ann. , 55:468-519. 1952. [8] W. Feller. Diffusion processes in one dimension. Trans. Amer. Math.

T h e collection of such fourtuples (m, n, v, A) is countable and G(rn, n, v, A) will be d e n o t e d by G , , u = 1, 2,. 9.. Set F,('w) = v + (fl(npa~ (w) )[f(w) - F;@w) - v] + F;~(w) and note t h a t for w E G,, Pc;, (w) = 0 36 and F , ( w ) = F ( w ) . (w) = w+G(~), T~(~) = ~+FA(~) T~(~) : ~ + ~('~Pc. (w) = w +v it is easily verified t h a t Tv o Tc o Ta = 7",. Now, Tx and Tv are bijeetive and from the definition of G , for a < 1/3 II VG(w) 112< 1. 3, Tc is also bijective and consequently T, is bijective.

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