Comptes Rendus
Probability Theory
Global gradient bounds for dissipative diffusion operators
Comptes Rendus. Mathématique, Volume 339 (2004) no. 4, pp. 277-282.

Let L be a second order elliptic operator on Rd with a constant diffusion matrix and a dissipative (in a weak sense) drift bLlocp with some p>d. We assume that L possesses a Lyapunov function, but no local boundedness of b is assumed. It is known that then there exists a unique probability measure μ satisfying the equation L*μ=0 and that the closure of L in L1(μ) generates a Markov semigroup {Tt}t0 with the resolvent {Gλ}λ>0. We prove that, for any Lipschitzian function fL1(μ) and all t,λ>0, the functions Ttf and Gλf are Lipschitzian and supx,t|Ttf(x)|supx|f(x)| and supx|Gλf(x)|1λsupx|f(x)|. In addition, we show that for every bounded Lipschitzian function g, the function Gλg is the unique bounded solution of the equation λfLf=g in the Sobolev class Hloc2,2(Rd).

Soit L un opérateur elliptique sur Rd tel que son terme du premier ordre bLlocp, p>d, soit dissipatif (mais pas nécessairement localement borné) et qu'il existe une fonction de Liapounoff. Il est connu qu'il existe une probabilité unique μ telle que L*μ=0 au sens faible et la fermeture de L dans L1(μ) est le générateur d'un semigroupe markovien {Tt}t0 de résolvante {Gλ}λ>0. Nous montrons que pour chaque fonction lipschitzienne fL1(μ) et tous t,λ>0 les fonctions Ttf et Gλf sont lipschitziennes et on a supx,t|Ttf(x)|supx|f(x)| et supx|Gλf(x)|1λsupx|f(x)|. De plus, nous montrons que pour chaque fonction bornée lipschitzienne g la fonction Gλg est la solution unique bornée de l'équation λfLf=g dans la classe de Sobolev Hloc2,2(Rd).

Published online:
DOI: 10.1016/j.crma.2004.05.016
Vladimir I. Bogachev 1; Giuseppe Da Prato 2; Michael Röckner 3; Zeev Sobol 4

1 Department of Mechanics and Mathematics, Moscow State University, 119992 Moscow, Russia
2 Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56125 Pisa, Italy
3 Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany
4 Department of Mathematics, Imperial College, 180, Queens Gate, London SW7 2BZ, UK
     author = {Vladimir I. Bogachev and Giuseppe Da Prato and Michael R\"ockner and Zeev Sobol},
     title = {Global gradient bounds for dissipative diffusion operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {277--282},
     publisher = {Elsevier},
     volume = {339},
     number = {4},
     year = {2004},
     doi = {10.1016/j.crma.2004.05.016},
     language = {en},
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AU  - Giuseppe Da Prato
AU  - Michael Röckner
AU  - Zeev Sobol
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PB  - Elsevier
DO  - 10.1016/j.crma.2004.05.016
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%A Giuseppe Da Prato
%A Michael Röckner
%A Zeev Sobol
%T Global gradient bounds for dissipative diffusion operators
%J Comptes Rendus. Mathématique
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Vladimir I. Bogachev; Giuseppe Da Prato; Michael Röckner; Zeev Sobol. Global gradient bounds for dissipative diffusion operators. Comptes Rendus. Mathématique, Volume 339 (2004) no. 4, pp. 277-282. doi : 10.1016/j.crma.2004.05.016.

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[2] V.I. Bogachev; N.V. Krylov; M. Röckner Regularity of invariant measures: the case of non-constant diffusion part, J. Funct. Anal., Volume 138 (1996) no. 1, pp. 223-242

[3] V.I. Bogachev; N.V. Krylov; M. Röckner On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. Partial Differential Equations, Volume 26 (2001) no. 11/12, pp. 2037-2080

[4] V.I. Bogachev; M. Röckner; W. Stannat Uniqueness of invariant measures and maximal dissipativity of diffusion operators on L1 (P. Clement et al., eds.), Infinite Dimensional Stochastic Analysis, Royal Netherlands Academy of Arts and Sciences, Amsterdam, 2000, pp. 39-54

[5] V.I. Bogachev; M. Röckner; W. Stannat Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions, Sb. Math., Volume 193 (2002) no. 7, pp. 945-976

[6] H. Brézis Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973

[7] M. Chicco Solvability of the Dirichlet problem in H2,p(Ω) for a class of linear second order elliptic partial differential equations, Boll. Un. Mat. Ital., Volume 4 (1971) no. 4, pp. 374-387

[8] G. Da Prato Elliptic operators with unbounded coefficients: construction of a maximal dissipative extension, J. Evol. Eq., Volume 1 (2001) no. 1, pp. 1-18

[9] G. Da Prato; M. Röckner Singular dissipative stochastic equations in Hilbert spaces, Probab. Theory Relat. Fields, Volume 124 (2002) no. 2, pp. 261-303

[10] N.V. Krylov Lectures on Elliptic and Parabolic Equations in Hölder Spaces, American Mathematical Society, Providence, RI, 1996

[11] M. Röckner; F.-Y. Wang On the spectrum of a class of (non-symmetric) diffusion operators, Bull. Lond. Math. Soc., Volume 36 (2004), pp. 95-104

[12] W. Stannat (Nonsymmetric) Dirichlet operators on L1: existence, uniqueness and associated Markov processes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 28 (1999) no. 1, pp. 99-140

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