Global gradient bounds for dissipative diffusion operators

Bogachev, Vladimir I.; Da Prato, Giuseppe; Röckner, Michael; Sobol, Zeev

doi:10.1016/j.crma.2004.05.016

Probability Theory

Global gradient bounds for dissipative diffusion operators
[Estimations globales de gradient pour les opérateurs de diffusion.]

Présenté par : Paul Malliavin

Bogachev, Vladimir I.¹ ; Da Prato, Giuseppe ² ; Röckner, Michael ³ ; Sobol, Zeev 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

Comptes Rendus. Mathématique, Tome 339 (2004) no. 4, pp. 277-282.

Résumé
Abstract

Soit L un opérateur elliptique sur $R^{d}$ tel que son terme du premier ordre $b \in L_{loc}^{p}$ , $p > d$ , 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 $L^{1} (μ)$ est le générateur d'un semigroupe markovien ${T_{t}}_{t ⩾ 0}$ de résolvante ${G_{λ}}_{λ > 0}$ . Nous montrons que pour chaque fonction lipschitzienne $f \in L^{1} (μ)$ et tous $t, λ > 0$ les fonctions $T_{t} f$ et $G_{λ} f$ sont lipschitziennes et on a ${sup}_{x, t} | \nabla T_{t} f (x) | ⩽ {sup}_{x} | \nabla f (x) |$ et ${sup}_{x} | \nabla G_{λ} f (x) | ⩽ \frac{1}{λ} {sup}_{x} | \nabla f (x) |$ . De plus, nous montrons que pour chaque fonction bornée lipschitzienne g la fonction $G_{λ} g$ est la solution unique bornée de l'équation $λ f - L f = g$ dans la classe de Sobolev $H_{loc}^{2, 2} (R^{d})$ .

Let L be a second order elliptic operator on $R^{d}$ with a constant diffusion matrix and a dissipative (in a weak sense) drift $b \in L_{loc}^{p}$ 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 $L^{1} (μ)$ generates a Markov semigroup ${T_{t}}_{t ⩾ 0}$ with the resolvent ${G_{λ}}_{λ > 0}$ . We prove that, for any Lipschitzian function $f \in L^{1} (μ)$ and all $t, λ > 0$ , the functions $T_{t} f$ and $G_{λ} f$ are Lipschitzian and ${sup}_{x, t} | \nabla T_{t} f (x) | ⩽ {sup}_{x} | \nabla f (x) |$ and ${sup}_{x} | \nabla G_{λ} f (x) | ⩽ \frac{1}{λ} {sup}_{x} | \nabla f (x) |$ . In addition, we show that for every bounded Lipschitzian function g, the function $G_{λ} g$ is the unique bounded solution of the equation $λ f - L f = g$ in the Sobolev class $H_{loc}^{2, 2} (R^{d})$ .

Reçu le : 2004-05-04
Accepté le : 2004-05-13
Publié le : 2004-07-28

DOI : 10.1016/j.crma.2004.05.016

Affiliations des auteurs :

Bogachev, Vladimir I. ¹ ; Da Prato, Giuseppe ² ; Röckner, Michael ³ ; Sobol, Zeev ⁴

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     title = {Global gradient bounds for dissipative diffusion operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {277--282},
     publisher = {Elsevier},
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     language = {en},
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Bogachev, Vladimir I.; Da Prato, Giuseppe; Röckner, Michael; Sobol, Zeev. Global gradient bounds for dissipative diffusion operators. Comptes Rendus. Mathématique, Tome 339 (2004) no. 4, pp. 277-282. doi : 10.1016/j.crma.2004.05.016. http://www.numdam.org/articles/10.1016/j.crma.2004.05.016/

Bibliographie
Cité par

[1] Bogachev, V.I.; Röckner, M. A generalization of Hasminskii's theorem on existence of invariant measures for locally integrable drifts, Theory Probab. Appl., Volume 45 (2000) no. 3, pp. 417-436

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

[3] Bogachev, V.I.; Krylov, N.V.; Röckner, M. 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] Bogachev, V.I.; Röckner, M.; Stannat, W. Uniqueness of invariant measures and maximal dissipativity of diffusion operators on $L^{1}$ (Clement, P. et al., eds.), Infinite Dimensional Stochastic Analysis, Royal Netherlands Academy of Arts and Sciences, Amsterdam, 2000, pp. 39-54

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

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

[7] Chicco, M. Solvability of the Dirichlet problem in $H^{2, p} (Ω)$ for a class of linear second order elliptic partial differential equations, Boll. Un. Mat. Ital., Volume 4 (1971) no. 4, pp. 374-387

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

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

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

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

[12] Stannat, W. (Nonsymmetric) Dirichlet operators on $L^{1}$ : 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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