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).

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DOI: 10.1016/j.crma.2004.05.016
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
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Bogachev, Vladimir I.; Da Prato, Giuseppe; Röckner, Michael; Sobol, Zeev. 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. http://www.numdam.org/articles/10.1016/j.crma.2004.05.016/

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