The domain of the Ornstein-Uhlenbeck operator on an L p -space with invariant measure
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 2, pp. 471-485.

We show that the domain of the Ornstein-Uhlenbeck operator on L p ( N ,μdx) equals the weighted Sobolev space W 2,p ( N ,μdx), where μdx is the corresponding invariant measure. Our approach relies on the operator sum method, namely the commutative and the non commutative Dore-Venni theorems.

Classification : 35J15, 35K10, 47A55, 47D06
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     title = {The domain of the {Ornstein-Uhlenbeck} operator on an $L^p$-space with invariant measure},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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Metafune, Giorgio; Prüss, Jan; Rhandi, Abdelaziz; Schnaubelt, Roland. The domain of the Ornstein-Uhlenbeck operator on an $L^p$-space with invariant measure. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 2, pp. 471-485. http://archive.numdam.org/item/ASNSP_2002_5_1_2_471_0/

[1] P. Cannarsa - V. Vespri, Generation of analytic semigroups in the L p topology by elliptic operators in n , Israel J. Math. 61 (1988), 235-255. | MR | Zbl

[2] A. Chojnowska-Michalik - B. Goldys, Generalized symmetric Ornstein-Uhlenbeck operators in L p : Littlewood-Paley-Stein inequalities and domains of generators, to appear in J. Funct. Anal. | MR

[3] A. Chojnowska-Michalik - B. Goldys, Symmetric Ornstein-Uhlenbeck generators: Characterizations and identification of domains, preprint.

[4] P. Clément - J. Prüss, Completely positive measures and Feller semigroups, Math. Ann. 287 (1990), 73-105. | MR | Zbl

[5] R. R. Coifman - G. Weiss, Transference Methods in Analysis, Amer. Math. Society, 1977. | MR | Zbl

[6] G. Da Prato, Characterization of the domain of an elliptic operator of infinitely many variables in L 2 (μ) spaces, Rend. Mat. Acc. Lincei 8 (1997), 101-105. | MR | Zbl

[7] G. Da Prato, Perturbation of Ornstein-Uhlenbeck semigroups, Rend. Istit. Mat. Univ. Trieste 28 (1997), 101-126. | MR | Zbl

[8] G. Da Prato - A. Lunardi, On the Ornstein-Uhlenbeck operator in spaces of continuous functions, J. Funct. Anal. 131 (1995), 94-114. | MR | Zbl

[9] G. Da Prato - B. Goldys, On perturbations of symmetric Gaussian diffusions, Stochastic Anal. Appl. 17 (1999), 369-382. | MR | Zbl

[10] G. Da Prato - V. Vespri, Maximal L p regularity for elliptic equations with unbounded coefficients, to appear in Nonlinear Analysis TMA. | Zbl

[11] G. Da Prato - J. Zabczyk, “Stochastic Equations in Infinite Dimensions”, Cambridge University Press, 1992. | MR | Zbl

[12] G. Da Prato - J. Zabczyk, Regular densities of invariant measures in Hilbert spaces, J. Funct. Anal. 130 (1995), 427-449. | MR | Zbl

[13] G. Dore - A. Venni, On the closedness of the sum of two closed operators, Math. Z. 196 (1987), 189-201. | MR | Zbl

[14] A. Lunardi, On the Ornstein-Uhlenbeck operator in L 2 spaces with respect to invariant measures, Trans. Amer. Math. Soc. 349 (1997), 155-169. | MR | Zbl

[15] G. Metafune, L p -spectrum of Ornstein-Uhlenbeck operators, Ann. Sc. Norm. Sup. Pisa 30 (2001), 97-124. | Numdam | MR | Zbl

[16] G. Metafune - D. Pallara - E. Priola, Spectrum of Ornstein-Uhlenbeck operators in L p spaces with respect to invariant measures, preprint. | Zbl

[17] S. Monniaux - J. Prüss, A theorem of the Dore-Venni type for noncommuting operators, Trans. Amer. Math. Soc. 349 (1997), 4787-4814. | MR | Zbl

[18] J. Prüss - H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z. 203 (1990), 429-452. | MR | Zbl

[19] J. Prüss - H. Sohr, Imaginary powers of elliptic second order differential operators in L p -space, Hiroshima Math. J. 23 (1993), 161-192. | MR | Zbl

[20] I. Shigekawa, Sobolev spaces over the Wiener space based on an Ornstein-Uhlenbeck operator, J. Math. Kyoto Univ. 32 (1992), 731-748. | MR | Zbl