The domain of the Ornstein-Uhlenbeck operator on an $L^p$-space with invariant measure

Metafune, Giorgio; Prüss, Jan; Rhandi, Abdelaziz; Schnaubelt, Roland

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 $L^p$ $({ℝ}^N,\mu dx)$ equals the weighted Sobolev space $W^{2,p}({ℝ}^N,\mu dx)$, where $\mu dx$ is the corresponding invariant measure. Our approach relies on the operator sum method, namely the commutative and the non commutative Dore-Venni theorems.

[1] P. Cannarsa - V. Vespri, Generation of analytic semigroups in the $L^p$ topology by elliptic operators in ${ℝ}^n$, 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 $L^2\left(\mu \right)$ 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