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

Giorgio Metafune; Jan Prüss; Abdelaziz Rhandi; Roland Schnaubelt

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2002)

  • Volume: 1, Issue: 2, page 471-485
  • ISSN: 0391-173X

Abstract

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

How to cite

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Metafune, Giorgio, et al. "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 1.2 (2002): 471-485. <http://eudml.org/doc/84478>.

@article{Metafune2002,
abstract = {We show that the domain of the Ornstein-Uhlenbeck operator on $L^p$$(\mathbb \{R\}^N,\mu dx)$ equals the weighted Sobolev space $W^\{2,p\}(\mathbb \{R\}^N,\mu dx)$, 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.},
author = {Metafune, Giorgio, Prüss, Jan, Rhandi, Abdelaziz, Schnaubelt, Roland},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {471-485},
publisher = {Scuola normale superiore},
title = {The domain of the Ornstein-Uhlenbeck operator on an $L^p$-space with invariant measure},
url = {http://eudml.org/doc/84478},
volume = {1},
year = {2002},
}

TY - JOUR
AU - Metafune, Giorgio
AU - Prüss, Jan
AU - Rhandi, Abdelaziz
AU - Schnaubelt, Roland
TI - The domain of the Ornstein-Uhlenbeck operator on an $L^p$-space with invariant measure
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2002
PB - Scuola normale superiore
VL - 1
IS - 2
SP - 471
EP - 485
AB - We show that the domain of the Ornstein-Uhlenbeck operator on $L^p$$(\mathbb {R}^N,\mu dx)$ equals the weighted Sobolev space $W^{2,p}(\mathbb {R}^N,\mu dx)$, 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.
LA - eng
UR - http://eudml.org/doc/84478
ER -

References

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  1. [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. Zbl0669.35026MR941240
  2. [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. MR1828795
  3. [3] A. Chojnowska-Michalik – B. Goldys, Symmetric Ornstein-Uhlenbeck generators: Characterizations and identification of domains, preprint. 
  4. [4] P. Clément – J. Prüss, Completely positive measures and Feller semigroups, Math. Ann. 287 (1990), 73-105. Zbl0717.47013MR1048282
  5. [5] R. R. Coifman – G. Weiss, Transference Methods in Analysis, Amer. Math. Society, 1977. Zbl0377.43001MR481928
  6. [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. Zbl0899.47035MR1485321
  7. [7] G. Da Prato, Perturbation of Ornstein–Uhlenbeck semigroups, Rend. Istit. Mat. Univ. Trieste 28 (1997), 101-126. Zbl0897.60070MR1602247
  8. [8] G. Da Prato – A. Lunardi, On the Ornstein-Uhlenbeck operator in spaces of continuous functions, J. Funct. Anal. 131 (1995), 94-114. Zbl0846.47004MR1343161
  9. [9] G. Da Prato – B. Goldys, On perturbations of symmetric Gaussian diffusions, Stochastic Anal. Appl. 17 (1999), 369-382. Zbl0935.60062MR1686991
  10. [10] G. Da Prato – V. Vespri, Maximal L p regularity for elliptic equations with unbounded coefficients, to appear in Nonlinear Analysis TMA. Zbl1012.35027
  11. [11] G. Da Prato – J. Zabczyk, “Stochastic Equations in Infinite Dimensions”, Cambridge University Press, 1992. Zbl0761.60052MR1207136
  12. [12] G. Da Prato – J. Zabczyk, Regular densities of invariant measures in Hilbert spaces, J. Funct. Anal. 130 (1995), 427-449. Zbl0832.60069MR1335387
  13. [13] G. Dore – A. Venni, On the closedness of the sum of two closed operators, Math. Z. 196 (1987), 189-201. Zbl0615.47002MR910825
  14. [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. Zbl0890.35030MR1389786
  15. [15] G. Metafune, L p -spectrum of Ornstein-Uhlenbeck operators, Ann. Sc. Norm. Sup. Pisa 30 (2001), 97-124. Zbl1065.35216MR1882026
  16. [16] G. Metafune – D. Pallara – E. Priola, Spectrum of Ornstein-Uhlenbeck operators in L p spaces with respect to invariant measures, preprint. Zbl1027.47036
  17. [17] S. Monniaux – J. Prüss, A theorem of the Dore-Venni type for noncommuting operators, Trans. Amer. Math. Soc. 349 (1997), 4787-4814. Zbl0887.47015MR1433125
  18. [18] J. Prüss – H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z. 203 (1990), 429-452. Zbl0665.47015MR1038710
  19. [19] J. Prüss – H. Sohr, Imaginary powers of elliptic second order differential operators in L p -space, Hiroshima Math. J. 23 (1993), 161-192. Zbl0790.35023MR1211773
  20. [20] I. Shigekawa, Sobolev spaces over the Wiener space based on an Ornstein-Uhlenbeck operator, J. Math. Kyoto Univ. 32 (1992), 731-748. Zbl0777.60047MR1194112

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