Kolmogorov kernel estimates for the Ornstein-Uhlenbeck operator
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 4, p. 729-748
Replacing the gaussian semigroup in the heat kernel estimates by the Ornstein-Uhlenbeck semigroup on d , we define the notion of Kolmogorov kernel estimates. This allows us to show that under Dirichlet boundary conditions Ornstein-Uhlenbeck operators are generators of consistent, positive, (quasi-) contractive C 0 -semigroups on L p (Ω) for all 1p< and for every domain Ω d . For exterior domains with sufficiently smooth boundary a result on the location of the spectrum of these operators is also given.
Classification:  47D06,  35K20
@article{ASNSP_2005_5_4_4_729_0,
     author = {Haller-Dintelmann, Robert and Wiedl, Julian},
     title = {Kolmogorov kernel estimates for the Ornstein-Uhlenbeck operator},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 4},
     number = {4},
     year = {2005},
     pages = {729-748},
     zbl = {1171.47302},
     mrnumber = {2207741},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2005_5_4_4_729_0}
}
Haller-Dintelmann, Robert; Wiedl, Julian. Kolmogorov kernel estimates for the Ornstein-Uhlenbeck operator. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 4, pp. 729-748. http://www.numdam.org/item/ASNSP_2005_5_4_4_729_0/

[1] R. A. Adams, “Sobolev Spaces”, Academic Press, New York, 1978. | MR 450957

[2] W. Arendt and P. Bénilan, Wiener regularity and heat semigroups on spaces of continuous functions, In: “Topics in Nonlinear Analysis”, Vol. 35, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 1999, 29-49. | MR 1724790 | Zbl 0920.35041

[3] M. Bertoldi and L. Lorenzi, Analytic methods for markov semigroups, preprint. | MR 2313847

[4] G. Da Prato and A. Lunardi, Elliptic operators with unbounded drift coefficients and Neumann boundary condition, J. Differential Equations 198 (2004), 35-52. | MR 2037749 | Zbl 1046.35025

[5] R. Dautray and J.-L. Lions, “Mathematical Analysis and Numerical Methods for Science and Technology”, Vol. 1, Springer-Verlag, Berlin, 1990. | MR 1036731 | Zbl 0683.35001

[6] E.B. Davies and B. Simon, L 1 -properties of intrinsic Schrödinger semigroups, J. Funct. Anal. 65 (1986), 126-146. | MR 819177 | Zbl 0613.47039

[7] K.-J. Engel and R. Nagel, “One-Parameter Semigroups for Linear Evolution Equations”, Vol. 194, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. | MR 1721989 | Zbl 0952.47036

[8] S. Fornaro, G. Metafune and E. Priola, Gradient estimates for Dirichlet parabolic problems in unbounded domains, J. Differential Equations 205 (2004), 329-353. | MR 2092861 | Zbl 1061.35022

[9] M. Geißert, H. Heck and M. Hieber, L p -theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle, 2005, preprint. | MR 2254804

[10] M. Geißert, H. Heck, M. Hieber and I. Wood, The Ornstein-Uhlenbeck semigroup in exterior domains, Arch. Math. (Basel), to appear. | MR 2191665 | Zbl 1114.47043

[11] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Operators of Second Order. Second Edition”, Vol. 224, A Series of Comprehensive Studies in Mathematics, Springer-Verlag, Berlin, 1983. | MR 737190 | Zbl 0562.35001

[12] M. Hieber and J. Prüss, Functional calculi for linear operators in vector-valued L p -spaces via the transference principle, Adv. Differential Equations 3 (1998), 847-872. | MR 1659281 | Zbl 0956.47008

[13] M. Hieber - O. Sawada, The Navier-Stokes equations in n with linearly growing initial data, Arch. Ration. Mech. Anal. 175 (2005), 269-285. | MR 2118478 | Zbl 1072.35144

[14] T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Ration. Mech. Anal. 150 (1999), 307-348. | MR 1741259 | Zbl 0949.35106

[15] G. Metafune, L p -spectrum of Ornstein-Uhlenbeck operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), 97-124. | Numdam | MR 1882026 | Zbl 1065.35216

[16] G. Metafune, D. Pallara and E. Priola, Spectrum of Ornstein-Uhlenbeck operators in L p spaces with respect to invariant measures, J. Funct. Anal. 196 (2002), 40-60. | MR 1941990 | Zbl 1027.47036

[17] G. Metafune, D. Pallara and V. Vespri, L p -estimates for a class of elliptic operators with unbounded coefficients in n , Houston Math. J. 31 (2005), 605-620. | MR 2132853 | Zbl 1129.35420

[18] G. Metafune, J. Prüss, A. Rhandi and R. Schnaubelt, The domain of the Ornstein-Uhlenbeck operator on an L p -space with invariant measure, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 1 (2002), 471-485. | Numdam | MR 1991148 | Zbl 1170.35375

[19] H. H. Schaefer, “Banach Lattices and Positive Operators”, Vol. 215 of Die Grundlehren der mathematischen Wissenschaften, Springer-Verlag, New York, 1974. | MR 423039 | Zbl 0296.47023