Kolmogorov kernel estimates for the Ornstein-Uhlenbeck operator
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 4 (2005) no. 4, pp. 729-748.

Replacing the gaussian semigroup in the heat kernel estimates by the Ornstein-Uhlenbeck semigroup on ${ℝ}^{d}$, 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 ${L}^{p}\left(\Omega \right)$ for all $1\le p<\infty$ and for every domain $\Omega \subseteq {ℝ}^{d}$. For exterior domains with sufficiently smooth boundary a result on the location of the spectrum of these operators is also given.

Classification: 47D06,  35K20
Haller-Dintelmann, Robert 1; Wiedl, Julian 1

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Haller-Dintelmann, Robert; Wiedl, Julian. Kolmogorov kernel estimates for the Ornstein-Uhlenbeck operator. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 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

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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