On the Ornstein Uhlenbeck operator perturbed by singular potentials in L^p--spaces

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November 2013, 33(11&12): 5049-5058. doi: 10.3934/dcds.2013.33.5049

On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces

Simona Fornaro ^1, and Abdelaziz Rhandi ^2,

1.	Università degli Studi di Pavia, Dipartimento di Matematica “F. Casorati”, via Ferrata 1, 27100 Pavia
2.	Dipartimento di Ingegneria dell'Informazione, Ingegneria Elletrica e Matematica Applicata, Università degli Studi di Salerno, Via Ponte Don Melillo, 84084 Fisciano (Sa)

Received November 2011 Published May 2013

Abstract
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In this paper we give sufficient conditions ensuring that the space of test functions $C_c^{\infty}(R^N)$ is a core for the operator $$L_0u=\Delta u-Mx\cdot \nabla u+\frac{\alpha}{|x|^2}u=:Lu+\frac{\alpha}{|x|^2}u,$$ and $L_0$ with domain $W_\mu^{2,p}(R^N)$ generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p_\mu(R^N),\,1 < p < \infty$. Here $M$ is a positive definite $N\times N$ hermitian matrix and $\mu$ is the unique invariant measure for the Ornstein-Uhlenbeck operator $L$. The proofs are based on an $L^p$-weighted Hardy's inequality and perturbation techniques.

Keywords: Ornstein-Uhlenbeck operator., core, dissipative and dispersive operator, Hardy's inequality, Inverse square potential, positivity preserving $C_0$-semigroup.

Mathematics Subject Classification: Primary: 47D06, 35R05, 35B25; Secondary: 35K15, 35K65, 34G1.

Citation: Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5049-5058. doi: 10.3934/dcds.2013.33.5049

References:

[1]	M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," Chapman Hall/CRC, Boca Raton FL, 2007.
[2]	T. Durante and A. Rhandi, On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials, Discrete Cont. Dyn. Syst. S., 6 (2013), 649-655. doi: doi:10.3934/dcdss.2013.6.649.
[3]	D. E. Edmunds and W. E. Evans, "Spectral Theory and Differential Operators," Clarendon Press, Oxford, 1987.
[4]	K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 2000.
[5]	G. R. Goldstein, J. A. Goldstein and A. Rhandi, Kolmogorov equation perturbed by an inverse-square potential, Discrete Cont. Dyn. Syst. S., 4 (2011), 623-630. doi: 10.3934/dcdss.2011.4.623.
[6]	G. R. Goldstein, J. A. Goldstein and A. Rhandi, Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential, Applicable Analysis, 91 (2012), 2057-2071. doi: 10.1080/00036811.2011.587809.
[7]	D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, 1983.
[8]	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), I (2002), 471-485.
[9]	R. Nagel, "One-Parameter Semigroups of Positive Operators," Lecture Notes in Math., 1184, Springer-Verlag, 1986.
[10]	N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan, 34 (1982), 677-701. doi: 10.2969/jmsj/03440677.
[11]	N. Okazawa, $L^p$-theory of Schrödinger operators with strongly singular potentials, Japan. J. Math., 22 (1996), 199-239.
[12]	E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Math. Soc. Monographs, 31. Princeton Univ. Press 2004.
[13]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness," Academic Press, New York, 1975.
[14]	B. Simon, Essential self-adjointness of Schrödinger operators with singular potentials, Arch. Rational Mech. Anal., 52 (1973), 44-48.
[15]	J. Walter, Note on a paper by Stetkœr-Hansen concerning essential self-adjointness of Schrödinger operators, Math. Scand., 25 (1969), 94-96.

show all references

References:

[1]	M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," Chapman Hall/CRC, Boca Raton FL, 2007.
[2]	T. Durante and A. Rhandi, On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials, Discrete Cont. Dyn. Syst. S., 6 (2013), 649-655. doi: doi:10.3934/dcdss.2013.6.649.
[3]	D. E. Edmunds and W. E. Evans, "Spectral Theory and Differential Operators," Clarendon Press, Oxford, 1987.
[4]	K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 2000.
[5]	G. R. Goldstein, J. A. Goldstein and A. Rhandi, Kolmogorov equation perturbed by an inverse-square potential, Discrete Cont. Dyn. Syst. S., 4 (2011), 623-630. doi: 10.3934/dcdss.2011.4.623.
[6]	G. R. Goldstein, J. A. Goldstein and A. Rhandi, Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential, Applicable Analysis, 91 (2012), 2057-2071. doi: 10.1080/00036811.2011.587809.
[7]	D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, 1983.
[8]	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), I (2002), 471-485.
[9]	R. Nagel, "One-Parameter Semigroups of Positive Operators," Lecture Notes in Math., 1184, Springer-Verlag, 1986.
[10]	N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan, 34 (1982), 677-701. doi: 10.2969/jmsj/03440677.
[11]	N. Okazawa, $L^p$-theory of Schrödinger operators with strongly singular potentials, Japan. J. Math., 22 (1996), 199-239.
[12]	E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Math. Soc. Monographs, 31. Princeton Univ. Press 2004.
[13]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness," Academic Press, New York, 1975.
[14]	B. Simon, Essential self-adjointness of Schrödinger operators with singular potentials, Arch. Rational Mech. Anal., 52 (1973), 44-48.
[15]	J. Walter, Note on a paper by Stetkœr-Hansen concerning essential self-adjointness of Schrödinger operators, Math. Scand., 25 (1969), 94-96.

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