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

Simona Fornaro; Abdelaziz Rhandi

doi:10.3934/dcds.2013.33.5049

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2013, Volume 33, Issue 11&12: 5049-5058. Doi: 10.3934/dcds.2013.33.5049

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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: October 31, 2011

Published: October 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 47D06, 35R05, 35B25; Secondary: 35K15, 35K65, 34G10.

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