Logarithmic Sobolev inequalities: Regularizing effect of Lévy operators and asymptotic convergence in the Lévy–Fokker–Planck equation

Abstract

In this paper we study some applications of the Lévy logarithmic Sobolev inequality to the study of the regularity of the solution of the fractal heat equation, i.e. the heat equation where the Laplacian is replaced with the fractional Laplacian. It is also used to the study of the asymptotic behaviour of the Lévy–Ornstein–Uhlenbeck process.

Keywords:

• Fokker–Planck equation
• Ornstein–Uhlenbeck equation
• Lévy operator
• Φ-entropy inequalities
• entropy production method
• logarithmic Sobolev inequalities
• fractional Laplacian
• ultracontractivity

Mathematics Subject Classification:

• 46N20
• 47G20
• 35K15

Acknowledgement

The first author sincerely thanks H. Ouerdiane for his hospitality during the autumn in Hammamet.

Notes

^1. [email protected]

Additional information

Notes on contributors

Cyril Imbert

1. 1. [email protected]