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Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential

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Abstract.

We consider the Fokker-Planck equation with a confining or anti-confining potential which behaves at infinity like a possibly high-degree homogeneous function. Hypoellipticity techniques provide the well-posedness of the weak Cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium. Lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding Witten Laplacian, with detailed applications.

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Authors and Affiliations

  1. Laboratoire de Mathématiques UFR Sciences exactes et naturelles, Université de Reims, Moulin de la Housse, 1039, 51687, Reims cedex 9, FRANCE

    Frédéric Hérau

  2. IRMAR, UMR-CNRS 6625 Campus de beaulieu, Université de Rennes, 1, 35042, Rennes cedex, FRANCE

    Francis Nier

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  1. Frédéric Hérau
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  2. Francis Nier
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Correspondence to Francis Nier.

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Communicated by Y. Brenier

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Hérau, F., Nier, F. Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential. Arch. Rational Mech. Anal. 171, 151–218 (2004). https://doi.org/10.1007/s00205-003-0276-3

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