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Isotropic hypoelliptic and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. (English) Zbl 1139.82323

Summary: “We consider the Fokker-Planck equation \[ \partial h/\partial t+v\cdot\nabla_x h-\nabla V(x)\cdot\nabla_v h= \Delta_v h-v\cdot\nabla_v h \] 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.”
The results of the authors were extended by J.-P. Eckmann and M. Hairer [Commun. Math. Phys. 235, No. 2, 233–253 (2003; Zbl 1040.35016)] to more general hypoelliptic situations, and by more recent works by B. Helffer and F. Nier [Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Lect. Notes Math. 1862, Berlin: Springer (2005; Zbl 1072.35006)], and by C. Villani. See also C. Villani and L. Desvillettes [Commun. Pure Appl. Math. 54, No. 1, 1–42 (2001; Zbl 1029.82032)]; Invent. Math. 159, No. 2, 245–316 (2005; Zbl 1162.82316)].

MSC:

82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35F10 Initial value problems for linear first-order PDEs
35H10 Hypoelliptic equations
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