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)].


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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[1] Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., Scheffer, G.: Sur les inégalités de Sobolev logarithmiques (avec une préface de D. Bakry et M. Ledoux). Panoramas & Synthèses, SMF, No 10, 2000 · Zbl 0982.46026
[2] Amrein, W., Boutet de Monv
[3] Beals, R.: Weighted distributions spaces and pseudodifferential · Zbl 0474.35089
[4] Bony, J.M., Chemin, J.Y.: Espaces fonctionnels associés au calcul de We
[5] Bony, J.M., Lerner, N.: Quantification asymptotique et microlocalisation d’ordre supérieur I. Ann. S · Zbl 0753.35005
[6] Bouchut, F., Dolbeault, F.: On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian pot · Zbl 0830.35129
[7] Boulton, L.S.: Non-self-adjoint harmonic oscillator, compact semigroups, and pseudospe · Zbl 1034.34099
[8] Crouzeix, M., Delyon, B.: Some estimates for analytic functions of band or sectorial operators. To appear in Archiv der Mathematik · Zbl 1060.47006
[9] Cycon, H.L., Froese, R. G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Text and Monographs in Physics. Springer-Verlag, 1987 · Zbl 0619.47005
[10] Davies, E.B.: One-parameter semigroups. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1980 · Zbl 0457.47030
[11] Davies, E.B.: Pseudospectra, the harmonic oscillator and complex resonances. R. Soc. London Ser. · Zbl 0931.70016
[12] Davies, E.B.: Semi-classical states for non self-adjoint Schrödinger oper · Zbl 0921.47060
[13] Desvillettes, L., Villani, C.: On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation · Zbl 1029.82032
[14] Dunford, N., Schwartz, J.T.: Linear operators. Part I. John Wiley & Sons Inc., New York, 1988 · Zbl 0635.47001
[15] Eckmann, J.P., Hairer, M.: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscill · Zbl 1044.82008
[16] Eckmann, J.P., Hairer, M.: Spectral properties of hypoelliptic operat · Zbl 1040.35016
[17] Eckmann, J.P., Pillet, C.A., Rey-Bellet, L.: Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different tempera · Zbl 0932.60103
[18] Georgescu, V., Gérard, C.: On the virial theorem in quantum mech · Zbl 0961.81009
[19] Helffer, B.: Semi-classical analysis for the Schrödinger operator and applications. Volume 1336 of Lect. Notes in Mathematics. Springer-Verlag, Berlin, 1988 · Zbl 0647.35002
[20] Helffer, B., Mohamed, A.: Caractérisation du spectre essentiel de l’opérateur de Schrödinger avec un champ magnétique. Ann. · Zbl 0638.47047
[21] Helffer, B., Mohamed, A.: Semiclassical analysis for the ground state energy of a Schrödinger operator with magneti · Zbl 0851.58046
[22] Helffer, B., Nier, F.: Criteria to the Poincaré inequality for some Dirichlet forms in dimension d · Zbl 1098.31005
[23] Helffer, B., Nourrigat, J.: Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs. Birkhäuser Boston Inc., Boston, MA, 1985 · Zbl 0568.35003
[24] Helffer, B., Sjöstrand, J.: Puits multiples en mécanique semi-classique, IV, étude du complexe de Witten. · Zbl 0597.35024
[25] Holley, R., Kusuoka, S., Stroock, D.: Asymptotics of the spectral gap with applications to the theory of simulated an · Zbl 0706.58075
[26] Hörmander, L.: Hypoelliptic second order differenti · Zbl 0156.10701
[27] Hörmander, L.: The Analysis of Linear Partial Differential Operators 3. Springer Verlag, 1985 · Zbl 0601.35001
[28] Hörmander, L.: Symplectic classification of quadratic forms, and general M · Zbl 0829.35150
[29] Johnsen, J.: On the spectral properties of Witten-Laplacians, their range projections and Brascamp-Lieb’s inequality · Zbl 1023.58012
[30] Kohn, J.J.: Lectures on degenerate elliptic problems. In: Pseudodifferential operator with applications (CIME 1977), Liguori, Naples, 1978, pp. 89–151
[31] Métivier, G.: équations aux dérivées partielles sur les groupes de Lie nilpotents. In: Bourbaki Seminar, Vol. 1981/1982, Soc. Math. France, Paris, 1982, pp. 75–99
[32] Nelson, E.: Dynamical theories of Brownian motion. Princeton, NJ: Princeton University Press, 1967 · Zbl 0165.58502
[33] Nourrigat, J.: Systèmes sous-elliptiques. In: Séminaire sur les équations aux dérivées partielles 1986–1987, pages Exp. No. V, 14. école Polytech., Palaiseau, 1987
[34] Nourrigat, J.: Systèmes sous-ellipt · Zbl 0771.35086
[35] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Volume 2. Acad. Press, 1975 · Zbl 0308.47002
[36] Rey-Bellet, L., Thomas, L.E.: Asymptotic Behavior of Thermal Nonequilibrium Steady States for a Driven Chain of Anharmonic Oscill · Zbl 1017.82028
[37] Rey-Bellet, L., Thomas, L.E.: Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mech · Zbl 0989.82023
[38] Rey-Bellet, L., Thomas, L.E.: Fluctuations of the Entropy Production in Anharmonic Cha · Zbl 1174.82314
[39] Riesz, F., Sz.-Nagy, B.: Functional analysis. New York: Dover Publications Inc., 1990 · Zbl 0732.47001
[40] Risken, H.: The Fokker-Planck equation. Springer-Verlag, Berlin, second edition, 1989. Methods of solution and applications · Zbl 0665.60084
[41] Robert, D.: Autour de l’approximation semi-classique. Volume 68 of Progress in Mathematics. Birkhaüser, 1987 · Zbl 0621.35001
[42] Rothschild, L.P., Stein, E.M.: Hypoelliptic differential operators and nilp · Zbl 0346.35030
[43] Sternberg, S.: Lectures on differential geometry. Chelsea Publishing Co., New York, second edition, 1983 · Zbl 0518.53001
[44] Talay, D.: Approximation of invariant measures of nonlinear Hamiltonian and dissipative stochastic differential equations. In: C. Soize R. Bouc, (eds.), Progress in Stochastic Structural Dynamics, volume 152 of Publication du L.M.A.-C.N.R.S., 1999, pp. 139–169
[45] Talay, D.: Stochastic Hamiltonian systems: exponential convergence to the invariant measure and discretization by the implicit Euler scheme. Markov Proc · Zbl 1011.60039
[46] Villani, C.: A review of mathematical topics in collisional kinetic theory. Handbook of Fluid Mechanics, Vol. 1, S. Friedlander and D. Serre, (eds.). North-Holland, 2002 · Zbl 1170.82369
[47] Witten, E.: Supersymmetry and Mors · Zbl 0499.53056
[48] Zworski, M.: Numerical linear algebra and solvability of partial differential equa · Zbl 1021.35077
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