In this paper we study the Fokker-Planck operator with potential , and analyze some kind of conditions imposed on the potential to ensure the validity of global hypoelliptic estimates (see Theorem 1.1). As a consequence, we obtain the compactness of resolvent of the Fokker-Planck operator if either the Witten Laplacian on 0-forms has a compact resolvent or some additional assumption on the behavior of the potential at infinity is fulfilled. This work improves the previous results of Hérau-Nier [5] and Helffer-Nier [3], by obtaining a better global hypoelliptic estimate under weaker assumptions on the potential.
@article{ASNSP_2012_5_11_4_789_0, author = {Li, Wei-Xi}, title = {Global hypoellipticity and compactness of resolvent for {Fokker-Planck} operator}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {789--815}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 11}, number = {4}, year = {2012}, mrnumber = {3060700}, zbl = {1261.35046}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2012_5_11_4_789_0/} }
TY - JOUR AU - Li, Wei-Xi TI - Global hypoellipticity and compactness of resolvent for Fokker-Planck operator JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2012 SP - 789 EP - 815 VL - 11 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2012_5_11_4_789_0/ LA - en ID - ASNSP_2012_5_11_4_789_0 ER -
%0 Journal Article %A Li, Wei-Xi %T Global hypoellipticity and compactness of resolvent for Fokker-Planck operator %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2012 %P 789-815 %V 11 %N 4 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2012_5_11_4_789_0/ %G en %F ASNSP_2012_5_11_4_789_0
Li, Wei-Xi. Global hypoellipticity and compactness of resolvent for Fokker-Planck operator. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 4, pp. 789-815. http://archive.numdam.org/item/ASNSP_2012_5_11_4_789_0/
[1] F. Bouchut, Hypoelliptic regularity in kinetic equations, J. Math. Pure Appl. 81 (2002), 1135–1159. | MR | Zbl
[2] L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation, Comm. Pure Appl. Math. 54 (2001), 1–42. | MR | Zbl
[3] B. Helffer and F. Nier, “Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians”, Lecture Notes in Mathematics, vol. 1862, Springer-Verlag, Berlin, 2005. | MR | Zbl
[4] B. Helffer and J. Nourrigat, “Hypoellipticité Maximale pour des Opérateurs Polynômes de Champs de Vecteurs”, Progress in Mathematics, vol. 58, Birkhäuser Boston Inc., 1985. | MR | Zbl
[5] F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2004), 151–218. | MR | Zbl
[6] L. Hörmander, “The Analysis of Linear Partial Differential Operators. III”, Springer-Verlag, Berlin, 1985. | MR | Zbl
[7] J. J. Kohn, Lectures on degenerate elliptic problems, In: “Pseudodifferential Operator with Applications (Bressanone, 1977)”, Liguori, Naples, 1978, 89–151. | MR | Zbl
[8] L. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247–320. | MR | Zbl