The Langevin Equation

  • Grigorios A. Pavliotis
Chapter
First Online:
Part of the Texts in Applied Mathematics book series (TAM, volume 60)

Abstract

In this chapter, we study the Langevin equation and the associated Fokker –Planck equation. In Sect. 6.1, we introduce the equation and study some of the main properties of the corresponding Fokker–Planck equation. In Sect. 6.2 we give an elementary introduction to the theories of hypoellipticity and hypocoercivity. In Sect. 6.3, we calculate the spectrum of the generator and Fokker–Planck operators for the Langevin equation in a harmonic potential. In Sect. 6.4, we study Hermite polynomial expansions of solutions to the Fokker–Planck equation. In Sect. 6.5, we study the overdamped and underdamped limits for the Langevin equation. In Sect. 6.6, we study the problem of Brownian motion in a periodic potential. Bibliographical remarks and exercises can be found in Sects. 6.7 and 6.8, respectively.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Grigorios A. Pavliotis
    • 1
  1. 1.Department of MathematicsImperial College LondonLondonUK

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