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The Langevin Equation

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Stochastic Processes and Applications

Part of the book series: Texts in Applied Mathematics ((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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Notes

  1. 1.

    Hamilton’s equations of motion are \(\dot{q} = \frac{\partial H} {\partial p} = p,\,\dot{p} = -\frac{\partial H} {\partial p} = -\nabla V (q)\). The Hamiltonian vector field is b(q, p) = (p, −∇V ). The corresponding Liouville operator is given by B. See Sect. 3.4.

  2. 2.

    Let A and B denote the first-order differential operators corresponding to the vector fields A(x) and B(x), i.e., \(A =\sum _{j}A_{j}(x) \frac{\partial } {\partial q_{i}},\;B =\sum _{j}B_{j}(x) \frac{\partial } {\partial q_{j}}\). The commutator between A and B is [A, B] = AB − BA.

  3. 3.

    This section was written in collaboration with M. Ottobre.

  4. 4.

    In other words, the exponentially fast convergence to equilibrium for the reversible Smoluchowski dynamics follows directly from the assumption that e −V satisfies a Poincaré inequality.

  5. 5.

    One can prove that the space \(\mathcal{K}^{\perp }\) is the same irrespective of whether we consider the scalar product \(\langle \cdot,\cdot \rangle\) of \(\mathcal{H}\) or the scalar product \(\langle \cdot,\cdot \rangle _{\mathcal{H}^{1}}\) associated with the norm \(\|\cdot \|_{\mathcal{H}^{1}}\).

  6. 6.

    We assume that ∫ e −V dq = 1.

  7. 7.

    Note that the invariant distribution of (6.95) is independent of \(\varepsilon\).

  8. 8.

    In order to avoid initial layers, we need to assume that the initial condition is a function of the Hamiltonian. We will not study the technical issue of initial layers.

  9. 9.

    \(\langle \ell^{2}\rangle\) will not, in general, be equal to the period of the potential, with the exception of the high-friction regime.

  10. 10.

    This is a familiar trick from the theory of homogenization for partial differential equations with periodic coefficients. For example, to study the PDE

    $$\displaystyle{-\nabla \cdot \left (A\left (x, \frac{x} {\varepsilon } \right )\nabla u^{\varepsilon }\right ) = f\,,}$$

    where the matrix-valued function A is periodic in its second argument, it is convenient to set \(z = \frac{x} {\varepsilon }\) and to treat x and z as independent variables.

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  1. Department of Mathematics, Imperial College London, London, UK

    Grigorios A. Pavliotis

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Pavliotis, G.A. (2014). The Langevin Equation. In: Stochastic Processes and Applications. Texts in Applied Mathematics, vol 60. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1323-7_6

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