Small Mass Limit of a Langevin Equation on a Manifold

Abstract

Abstract. We study damped geodesic motion of a particle of mass m on a Riemannian manifold, in the presence of an external force and noise. Lifting the resulting stochastic differential equation to the orthogonal frame bundle, we prove that, as m → 0, its solutions converge to solutions of a limiting equation which includes a noise-induced drift term. A very special case of the main result presents the Brownian motion on the manifold as a limit of inertial systems.Mathematics Subject Classification (2010). 58J65, 60H10, 82C31.

Homogenization for Generalized Langevin Equations with Applications to Anomalous Diffusion

“…One can verify that this is indeed the vanishing effective damping constant and effective diffusion constant case (i.e. 4). Also, for i = 2, 4, the memory kernel, κ 2 (t) and covariance function, R 4 (t), are of the following bi-exponential form:…”

“…• The last main result is Theorem A.6, on homogenization of a family of parametrized SDEs whose coefficients are state-dependent. These SDEs are variants of the ones studied in earlier works [28,4,6]. In comparison with all the earlier studies, the state-dependent coefficients of the prelimit SDEs (A.3)-(A.4) may depend on the parameter > 0 (to be taken to zero) explicitly.…”

“…ξ t ∈ R r is a mean-zero stationary Gaussian vector process, to be defined in detail later. The statistical properties of the process ξ t are completely determined by its (matrix-valued) covariance function, 4) or equivalently, by its spectral density, S(ω), i.e. the Fourier transform of R(t) defined as:…”

“…In this section, we study homogenization for a general class of perturbed SDEs with state-dependent coefficients. Homogenization of differential equations has been extensively studied, from the seminal works of Kurtz [37], Papanicolaou [56] and Khasminksy [33] to the more recent works [58,57,28,27,5,4,9].…”

Homogenisation for anisotropic kinetic random motions

“…In the isotropic case, it is a simple scalar example of the hypoelliptic Laplacian of Bismut; see [Bis15]. The anisotropic case is also treated in [BHVW17]. Here, I : W " R d Ñ R d is simply the identity, and hence does quite fit the hypotheses of Theorem 1.4.…”