The Small-Mass Limit and White-Noise Limit of an Infinite Dimensional Generalized Langevin Equation

2018/04/30 by Hung D. Nguyen
Mathematics · Physics and Astronomy · #Class (philosophy) #Convergence (economics) #Langevin dynamics #Langevin equation #Limit (mathematics) #Lipschitz continuity #Markov Chains and Monte Carlo Methods #Markov process #Space (punctuation) #Stochastic processes and statistical mechanics #math.PR #stochastic dynamics and bifurcation

paper · pdf · doi:10.1007/s10955-018-2139-1

published as Journal of Statistical Physics 173 (2), 411-437, 2018

openalex created_date 2018/05/07 · openalex publication_date 2018/08/28 · arxiv created 2021/03/08 · arxiv updated 2021/03/10 · openalex updated_date 2026/08/06

Abstract

We study asymptotic properties of the Generalized Langevin Equation (GLE) in the presence of a wide class of external potential wells with a power-law decay memory kernel. When the memory can be expressed as a sum of exponentials, a class of Markovian systems in infinite-dimensional spaces is used to represent the GLE. The solutions are shown to converge in probability in the small-mass limit and the white-noise limit to appropriate systems under minimal assumptions, of which no Lipschitz condition is required on the potentials. With further assumptions about space regularity and potentials, we obtain L¹ convergence in the white-noise limit.

Citations

• Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins
• The Smoluchowski-Kramers Limit of Stochastic Differential Equations with Arbitrary State-Dependent Friction
• Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations
• Noise-Induced Drift in Stochastic Differential Equations with Arbitrary Friction and Diffusion in the Smoluchowski-Kramers Limit