Journal of Statistical Physics

, Volume 169, Issue 2, pp 316–339 | Cite as

Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem

  • Lei LiEmail author
  • Jian-Guo Liu
  • Jianfeng Lu


We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the ‘fluctuation-dissipation theorem’, the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the ‘fluctuation-dissipation theorem’ is satisfied, and this verifies that satisfying ‘fluctuation-dissipation theorem’ indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.


Fractional SDE Fluctuation-dissipation-theorem Caputo derivative Fractional Brownian motion Generalized Langevin equation Subdiffusion 



The work of J.-G Liu is partially supported by KI-Net NSF RNMS11-07444 and NSF DMS-1514826. The work of J. Lu is supported in part by National Science Foundation under Grant DMS-1454939. J. Lu would also like to thank Eric Vanden-Eijnden for helpful discussions.


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.Departments of Physics and MathematicsDuke UniversityDurhamUSA
  3. 3.Department of Mathematics, Department of Physics, and Department of ChemistryDuke UniversityDurhamUSA

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