  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.
