Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem

Abstract

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.

Appendix: Proof of Theorem 1

Proof

We just consider a sample point $x_0$ and a sample path G with G being continuous. We then construct a path that satisfies the integral equation given this sample initial data.

By Proposition 1, G(t) is continuous. Consider the sequence given by

$$\begin{aligned} x^{(0)}=x_0, \end{aligned}$$

and $x^{(n)}, n\ge 1$ is given by

$$\begin{aligned} x^{(n)}(t)=x_0-\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1} V'(x^{(n-1)}(s))\,ds+G(t). \end{aligned}$$

Assume L is a Lipschitz constant for $V'(\cdot )$. Introducing $g_{\gamma }=\frac{\theta (t)}{\varGamma (\gamma )}t^{\gamma -1}$, we find that $\{g_{\gamma }\}_{\gamma >0}$ forms a convolution semigroup (Lemma 3). We define

$$\begin{aligned} e^n=x^{(n)}-x^{(n-1)}. \end{aligned}$$

Explicit formula tells us that

$$\begin{aligned} e^1=-V'(x_0)g_{\alpha +1}+G(t), \end{aligned}$$

and that

$$\begin{aligned} |e^n|=|-g_{\alpha }*(V'(x^{n-1})-V'(x^{n-2}))|\le L g_{\alpha }*|e^{n-1}|,\ \ n\ge 2. \end{aligned}$$

Hence,

$$\begin{aligned} |e^n|\le L^{n-1} g_{(n-1)\alpha }*|e^1|. \end{aligned}$$

Direct computation shows that $\sup _{0\le t\le T}g_{(n-1)\alpha }*|e^1|$ decays exponentially in n. Hence, $\sum _n |e^n|$ converges. It follows that $\sum _n e^n$ converges uniformly on any interval [0, T] with $T\in (0,\infty )$. The limit is also a continuous function. It turns out that the limit satisfies the integral equation.

For the uniqueness, assume that both x(t) and y(t) are solutions. Then, we take a sample where both x(t) and y(t) are continuous. For this sample, $\forall t>0$,

$$\begin{aligned} |x(t)-y(t)|=\frac{1}{\varGamma (\alpha )}\left| \int _0^t(t-s)^{\alpha -1} (V'(x(s))-V'(y(s)))\right| ds\le L (g_{\alpha }*|x-y|)(t). \end{aligned}$$

Applying this inequality iteratively and using the semi-group property of $g_{\gamma }$, we find

$$\begin{aligned} |x-y|(t)\le L^n g_{n\alpha }*|x-y|. \end{aligned}$$

Fixing $T>0$, the right hand side goes to zero uniformly on [0, T]. Then, we find that $x=y$ on [0, T] for this sample path. Since both solutions are continuous almost surely, then $x=y$ on [0, T] almost surely. By the arbitrariness of T, $x=y$ almost surely. The uniqueness then is shown. This then completes the proof of the theorem. $\square$