Typical dynamics and fluctuation analysis of slow–fast systems driven by fractional Brownian motion

Abstract

This paper studies typical dynamics and fluctuations for a slow–fast dynamical system perturbed by a small fractional Brownian noise. Based on an ergodic theorem with explicit rates of convergence, which may be of independent interest, we characterize the asymptotic dynamics of the slow component to two orders (i.e. the typical dynamics and the fluctuations). The limiting distribution of the fluctuations turns out to depend upon the manner in which the small-noise parameter is taken to zero relative to the scale-separation parameter. We study also an extension of the original model in which the relationship between the two small parameters leads to a qualitative difference in limiting behavior. The results of this paper provide an approximation, to two orders, to dynamical systems perturbed by small fractional Brownian noise and incorporating multiscale effects.

Functional limit theorems for the fractional Ornstein-Uhlenbeck process

“…For the above we follow [CNN20] and use Malliavin calculus to obtain suitable moment bounds on t 0 G(y ε s )ds. These results appeared in the previous version of the current paper [GL19], for the diffusion limit case these were presented recently also in [BGS19]. Equations driven by fractional Brownian motions are also studied for the averaging regime, see [LH19] and in [FK00].…”

Discrete-time inference for slow-fast systems driven by fractional Brownian motion

“…In this section we introduce notation, present the conditions that we will assume throughout the paper, and recall necessary results of [7]. We shall suppress the dependence on θ for expository purposes.…”

“…Whereas the standard Bm case has been studied extensively, the mathematical theory of multiscale models with H = 1/2 is considerably less developed. In our recent work [7], we obtained results on averaging, homogenization, and fluctuations corrections for models like (1) considered in the limit as ε := (ǫ, η) → 0; see also [23,45] for related averaging results.…”

Functional Limit Theorems for the Fractional Ornstein–Uhlenbeck Process

“…Equations driven by fractional Brownian motions are also studied for the averaging regime, see [21] and [16]. A fluctuation theorem around the effective average was obtained [3].…”