Sample Paths Estimates for Stochastic Fast-Slow Systems Driven by Fractional Brownian Motion

Abstract

We analyze the effect of additive fractional noise with Hurst parameter H > 1 2 on fastslow systems. Our strategy is based on sample paths estimates, similar to the approach by Berglund and Gentz in the Brownian motion case. Yet, the setting of fractional Brownian motion does not allow us to use the martingale methods from fast-slow systems with Brownian motion. We thoroughly investigate the case where the deterministic system permits a uniformly hyperbolic stable slow manifold. In this setting, we provide a neighborhood, tailored to the fast-slow structure of the system, that contains the process with high probability. We prove this assertion by providing exponential error estimates on the probability that the system leaves this neighborhood. We also illustrate our results in an example arising in climate modeling, where time-correlated noise processes have become of greater relevance recently.

Warning signs for non-Markovian bifurcations: colour blindness and scaling laws

“…with Hurst index H ∈ (1/2, 1), where c(H) is a H-dependent positive constant, which ensures that E(B H 1 ) 2 = 1. The previous computation holds for H := α + 1/2 (see also [51]). Another example is the Rosenblatt process (see [53]), which is a non-Gaussian processes with stationary increments and the same covariance as fractional Brownian motion:…”

“…, for an α-dependent positive constant c α . Next, we compute a linearized fast-slow process for the variance; see also [51] for a similar computation in the case of a fractional Brownian motion. In this case, regarding the scaling properties of the noise, we infer that the linearization ξ along an attracting critical manifold satisfies the equation…”

Introduction to the Special Issue on the Statistical Mechanics of Climate

“…A model reduction technique based on a stochastic variational approach for geophysical fluid dynamics is used for developing a new ensemble-based data assimilation methodology for highdimensional fluid dynamics models in [25]. Eichinger et al [29], instead, address the impact of adding noise in the form of additive fractional Brownian motion on fast-slow systems and investigate the problem of estimating how likely is for the trajectories to stay nearby the slow manifold of the deterministic system. • A new paradigm for climate science is suggested in three closely linked contributions.…”

Amplitude equations for SPDEs driven by fractional additive noise with small Hurst parameter

“…Regarding qualitative properties of slow-fast systems with fractional noise we mention: homogenization results [21], sample path estimates for additive fractional noise with H > 1/2 [18] and averaging principles [22]. Here the slow variable is perturbed by multiplicative fractional noise, with Hurst index H > 1/2 and the fast variable by a Brownian motion.…”