Escape Rates in a Stochastic Environment with Multiple Scales

Abstract

Abstract. We consider a stochastic environment with two time scales and outline a general theory that compares two methods to reduce the dimension of the original system. The first method involves the computation of the underlying deterministic center manifold followed by a "naïve" replacement of the stochastic term. The second method allows one to more accurately describe the stochastic effects and involves the derivation of a normal form coordinate transform that is used to find the stochastic center manifold. The results of both methods are used along with the path integral formalism of large fluctuation theory to predict the escape rate from one basin of attraction to another. The general theory is applied to the example of a surface flow described by a generic, singularly perturbed, damped, nonlinear oscillator with additive, Gaussian noise. We show how both nonlinear reduction methods compare in escape rate scaling. Additionally, the center manifolds are shown to predict high pre-history probability regions of escape. The theoretical results are confirmed using numerical computation of the mean escape time and escape prehistory, and we briefly discuss the extension of the theory to stochastic control.

Large rare fluctuations in systems with delayed dissipation

“…To first order in the delay and correlation perturbation, the correction to R(q,p) can be calculated using the zeroth-order trajectories (32). In the integrals over t in Eq.…”

“…Using the approach [24] and its extensions, the narrow peak in the distribution of the trajectories has indeed been seen in simulations and in the experiments (see Refs. [25][26][27][28][29][30][31][32]). …”

Approximation of Stochastic Invariant Manifolds

“…In [83,116,136], pursuing the works of [2,3], reduced stochastic equations involving also extrinsic memory terms have been derived mainly in the context of the stochastic slow manifold; see also [17]. By seeking for a random change of variables, which typically involves repeated stochastic convolutions, reduced equations (different from those derived in [37,Chap.…”

“…5]) are obtained to model the dynamics of the slow variables. These reduced equations are also non-Markovian but require a special care in their derivation to push the anticipative terms (arising in such an approach) to higher order albeit not eliminating them [83,136].…”

Accurate noise projection for reduced stochastic epidemic models

“…25 This technique has been modified and applied to the large fluctuations of multiscale problems. 17 Many publications [19][20][21][22] discuss the simplification of a stochastic dynamical system using a stochastic normal form transformation. In some of this work, 19,22 anticipative noise processes appeared in the normal form transformations, but these integrals of the noise process into the future were not dealt with rigorously.…”