Spectral Properties of Effective Dynamics from Conditional Expectations

Abstract

The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting.

Non-parametric Estimation of Stochastic Differential Equations from Stationary Time-Series

“…Finally, the non-parametric estimation framework discussed here can also be combined with regression techniques to perform parametric fitting of a nonlinear function to obtain the functional form of the drift and diffusion coefficients (e.g. [17]). In addition, it is also possible to utilize LASSO-type techniques to obtain the optimal functional form of the drift and diffusion.…”

“…We can compute the bias of the drift estimator in (17) in a manner totally similar to the computations of the bias for Â(x k ) in Sect. 4.1, i.e.…”

“…. Thus, there is an additional condition x 2 / t → 0 for the estimator in (17) to be asymptotically unbiased. In addition, the term r (x k ) x 2 / t can provide a significant contribution to the bias of the estimator (17) for finite x and t. Similar issue arises if we consider modified estimator for the diffusion coefficient.…”

“…In addition, the term r (x k ) x 2 / t can provide a significant contribution to the bias of the estimator (17) for finite x and t. Similar issue arises if we consider modified estimator for the diffusion coefficient. Thus, estimator (17) is inferior compared to the estimator (5) and we will consider (5) for the rest of this paper.…”

“…[3,8,12,15,16,23]), and biology (e.g. [1,17,25]). This is a very active area of research with many publications including results on parametric and non-parametric estimation of autoregressive processes and stochastic differential equations.…”