Ergodic properties of quasi-Markovian generalized Langevin equations with configuration dependent noise and non-conservative force

We discuss the ergodic properties of quasi-Markovian stochastic differential equations, providing general conditions that ensure existence and uniqueness of a smooth invariant distribution and exponential convergence of the evolution operator in suitably weighted $L^{\infty}$ spaces, which implies the validity of central limit theorem for the respective solution processes. The main new result is an ergodicity condition for the generalized Langevin equation with configuration-dependent noise and (non-)conservative force

Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems

We study a two-time-scale system of jump-diffusion stochastic differential equations. The main goal is to study the convergence rate of the slow components to the effective dynamics. The convergence established here is in the strong sense, i. e., uniformly in time. For the ergodicity assumptions, we use the existence of a Lyapunov function to control the return times. This assumption is weaker than the one- sided Lipschitz condition, frequently used for deriving rates

Extracting macroscopic dynamics: model problems and algorithms

In many applications, the primary objective of numerical simulation of time-evolving systems is the prediction of macroscopic, or coarse-grained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic approaches have been introduced to extract effective, lower-dimensional, models for the macroscopic dynamics; the starting point is the full, detailed, evolution equations. In many cases the effective low-dimensional dynamics may be stochastic, even when the original starting point is deterministic. This review surveys a number of these new approaches to the problem of extracting effective dynamics, highlighting similarities and differences between them. The importance of model problems for the evaluation of these new approaches is stressed, and a number of model problems are described. When the macroscopic dynamics is stochastic, these model problems are either obtained through a clear separation of time-scales, leading to a stochastic effect of the fast dynamics on the slow dynamics, or by considering high dimensional ordinary differential equations which, when projected onto a low dimensional subspace, exhibit stochastic behaviour through the presence of a broad frequency spectrum. Models whose stochastic microscopic behaviour leads to deterministic macroscopic dynamics are also introduced. The algorithms we overview include SVD-based methods for nonlinear problems, model reduction for linear control systems, optimal prediction techniques, asymptotics-based mode elimination, coarse timestepping methods and transfer-operator based methodologies