Effective dynamics using conditional expectations

The question of coarse-graining is ubiquitous in molecular dynamics. In this article, we are interested in deriving effective properties for the dynamics of a coarse-grained variable $\xi(x)$, where $x$ describes the configuration of the system in a high-dimensional space $\R^n$, and $\xi$ is a smooth function with value in $\R$ (typically a reaction coordinate). It is well known that, given a Boltzmann-Gibbs distribution on $x \in \R^n$, the equilibrium properties on $\xi(x)$ are completely determined by the free energy. On the other hand, the question of the effective dynamics on $\xi(x)$ is much more difficult to address. Starting from an overdamped Langevin equation on $x \in \R^n$, we propose an effective dynamics for $\xi(x) \in \R$ using conditional expectations. Using entropy methods, we give sufficient conditions for the time marginals of the effective dynamics to be close to the original ones. We check numerically on some toy examples that these sufficient conditions yield an effective dynamics which accurately reproduces the residence times in the potential energy wells. We also discuss the accuracy of the effective dynamics in a pathwise sense, and the relevance of the free energy to build a coarse-grained dynamics

HITTING TIMES, FUNCTIONAL INEQUALITIES, LYAPUNOV CONDITIONS AND UNIFORM ERGODICITY

International audienceThe use of Lyapunov conditions for proving functional inequalities was initiated in [5]. It was shown in [4, 30] that there is an equivalence between a Poincaré inequality, the existence of some Lyapunov function and the exponential integrability of hitting times. In the present paper, we close the scheme of the interplay between Lyapunov conditions and functional inequalities by • showing that strong functional inequalities are equivalent to Lyapunov type conditions; • showing that these Lyapunov conditions are characterized by the finiteness of generalized exponential moments of hitting times. We also give some complement concerning the link between Lyapunov conditions and in-tegrability property of the invariant probability measure and as such transportation inequalities , and we show that some " unbounded Lyapunov conditions " can lead to uniform ergodicity, and coming down from infinity property

Asymptotic behaviours of stochastic differential delay equations

Most of the existing results on stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in this paper. We shall establish the sufficient condition, in terms of multiple Lyapunov functions, for the asymptotic behaviours of solutions of stochastic differential delay equations. Moreover, from them follow many effective criteria on stochastic asymptotic stability, which enable us to construct the Lyapunov functions much more easily in applications. In particular, the well-known classical theorem on stochastic asymptotic stability is a special case of our more general results. These show clearly the power of our new results. Two examples are also given for illustration

Backward Fokker-Planck equation for determining model valid prediction period

Journal of Geophysical Research, American Geophysical Union, 107, C6, 10.1029/2001JC000879.new concept, valid prediction period (VPP), is presented here to evaluate ocean (or atmospheric) model predictability. VPP is defined as the time period when the prediction error first exceeds a predetermined criterion (i.e., the tolerance level). It depends not only on the instantaneous error growth but also on the noise level, the initial error, and the tolerance level. The model predictability skill is then represented by a single scalar, VPP. The longer the VPP, the higher the model predictability skill is. A theoretical framework on the basis of the backward Fokker-Planck equation is developed to determine the mean and variance of VPP. A one-dimensional stochastic dynamical system [Nicolis, 1992] is taken as an example to illustrate the benefits of using VPP for model evaluation