The simulation of SDEs with a known equilibrium distribution is a common problem in computational statistical mechanics with applications e.g. to molecular dynamics simulations. 

The problem is made nontrivial by the requirements that the time integrator be not only finite-time accurate but also ergodic with respect to the target distribution - this second requirement rules out e.g. most standard explicit integration schemes that are typically unstable on an infinite time window (i.e. non-ergodic). 
It has been realized recently, however, that by incorporating a Metropolis-Hasting Monte-Carlo step in such an explicit integration scheme one can design time integrators that meet both requirements (for an expository introduction see: Bou-Rabee [2014]. Time Integrators for Molecular Dynamics, Entropy, vol.~16, pg.~138-162). 

Since these integrators accurately capture the dynamics on any piece of this trajectory of finite duration, they can be used to estimate not only equilibrium expectations of an observable, but also multi-time expectations such as equilibrium auto-correlation functions. 
With the additional structure of geometric ergodicity, a simple bootstrapping argument shows that such finite-time statements hold on infinite-time windows, thereby permitting to estimate transport coefficients for molecular systems. 
This article uses a different technique to sharpen this error -- from O(h^{3/4}) to O(h) -- based on a direct analysis of the Markov operator of the time integrator. 

The authors derive these estimates in the special case of a Metropolis integrator applied to a self-adjoint diffusion, however, the idea is more general.



Nawaf Bou-Rabee (Rutgers University)