Measuring the Irreversibility of Numerical Schemes for Reversible Stochastic Differential Equations^∗

Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, USA.
luc@math.umass.edu

Received: 23 July 2012
Revised: 1 September 2013

Abstract

For a stationary Markov process the detailed balance condition is equivalent to the time-reversibility of the process. For stochastic differential equations (SDE’s), the time discretization of numerical schemes usually destroys the time-reversibility property. Despite an extensive literature on the numerical analysis for SDE’s, their stability properties, strong and/or weak error estimates, large deviations and infinite-time estimates, no quantitative results are known on the lack of reversibility of discrete-time approximation processes. In this paper we provide such quantitative estimates by using the concept of entropy production rate, inspired by ideas from non-equilibrium statistical mechanics. The entropy production rate for a stochastic process is defined as the relative entropy (per unit time) of the path measure of the process with respect to the path measure of the time-reversed process. By construction the entropy production rate is nonnegative and it vanishes if and only if the process is reversible. Crucially, from a numerical point of view, the entropy production rate is an a posteriori quantity, hence it can be computed in the course of a simulation as the ergodic average of a certain functional of the process (the so-called Gallavotti−Cohen (GC) action functional). We compute the entropy production for various numerical schemes such as explicit Euler−Maruyama and explicit Milstein’s for reversible SDEs with additive or multiplicative noise. In addition we analyze the entropy production for the BBK integrator for the Langevin equation. The order (in the time-discretization step Δt) of the entropy production rate provides a tool to classify numerical schemes in terms of their (discretization-induced) irreversibility. Our results show that the type of the noise critically affects the behavior of the entropy production rate. As a striking example of our results we show that the Euler scheme for multiplicative noise is not an adequate scheme from a reversibility point of view since its entropy production rate does not decrease with Δt.