Convergence of the likelihood ratio method for linear response of non-equilibrium stationary states

TL;DR

This paper develops and analyzes numerical schemes based on Girsanov's change-of-measure for accurately computing the linear response of non-equilibrium steady states in stochastic dynamics, emphasizing bounded variance and second-order accuracy.

Contribution

It introduces a new class of numerical schemes that improve the accuracy and variance properties for linear response calculations in stochastic systems.

Findings

Schemes have bounded variance over long simulation times

Discretization error can be reduced to second order in time step

Methods effectively compute linear response in non-equilibrium states

Abstract

We consider numerical schemes for computing the linear response of steady-state averages of stochastic dynamics with respect to a perturbation of the drift part of the stochastic differential equation. The schemes are based on Girsanov's change-of-measure theory to reweight trajectories with factors derived from a linearization of the Girsanov weights. We investigate both the discretization error and the finite time approximation error. The designed numerical schemes are shown to be of bounded variance with respect to the integration time, which is a desirable feature for long time simulation. We also show how the discretization error can be improved to second order accuracy in the time step by modifying the weight process in an appropriate way.