Colloquium

In many applications one is interested in estimating the response of the steady-state distribution of a stochastic dynamical system under a perturbation to the dynamics. For example, such computations form a basis for using the linear response theory of statistical mechanics in estimating transport coefficients,  e.g., the mobility, the shear viscosity or the thermal conductivity,  that relate the average response of the system at its steady state to an external forcing applied to the system.  We briefly review a background of linear response theory and then we discuss schemes based on Girsanov's change-of-measure theory for  computing the sensitivity or linear response of steady-state averages of stochastic dynamics.   We discuss application of this approach to dynamics described by continuous time Markov chains as well as time homogenous Ito diffusions. We also present new schemes for estimating linear response from equilibrium fluctuations for invariant measures of Langevin dynamics. The schemes  apply reweighting of trajectories by factors derived from a linearization of the Girsanov weights. We explain numerical analysis of such schemes by presenting 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 simulations needed for steady-state sampling.

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