Martingale product estimators for sensitivity analysis in computational statistical physics

We introduce a new class of estimators for the linear response of steady states of stochastic dynamics. We generalize the likelihood ratio approach and formulate the linear response as a product of two martingales, hence the name ‘martingale product estimators’. We present a systematic derivation of the martingale product estimator, and show how to construct such estimator so that its bias is consistent with the weak order of the numerical scheme that approximates the underlying stochastic differential equation. Motivated by the estimation of transport properties in molecular systems, we present a rigorous numerical analysis of the bias and variance for these new estimators in the case of Langevin dynamics. We prove that the variance is uniformly bounded in time and derive a specific form of the estimator for second-order splitting schemes for Langevin dynamics. For comparison, we also study the bias and variance of a Green–Kubo (GK) estimator, motivated, in part, by its variance growing linearly in time. We compare on illustrative numerical tests the new estimators with results obtained by the GK method.