Path-reversal, Doi-Peliti generating functionals, and dualities between dynamics and inference for stochastic processes

Abstract

Fluctuation theorems may be partitioned into those that apply the probability measure under the original stochastic process to reversed paths, and those that construct a new, adjoint measure by similarity transform, which locally reverses probability currents. Results that use the original measure have a natural interpretation in terms of time-reversal of the dynamics. Here we develop a general interpretation of fluctuation theorems based on the adjoint process by considering the duality of the Kolmogorov-forward and backward equations, acting on distributions versus observables. The backward propagation of the dependency of observables is related to problems of statistical inference, so we characterize the adjoint construction as a duality between dynamics and inference. The adjoint process corresponds to the Kolmogorov backward equation in a generating functional that erases memory from the dynamics of its underlying distribution. We show how erasure affects general correlation functions by showing that duality under the adjoint fluctuation theorems exchanges the roles of advanced and retarded Green's functions. We derive results for the class of discrete-state stochastic processes corresponding to Chemical Reaction Networks (CRNs), and show that dualization acts on the \emph{finite} representation of the generating event-set, in a manner similar to the usual similarity transform acting on the (potentially infinite) set of state transitions. We construct generating functionals within the Doi-Peliti (DP) functional integral framework, within which duality transformation takes a remarkably simple form as a change of integration variable. Our Green's function analysis recovers the Extended Fluctuation-Dissipation Theorem of Seifert and Speck for non-equilibrium steady states, shows that the causal structure responsible for it applies also to dualization about non-steady states.