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Borrowing ideas from open quantum systems, we describe a formalism to encodeensembles of trajectories of classical stochastic dynamics in terms ofcontinuous matrix product states (cMPSs). We show how to define in thisapproach "biased" or "conditioned" ensembles where the probability oftrajectories is biased from that of the natural dynamics by some condition ontrajectory observables. In particular, we show that the generalised Doobtransform which maps a conditioned process to an equivalent "auxiliary" or"driven" process (one where the same conditioned set of trajectories isgenerated by a proper stochastic dynamics) is just a gauge transformation ofthe corresponding cMPS. We also discuss how within this framework one caneasily prove properties of the dynamics such as trajectory ensemble equivalenceand fluctuation theorems.

journal of statistical mechanics

Classical stochastic dynamics and continuous matrix product states:  gauge transformations, conditioned and driven processes, and equivalence of  trajectory ensembles