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We describe a simple method of umbrella trajectory sampling for Markovchains. The method allows the estimation of large-deviation rate functions, forpath-extensive dynamic observables, for an arbitrary number of models within acertain family. The general relationship between probability distributions ofdynamic observables of members of this family is an extended fluctuationrelation. When the dynamic observable is chosen to be entropy production,members of this family include the forward Markov chain and its time reverse,whose probability distributions are related by the expected simple fluctuationrelation.

Multi-point nonequilibrium umbrella sampling and associated fluctuation relations