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In the field of large deviations for stochastic dynamics, the canonicalconditioning of a given Markov process with respect to a given time-localtrajectory observable over a large time-window has attracted a lot of interestrecently. In the present paper, we analyze the following inverse problem: whentwo Markov generators are given, is it possible to connect them via somecanonical conditioning and to construct the corresponding time-local trajectoryobservable? We focus on continuous-time Markov processes and obtain thefollowing necessary and sufficient conditions: (i) for continuous-time Markovjump processes, the two generators should involve the same possible elementaryjumps in configuration space, i.e. only the values of the corresponding ratescan differ; (ii) for diffusion processes, the two Fokker-Planck generatorsshould involve the same diffusion coefficients, i.e. only the two forces candiffer. In both settings, we then construct explicitly the various time-localtrajectory observables that can be used to connect the two given generators viacanonical conditioning. This general framework is illustrated with variousapplications involving a single particle or many-body spin models. Inparticular, we describe several examples to show how non-equilibrium Markovprocesses with non-vanishing steady currents can be interpreted as thecanonical conditionings of detailed-balance processes with respect to explicittime-local trajectory observables.

journal of statistical mechanics

Inverse problem in the conditioning of Markov processes on trajectory  observables : what canonical conditionings can connect two given Markov  generators ?