Microcanonical conditioning of Markov processes on time-additive observables

The recent study by B. De Bruyne, S. N. Majumdar, H. Orland and G. Schehr [arXiv:2110.07573], concerning the conditioning of Brownian motion random walks on global dynamical constraints over a finite time-window $T$, is reformulated as general framework for 'microcanonical conditioning' Markov processes time-additive observables. This formalism applied to various types processes, namely discrete-time chains, continuous-time jump diffusion in arbitrary dimension. In each setting, observable also fully general, i.e. it can involve both time spent configuration elementary increments process. cases are illustrated via simple explicit examples. Finally, we describe link with 'canonical based generating function while regime large $T$ allows recover standard deviation analysis observables deformed operator approach.