Intrinsic timing in classical master equation dynamics from an extended quadratic format of the evolution law

The paradigm of Markov jump process is widely employed in disparate contexts, like for instance in ecology, epidemiology and chemistry, any time the state-space is discrete and a tracked entity (the system) stochastically jumps from site to site. The classical master equation describes the system's evolution in terms of site occupation probabilities starting from a given initial condition associated with the initial knowledge about the system. For the master equation of Markov processes admitting stationary occupation probabilities, it is here derived an equivalent quadratic format in an extended space of mutually interrelated variables having physical dimension of inverse of time. The evolution in the probability space is thus mirrored by the evolution in the extended space. This universal format potentially allows one to unveil general traits underlying the master equation dynamics. Here we specifically consider the emergence of an intrinsic rate which, behaving as a state function in the probability space, introduces a timing during the relaxation process. This specific feature has to be taken an empirical discovery which derives from the analysis of numerical calculations; a possible direction towards a formal proof is however proposed. The conjecture made here is that such intrinsic timing is a typical trait (i.e., normally present) of the Markov jump processes.