Role of conditional probability in multi-scale stationary Markovian processes

The aim of the paper is to understand how the inclusion of more and more time scales into a stochastic stationary Markovian process affects its conditional probability. To this end, we consider two Gaussian processes: (i) a short-range correlated process with an infinite set of time scales bounded from above and (ii) a power-law correlated process with an infinite and unbounded set of time scales. For these processes we investigate the equal position conditional probability P(x,t∣x,0) and the mean first passage time Tx(Λ). The function P(x,t∣x,0) can be considered as a proxy of the persistence, i.e., the fact that when a process reaches a position x then it spends some time around that position value. The mean first passage time can be considered as a proxy of how fast is the process in reaching a position at distance Λ starting from position x. In the first investigation we show that the more time scales the process includes, the larger is the persistence. Specifically, we show that the power-law correlated process shows a slow power-law decay of P(x,t∣x,0) to the stationary probability density function. By contrast, the short-range correlated process shows a decay dominated by an exponential cutoff. Moreover, we also show that the existence of an infinite and unbounded set of time scales is a necessary but not sufficient condition for observing a slow power-law decay of P(x,t∣x,0). In fact, in the context of stationary Markovian processes such a form of persistence seems to be associated with the existence of an algebraic decay of the autocorrelation function. In the second investigation, we show that for large values of Λ the more time scales the process includes, the larger is the mean first passage time, i.e., the slower is the process. On the other hand, for small values of Λ, the more time scales the process includes, the smaller is the mean first passage time, i.e., when a process statistically spends more time in a given position the likelihood that it reached nearby positions by chance may also be enhanced.