THE ROLE OF UNBOUNDED TIME-SCALES IN GENERATING LONG-RANGE MEMORY IN ADDITIVE MARKOVIAN PROCESSES

Abstract

Any additive stationary and continuos Markovian process described by a Fokker–Planck equation, can also be described in terms of a Schr ̈odinger equation with an appropriate quantum potential. By using such analogy, it has been proved that a power-law correlated stationary Markovian process can stem from a quantum potential that (i) shows an $x^{−2}$ decay for large x values (ii) and whose eigenvalue spectrum admits a null eigenvalue and a continuum part of positive eigenvalues attached to it. In this paper we show that such two features are both necessary. Specifically, we show that considering a potential with tails decaying like $x^{−μ}$ give rise to a stationary Markovian process which is not power-law autocorrelated, despite the fact that the process has an unbounded set of time scales. Moreover, we present an exactly solvable example where the potential decays as $x^{−2}$ but there is a gap between the continuum spectrum of eigenvalues and the null eigenvalue. We show that the process is not power law autocorrelated, but by decreasing the gap one can arbitrarily well approximate it. A crucial role in obtaining a power-law autocorrelated process is played by the weights $C_λ^2$ by which each time-scales contribute to the autocorrelation function. In fact, we will see that such weights must behave like a power-law for small energy values λ. This is only possible if the potential V_S(x) shows an $x^{−2}$ decay to zero for large x values.