Correlation and Relaxation Times for a Stochastic Process with a Fat-Tailed Steady-State Distribution

TL;DR

This paper analyzes a stochastic process with a fat-tailed steady-state distribution, deriving its correlation and relaxation times, and revealing how stochasticity influences cumulant divergence and relaxation dynamics.

Contribution

It provides an analytical characterization of the correlation function, relaxation times, and distribution of relaxation times for a process with an Inverse Gamma steady-state.

Findings

Correlation function determined by inverse interaction strength

Relaxation times diverge with increasing stochasticity

Distribution of relaxation times is Inverse Gaussian

Abstract

We study a stochastic process defined by the interaction strength for the return to the mean and a stochastic term proportional to the magnitude of the variable. Its steady-state distribution is the Inverse Gamma distribution, whose power-law tail exponent is determined by the ratio of the interaction strength to stochasticity. Its time-dependence is characterized by a set of discrete times describing relaxation of respective cumulants to their steady-state values. We show that as the progressively lower cumulants diverge with the increase of stochasticity, so do their relaxation times. We analytically evaluate the correlation function and show that it is determined by the longest of these times, namely the inverse interaction strength, which is also the relaxation time of the mean. We also investigate relaxation of the entire distribution to the steady state and the distribution of relaxations times, which we argue to be Inverse Gaussian.