Persistence exponents of non-Gaussian processes in statistical mechanics

TL;DR

This paper studies the persistence properties of non-Gaussian stochastic processes with jumps, providing exact calculations and an expansion method for small curvature, relevant for statistical physics models including Langevin dynamics.

Contribution

It introduces an exact calculation of persistence exponents for linear decay processes and develops a novel expansion method for small curvature corrections.

Findings

Exact persistence exponent for linear decay processes.

Expansion method for small curvature corrections.

Improved approximation for large level exponential decay.

Abstract

Motivated by certain problems of statistical physics we consider a stationary stochastic process in which deterministic evolution is interrupted at random times by upward jumps of a fixed size. If the evolution consists of linear decay, the sample functions are of the "random sawtooth" type and the level dependent persistence exponent \theta can be calculated exactly. We then develop an expansion method valid for small curvature of the deterministic curve. The curvature parameter g plays the role of the coupling constant of an interacting particle system. The leading order curvature correction to \theta is proportional to g^{2/3}. The expansion applies in particular to exponential decay in the limit of large level, where the curvature correction considerably improves the linear approximation. The Langevin equation, with Gaussian white noise, is recovered as a singular limiting case.