Residence time distribution for a class of Gaussian Markov processes

1999/02/07 by Abhishek Dhar, Satya N. Majumdar
Mathematics · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech

paper · pdf · doi:10.1103/physreve.59.6413

7 pages, RevTex, 1 figure, Journal references added

arxiv created 1999/02/07 · openalex publication_date 1999/06/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the distribution of residence time or equivalently that of "mean magnetization" for a family of Gaussian Markov processes indexed by a positive parameter alpha. The persistence exponent for these processes is simply given by theta=alpha but the residence time distribution is nontrivial. The shape of this distribution undergoes a qualitative change as theta increases, indicating a sharp change in the ergodic properties of the process. We develop two alternate methods to calculate exactly but recursively the moments of the distribution for arbitrary alpha. For some special values of alpha, we obtain closed form expressions of the distribution function.

Citations

• Non-trivial exponents in the zero temperature dynamics of the 1D Ising and Potts models
• Persistent Spins in the Linear Diffusion Approximation of Phase Ordering and Zeros of Stationary Gaussian Processes
• Non-Markovian persistence and nonequilibrium critical dynamics
• The One-Sided Barrier Problem for Gaussian Noise
• Mathematical Analysis of Random Noise