Ornstein-Uhlenbeck Process Driven by Multiple Dichotomous Noises
Abstract
We study a generalized Ornstein-Uhlenbeck process driven by a superposition of K independent dichotomous noises with arbitrary fixed amplitudes and switching rates. Unlike the classical Ornstein-Uhlenbeck process driven by equilibrium Gaussian white noise, the present system is governed by bounded nonequilibrium fluctuations with finite correlation times. We obtain exact expressions for the stationary position distribution and all cumulants, and show that the stationary state possesses an unexpectedly rich structure, including compact support, algebraic branch-point singularities, edge divergences, and multiple extrema. We establish a mapping onto a heterogeneous random-flight process with bounded jumps, yielding a transparent probabilistic interpretation of the stationary measure. We further analyze several limiting regimes, including the crossover to Gaussian statistics for large numbers of noise sources. For ensembles with exponentially-distributed quenched amplitudes, we derive exact disorder-averaged stationary distributions and show that disorder fundamentally alters the stationary state, producing exponential tails decorated by algebraic prefactors with non-trivial exponents.
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BibTeX
@misc{kalaj2026ornsteinuhlenbeck,
title = {{Ornstein-Uhlenbeck Process Driven by Multiple Dichotomous Noises}},
author = {Silvio Kalaj and Dongho Lee and Enzo Marinari and Jae-Hyung Jeon and Pascal Viot and Gleb Oshanin},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29226},
eprint = {2608.29226},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29226},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Ornstein-Uhlenbeck Process Driven by Multiple Dichotomous Noises
AU - Silvio Kalaj
AU - Dongho Lee
AU - Enzo Marinari
AU - Jae-Hyung Jeon
AU - Pascal Viot
AU - Gleb Oshanin
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29226
UR - https://arxiv.org/abs/2608.29226
AB - We study a generalized Ornstein-Uhlenbeck process driven by a superposition of K independent dichotomous noises with arbitrary fixed amplitudes and switching rates. Unlike the classical Ornstein-Uhlenbeck process driven by equilibrium Gaussian white noise, the present system is governed by bounded nonequilibrium fluctuations with finite correlation times. We obtain exact expressions for the stationary position distribution and all cumulants, and show that the stationary state possesses an unexpectedly rich structure, including compact support, algebraic branch-point singularities, edge divergences, and multiple extrema. We establish a mapping onto a heterogeneous random-flight process with bounded jumps, yielding a transparent probabilistic interpretation of the stationary measure. We further analyze several limiting regimes, including the crossover to Gaussian statistics for large numbers of noise sources. For ensembles with exponentially-distributed quenched amplitudes, we derive exact disorder-averaged stationary distributions and show that disorder fundamentally alters the stationary state, producing exponential tails decorated by algebraic prefactors with non-trivial exponents.
ER -