A Small Delay and Correlation Time Limit of Stochastic Differential Delay Equations with State-Dependent Colored Noise

Scott Hottovy, Austin McDaniel, Jan Wehr

Research output: Contribution to journalArticle

Abstract

We consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. An Ornstein–Uhlenbeck process is used to model the colored noise. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.

Original languageEnglish (US)
JournalJournal of Statistical Physics
DOIs
StatePublished - Jan 1 2019

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Stochastic Differential Delay Equations
Colored Noise
Maximal Inequality
Electrical Circuits
Stationary Sequences
Delayed Feedback
Convergence of Solutions
Dependent
Stochastic Equations
Time Delay
Random variable
random variables
Differential equation
Zero
differential equations
time lag
theorems
Theorem
Experiment
Model

Keywords

  • Itô–Stratonovich transition
  • Noise-induced drift
  • Stochastic differential delay equations

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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AB - We consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. An Ornstein–Uhlenbeck process is used to model the colored noise. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.

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