### 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 language | English (US) |
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Journal | Journal of Statistical Physics |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**A Small Delay and Correlation Time Limit of Stochastic Differential Delay Equations with State-Dependent Colored Noise.** / Hottovy, Scott; McDaniel, Austin; Wehr, Jan.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - A Small Delay and Correlation Time Limit of Stochastic Differential Delay Equations with State-Dependent Colored Noise

AU - Hottovy, Scott

AU - McDaniel, Austin

AU - Wehr, Jan

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

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.

KW - Itô–Stratonovich transition

KW - Noise-induced drift

KW - Stochastic differential delay equations

UR - http://www.scopus.com/inward/record.url?scp=85061192666&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061192666&partnerID=8YFLogxK

U2 - 10.1007/s10955-019-02242-2

DO - 10.1007/s10955-019-02242-2

M3 - Article

AN - SCOPUS:85061192666

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

ER -