Functional characterization of linear delay Langevin equations

Abstract

We present an exact functional characterization of linear delay Langevin equations driven by any noise structure defined through its characteristic functional. This method relies on the possibility of finding an explicitly analytical expression for each realization of the delayed stochastic process in terms of those of the driving noise. General properties of the transient dissipative dynamics are analyzed. The corresponding interplay with a color Gaussian noise is presented. As a full application of our functional method we study a model for population growth with non-Gaussian fluctuations: the Gompertz model driven by multiplicative white shot noise.

Time-Delay Induced Reentrance Phenomenon in a Triple-Well Potential System Driven by Cross-Correlated Noises

“…In many natural and physical situations, the time delay is usually used to describe an intrinsic delay mechanism or an introduction of the time-delayed feedback, which implies that the dissipative evolution depends on the state of the system in a shifted previous time [1]. Thus, the effects of the time delay on the nonlinear stochastic systems have recently gained considerable attention [9,16,17].…”

Critical Fluctuations and 1/fα-Activity of Neural Fields Involving Transmission Delays

“…Likewise, for k = 0 and Q > 0 Eq. (17) exhibits stationary distributions for delays t 0 < t 0,c , whereas for delays t 0 > t 0,c stationary distributions do not exist [51,52,[70][71][72].…”

Adiabatic small noise fluctuations around anticipated synchronization: A perspective from scalar master–slave dynamics

“…(32) strongly depends on the stability region determined by the parameters K and a (see Ref. [23] were we have given and exhaustive analysis of the oscillatory and non-oscillatory-like behaviors of Λ (t) as a function of the stability region).…”

“…As in the previous case, this Langevin equation can be integrated for each realization of the noises as [23] (t)…”

“…Then, in this context, the master and slave fluctuations are described after linearizing the scalar function f (x) and assuming that their relaxing dynamics is much faster than the characteristic variations of the synchronized master-slave average dynamics. Under this linear adiabatic approximation, the dynamical fluctuations can be described in terms of linear delay Langevin equations whose statistical properties can be found in an exact analytical way [11][12][13][14][15][16][17][18][19][20][21][22][23].…”