2013/11/18 by Nishanth Lingala, Lingala, Nishanth, N. Sri Namachchivaya +1
Computer Science · Economics, Econometrics and Finance · Engineering · Mathematics · #34K06 #34K27 #34K33 #34K50 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications #math.PR #msc:34K06 #msc:34K27 #msc:34K33 #msc:34K50
paper · pdf · doi:10.48550/arxiv.1311.4532
For v3: revised section 5 and appendix. For v2: corrected the assumption on coefficient G. Included the case of multiplicative noise in appendix. Deleted a section on oscillators, because we found that in multidiemnsional case it is easier to work with complexifications (see 1403.3029)
openalex publication_date 2013/11/18 · arxiv created 2014/04/03 · arxiv updated 2014/04/07 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. We deal with linear DDEs that are on the verge of instability, i.e. a pair of roots of the characteristic equation (eigenvalues) lie on the imaginary axis of the complex plane, and all other roots have negative real parts. We show that, when the system is perturbed by small noise, under an appropriate change of time scale, the law of the amplitude of projection onto the critical eigenspace is close to the law of a certain one-dimensional stochastic differential equation (SDE) without delay. Further, we show that the projection onto the stable eigenspace is small. These results allow us to give an approximate description of the delay-system using an SDE (without delay) of just one dimension. The proof is based on the martingale problem technique.