Large deviations in presence of small noise for delay differential equations at an instability

2017/06/01 by Nishanth Lingala, Lingala, Nishanth
Computer Science · Engineering · Mathematics · #34K27 #60F10 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical methods for differential equations #Probability (math.PR) #Stability and Controllability of Differential Equations #math.PR #msc:34K27 #msc:60F10

paper · pdf · doi:10.48550/arxiv.1706.00408

arxiv created 2017/06/01 · openalex publication_date 2017/06/01 · arxiv updated 2017/06/02 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We consider delay differential equations (DDE) that are on the verge of an instability, i.e. the characteristic equation for the linearized equation has one root as zero and all other roots have negative real parts. In presence of small mean-zero noise, we study the large deviations from the corresponding deterministic system. Using spectral theory for DDE it is easy to see that, the projection on to the one dimensional space corresponding to the zero root is exponentially equivalent with the original process. For the one-dimensional process we make the observation that the results of Freidlin-Wentzell apply.

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