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June 2018, 38(6): 3099-3138. doi: 10.3934/dcds.2018135
Exit time asymptotics for small noise stochastic delay differential equations
David Lipshutz ,
Faculty of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
* Corresponding author: David Lipshutz
Received October 2017 Revised January 2018 Published April 2018
Fund Project: This research was supported in part by NSF grants DMS-1206772 and DMS-1148284
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Dynamical system models with delayed dynamics and small noise arise in a variety of applications in science and engineering. In many applications, stable equilibrium or periodic behavior is critical to a well functioning system. Sufficient conditions for the stability of equilibrium points or periodic orbits of certain deterministic dynamical systems with delayed dynamics are known and it is of interest to understand the sample path behavior of such systems under the addition of small noise. We consider a small noise stochastic delay differential equation (SDDE). We obtain asymptotic estimates, as the noise vanishes, on the time it takes a solution of the stochastic equation to exit a bounded domain that is attracted to a stable equilibrium point or periodic orbit of the corresponding deterministic equation. To obtain these asymptotics, we prove a sample path large deviation principle (LDP) for the SDDE that is uniform over initial conditions in bounded sets. The proof of the uniform sample path LDP uses a variational representation for exponential functionals of strong solutions of the SDDE. We anticipate that the overall approach may be useful in proving uniform sample path LDPs for other infinite-dimensional small noise stochastic equations.
Keywords: Delayed dynamics, small noise asymptotics, exit time, uniform large deviation principle, stochastic delay differential equation.
Mathematics Subject Classification: Primary: 60F10, 34K50; Secondary: 60J25.
Citation: David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135
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