The first passage problem for stable linear delay equations perturbed by power law Lévy noise

Area of Science:

• Stochastic analysis
• Non-Markovian systems
• Delay differential equations

Background:

• Investigating the behavior of linear scalar delay differential equations under noise is crucial for understanding complex systems.
• The influence of multiplicative power-tail Lévy noise on such systems presents unique challenges due to its non-Gaussian nature and heavy tails.
• Understanding first passage time problems (Kramers problem) is essential for predicting system stability and escape dynamics.

Purpose of the Study:

• To analyze the first passage problem for a linear scalar delay differential equation driven by small multiplicative power-tail Lévy noise.
• To investigate the asymptotic behavior and memory effects in this non-Markovian system.
• To explore the impact of noise amplitude and delay on the system's exit time and dynamics.

Main Methods:

• Probabilistic methods were employed to solve the first passage (Kramers) problem.
• Asymptotic analysis was used to identify memory loss and system behavior.
• Mathematical modeling and simulation, illustrated with a linear delay oscillator example, were utilized.

Main Results:

• An asymptotic loss of memory was discovered in the non-Markovian system.
• The mean exit time was found to increase with the power of the small noise amplitude.
• A nonlinear delay-induced exit acceleration was observed, attributed to non-normal growth phenomena.

Conclusions:

• The study demonstrates an asymptotic memory loss in delay differential equations with Lévy noise.
• Noise amplitude and system delay significantly influence exit time and dynamics, leading to phenomena like exit acceleration.
• The findings provide insights into the behavior of stochastic delay systems, with implications for fields utilizing such models.