Extracting non-Gaussian governing laws from data on mean exit time

Area of Science:

• Stochastic processes
• Dynamical systems theory
• Mathematical modeling

Background:

• Complex systems often lack observable state time series, hindering mathematical modeling.
• Systems driven by non-Gaussian Lévy motion present unique modeling challenges.
• Mean exit time is an observable quantity for certain complex systems.

Purpose of the Study:

• To develop a method for extracting non-Gaussian governing laws from mean exit time observations.
• To address limitations in modeling complex systems driven by non-Gaussian Lévy motion.
• To identify stochastic differential equations from limited observational data.

Main Methods:

• Utilizing sparse regression in the least squares sense to approximate mean exit time functions.
• Learning the generator of the system.
• Solving an inverse problem for a nonlocal partial differential equation.
• Minimizing an error objective function to identify the governing stochastic differential equation.

Main Results:

• The proposed method accurately extracts non-Gaussian governing laws.
• The technique is effective even for systems with complex rational drift.
• The method demonstrates applicability to both Gaussian Brownian motion and non-Gaussian Lévy motion driven systems.

Conclusions:

• The developed method provides a feasible approach for modeling complex systems with limited observations.
• It expands the capability to model systems driven by non-Gaussian processes.
• This technique offers a robust tool for analyzing stochastic dynamical systems.