CEE 4215

Stochastic Modeling of Complex Systems

Course Description

Course information provided by the Courses of Study 2024-2025.

The theory of stochastic processes is introduced through examples of complex systems from the natural and applied sciences, with an emphasis on applications rather than mathematical abstraction. Students will learn how to model dynamical systems with intrinsic and extrinsic noise sources, simulate their dynamics, and explore how the interplay between stochasticity and nonlinear dynamics can impact their behavior. Topics covered include generating processes for heavy-tailed distributions, diffusion processes (gaussian white noise), jump processes (white shot noise, birth/death processes, dichotomous Markov noise), stochastic hybrid systems, stochastic differential equations, first passage times, and noise-induced transitions. Analytical tools developed in the class include stochastic differential equations, the differential Chapman-Kolmogorov equation and its derivatives (Fokker-Planck and master equation), the use of transforms to solve master and Fokker-Planck equations, and the system size expansion to approximate solutions to master equations with nonlinear transition rates. Applications include examples from biophysics, climate, environmental sciences, and various areas of biology including systems, synthetic, molecular, and cellular biology.

When Offered Fall.

Prerequisites/Corequisites Prerequisite: MATH 2930 or MATH 3230.

Comments Students should have at least one course in introductory probability and statistics, such as CEE 3200 or equivalent, and should know how to write simple code in either Mathematica, MATLAB, or Python.

• Identify the sources of stochasticity in a variety of processes in the applied and natural sciences.
• Model complex stochastic processes using different types of stochastic processes including random walks, Lévy flights and hybrid stochastic processes.
• Determine and analyze the equation(s) describing the temporal evolution of probabilities and probability densities in continuous-state and discrete-state stochastic processes.
• Analyze the stationary probability distribution of a stochastic process (if it exists) and the temporal evolution of moments and cumulants.
• Understand the relationship between the Langevin and Fokker-Planck equations, and how to derive approximations to the solution of master equations with nonlinear transition rates via the system size expansion.
• Identify the fundamental processes that lead to noise-induced transitions and heavy-tailed distributions.

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