Stochastic Modeling of Complex Systems

Catalog description

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.