Office Hours: By appoitment

Email: kspiliop_at_dam.brown.edu

**Recommended textbooks: **

- For general averaging principle and metastability:
- E. Olivieri and M. E. Vares, Large Deviations and Metastability, Cambridge, 2005
- Mark I. Freidlin, Alexander D. Wentzell, Random Perturbations For Dynamical Systems, Springer, 1998
- N. Berglund and B. Gentz , Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach Springer, 2005.
- For multiscale methods and perturbation theory:
- G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization, Springer, 2007
- A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures (Studies in mathematics and its applications), Elsevier, 1978
- For stochastic calculus and the interplay between PDE's and stochastic processes:
- M. Freidlin, Functional Integration and Partial Differential Equations, Princeton University Press, 1985
- B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, 2007 (6th edition)
- I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 2nd edition

**Course Description:**
Metastability and stochastic resonance phenomena appear every often in various scientific disciplines. Typical examples include the dynamical behavior of proteins,
magnetic systems with magnetization opposite to the external field, etc. Roughly speaking, metastability refers to the phenomenon of rare transitions of a system
between different stable equilibriums. Even though these transitions are rare, they usually have very important effects on the behavior of the system.
A closely related phenomenon is stochastic resonance. Stochastic resonance occurs when a multistable system, subject to noise,
displays an organized behavior which is different from its behavior in the absence of noise. Therefore, establishing a mathematical theory of metastability is
extremely important for applications. One of the main tools in the mathematical theory of metastability is large deviations theory. The construction of a mathematical
theory of metastability not only provides interesting and physically relevant applications of the already established large deviation theory, but also poses new problems.

Emphasis will be placed on

- Review of probability theory, introduction to stochastic calculus (Brownian motion, stochastic differential equations, It\^{o} formula, Fokker-Planck eqs, Feynman-Kac formula, relation to PDE's)
- Introduction to large deviations theory for stochastc processes
- Averaging principle for stochastic dynamical systems and in particular for Hamiltonian systems
- Notions of metastability, the pathwise approach and physical examples of metastability
- Freidlin-Wentzell theory of metastability for It\^{o} processes (explicit computation of metastable states, first exit problem from particular sets of states, called cycles, etc.).

The course will focus on metastability and use the theory of large deviations as a tool, rather than developing the theory of large deviations.