Eric Vanden-Eijnden, New York University

Stochastic dynamics, rare events and large deviations (10.5h)

1. Law of large Numbers, Central Limit Theorem, and Large Deviation Principle (LDP): Cramer's theorem, tilted distribution, Varadhan's Lemma, Gartner-Ellis theorem, Laplace method, relative entropy, contraction principle.

2. Markov Jump Processes & Stochastic Differential Equations: Definition, characterization, basic properties, examples; forward and backward Kolmogorov equations, Feynman-Kac representation formula.

3. The small noise regime -- Wentzell-Freidlin Theory: Girsanov transform, Schilder's theorem, action functional, path integral interpretation, optimal control formulation, instanton equations.

4. The large particle number regime -- mean-field limit & beyond: WKB ansatz, Kurtz theorems, Sanov theorem, action functional, interacting particles systems, hydrodynamic limit (+ a detour by machine learning).

5. Reactions in Equilibrium and Nonequilibrium systems: Minimum energy path & string method; minimum action methods.

6. LDP-based importance sampling strategies and genealogical algorithms.

7. Fluctuation relations, Crooks theorem, Jarzynski equality, Entropy production, work relations, optimal driving protocols, macroscopic fluctuation theory.