## 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.

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