Syllabus Stochastic calculus and applications

Corsi di Laurea Magistrali in Matematica A.A. 2024/25, Prof. L. Bertini.

• Gaussian processes. Brownian motion. Martingales and Doob's inequality.
• Convergence of probability on the space of continuos functions. Invariance principle.
• Regularity of Brownian sample paths. Quadratic variation.
• Strong Markov property.
• Reflection principle and applications.
• Stochastic integrals and Ito's formula.
• Levy's theorem.
• Stochastic equation. Ito's theory.
• Smoluchowski approximation.
• McKean-Vlasov limit.
• Markov processes. Feller semigroups and their generators (sketch).
• Dynkin's martingale.
• Solution to stochastic equations as Markov processes.
• Invariant measures for Markov processes. Case of diffusion processes.
• Bakry-Emery crtiterion for the spectral gap.
• Central limit theorem for additive functionals of Markov processes (sketch).
• Probabilistic representation for elliptic problems.
• Fenyman-Kac formula.

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