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Dati Insegnamento Contesto Anno Accademico 2022/2023 Corso di Studi Dott. - MI (1385) Modelli e Metodi Matematici per l'Ingegneria / Mathematical Models and Methods in Engineering Anno di Corso 1 Scheda Insegnamento Codice Identificativo 058413 Denominazione Insegnamento STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS Tipo Insegnamento MONODISCIPLINARE Crediti Formativi Universitari (CFU) 5.0 Programma sintetico The first part of the course is devoted to the development of a general theory of stochastic analysis in infinite dimensions: this will cover, in particular, Gaussian measures in Hilbert spaces, stochastic processes and martingales in Hilbert spaces, quadratic variation and tensor quadratic variation, measurability of stochastic processes, stochastic compactness methods, stochastic integration with respect to a square-integrable continuous martingale, Wiener processes and cylindrical Wiener processes, maximal inequalities for square integrable martingales, Burkholder-Davis-Gundy inequality. The second part of the course covers applications of such tools to the variational theory of stochastic PDEs: existence/uniqueness results for SPDEs with Lipschitz nonlinearities, existence/uniqueness results for SPDEs of monotone type in Hilbert triplets. Eventually, selected explicit examples of stochastic partial differential equations will be discussed, with focus on the state of the art and possible open problems: these will be focused on phase-field-type SPDEs such as stochastic Allen-Cahn and Cahn-Hilliard equations. The course is highly interdisciplinary. On the one hand, it bridges from Functional Analysis and Partial Differential Equations to Probability theory. On the other hand, it balances theoretical knowledge of stochastic analysis in Hilbert spaces and direct applications to open problems on stochastic PDEs that arise in Physics and Engineering. For these reasons the course can be beneficial for students working both on Probability Theory and Mathematical Analysis of PDEs, on either an abstract or applicative level. Settori Scientifico Disciplinari (SSD) -- Dettaglio Scaglione Nome Programma dettagliato Da (compreso) A (escluso) A ZZZZ Scarpa Luca

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24/01/2025