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The lectures are at a beginning graduate level and assume only basic familiarity with Functional Analysis and Probability Theory. Topics covered include:
Random variables in Banach spaces: Gaussian random variables, contraction principles, Kahane-Khintchine inequality, Anderson’s inequality.
Stochastic integration in Banach spaces I: γ-Radonifying operators, γ-boundedness, Brownian motion, Wiener stochastic integral.
Stochastic evolution equations I: Linear stochastic evolution equations: existence and uniqueness, Hölder regularity.
Stochastic integral in Banach spaces II: UMD spaces, decoupling inequalities, Itô stochastic integral.
Stochastic evolution equations II: Nonlinear stochastic evolution equations: existence and uniqueness, Hölder regularity.
Study Goals: At the end of the course, the student understands the basic techniques of probability theory in infinite-dimensional spaces and their applications to stochastic partial differential equations. The student is able to model a stochastic partial differential equation as an abstract stochastic evolution equation on a suitably chosen infinite-dimensional state space and solve this equation using fixed point techniques and stochastic integration in infinite dimensions..
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