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, KahaneKhintchine inequality, Anderson’s inequality. Stochastic integration in Banach spaces I: γRadonifying operators, γboundedness, 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.
At the end of the course, the student understands the basic techniques of probability theory in infinitedimensional 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 infinitedimensional state space and solve this equation using fixed point techniques and stochastic integration in infinite dimensions.

Subjects
 Lecture 01: Integration in Banach spaces
 Lecture 02: Random variables in Banach spaces
 Lecture 03: Sums of independent random variables
 Lecture 04: Gaussian random variables
 Lecture 05: Radonifying operators
 Lecture 06: Stochastic integration I: the Wiener integral
 Lecture 07: Semigroups of linear operators
 Lecture 08: Linear equations with additive noise I
 Lecture 09: Gammaboundedness
 Lecture 10: Linear equations with additive noise II
 Lecture 11: Conditional expectations and martingales
 Lecture 12: UMDspaces
 Lecture 13: Stochastic integration II: the Ito integral
 Lecture 14: Linear equations with multiplicative noise
 Lecture 15: Applications to stochastic PDE
Stochastic Evolution Equations by TU Delft OpenCourseWare is licensed under a Creative Commons AttributionNonCommercialShareAlike 4.0 International License.
Based on a work at https://ocw.tudelft.nl/courses/stochasticevolutionequations/.