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.

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.

Creative Commons License
Stochastic Evolution Equations by TU Delft OpenCourseWare is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Based on a work at https://ocw.tudelft.nl/courses/stochastic-evolution-equations/.
Back to top

Help us improve TU Delft OpenCourseWare

Help us to get to know you and your ideas on how to improve the OCW website. If you take 5 minutes and fill out our survey, you would be very helpful.

Open Survey