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.
- 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: Gamma-boundedness
- Lecture 10: Linear equations with additive noise II
- Lecture 11: Conditional expectations and martingales
- Lecture 12: UMD-spaces
- 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 Attribution-NonCommercial-ShareAlike 4.0 International License.
Based on a work at https://ocw.tudelft.nl/courses/stochastic-evolution-equations/.