# Introduction to stochastic PDEs

## Summary

Stochastic PDEs are used to model systems that are spatially extended and include a random component. This course gives an introduction to this topic, including some Gaussian measure theory and some analytic semigroup theory.

## Content

Stochastic PDEs form a relatively recent area of mathematics that combines many different fields, including PDE theory, stochastic analysis, ergodic theory, functional analysis, etc. This course is an introduction to the area with the aim of being able to appreciate some 21st century developments towards the end of the course. We will mainly focus on the development of a rather general solution theory for linear and semilinear stochastic PDEs, including stochastically forced heat, Navier-Stokes, and reaction-diffusion equations.

Some of the tools developed in this course, in particular Gaussian measure theory and analytic semigroup theory, are of broader interest.

## Keywords

probability, partial differential equations, semigroups, Gaussian measures

## Learning Prerequisites

## Required courses

Analysis I-IV

Probability

## Recommended courses

Measure and integration

Probability theory

## Important concepts to start the course

Basic concepts in probability theory

Basic properties of Hilbert and Banch spaces

## Teaching methods

Weekly lectures (on blackboard) and exercise sessions with assistant

## Expected student activities

Attending the lectures and solving the exercises

## Assessment methods

Oral exam

## Supervision

Office hours | No |

Assistants | Yes |

Forum | No |

## Resources

## Virtual desktop infrastructure (VDI)

No

## Bibliography

G. DA PRATO and J. ZABCZYK. Stochastic equations in infinite dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1992.

A. LUNARDI. Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995.

V. I. BOGACHEV. Gaussian measures, vol. 62 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1998.

P. BILLINGSLEY. Convergence of probability measures. John Wiley & Sons Inc., New York, 1968.

K. YOSIDA. Functional analysis. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the sixth (1980) edition.

## Ressources en bibliothèque

- Stochastic equations in infinite dimensions / Da Prato
- Analytic semigroups and optimal regularity in parabolic problems / Lunardi
- Gaussian measures / Bogachev
- Convergence of probability measures / Billingsley
- Functional analysis / Yosida

## Notes/Handbook

The lecture will mainly follow the notes available at https://www.hairer.org/notes/SPDEs.pdf, but might cover additional material if time permits.

## Moodle Link

## In the programs

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Introduction to stochastic PDEs**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Introduction to stochastic PDEs**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Introduction to stochastic PDEs**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks