Instructor: Jan Maas

Teaching Assistant: Giovanni Zanco

Stochastic partial differential equations (SPDEs) are used to model a wide variety of physical systems subject to randomness or noise. In many interesting situations, solutions to SPDEs are very irregular, which makes their analysis particularly challenging. In the last couple of years, spectacular progress has been obtained through the use of "rough paths theory" and the development of the theory of "regularity structures", for which M. Hairer has been awarded a Fields Medal. In this course we aim to give a gentle introduction to SPDEs with a special emphasis on these recent developments.

**Prerequisites**

Part I of this course is a prerequisite for Part II.

**Literature**

* P. Friz and M. Hairer, A course on rough paths

* M. Hairer, An introduction to stochastic PDEs

* M. Hairer, Introduction to Regularity Structures

* M. Hairer, Regularity structures and the dynamical Φ_4^3 model

3

The final grade is based on homework (50%) and the oral exam (50%).

Date | Time | Topic |
---|---|---|

Nov 25 | 8:45 - 10:00 | Background material in probability theory (Giovanni Zanco) |

Nov 27 | 10:15 - 11:30 | Space-time regularity of the stochastic heat equation |

Nov 30 | 14:00 - 15:15 | Stochastic reaction-diffusion equations |

Dec 4 | 10:15 - 11:30 | Stochastic Burgers equations |

Dec 7 | 14:00 - 15:15 | Stochastic heat equation as a rough path |

Dec 9 | 10:15 - 11:30 | Wiener chaos, hypercontractivity, and equivalence of moments I |

Dec 18 | 14:00 - 15:15 | Wiener chaos, hypercontractivity, and equivalence of moments II |

Jan 11 | 10:15 - 11:30 | Rough stochastic Burgers equations |

Jan 13 | 10:15 - 11:30 | Introduction to regularity structures |

Jan 18 | 10:15 - 11:30 | Regularity structures, models, and modelled distributions |

Jan 20 | 10:15 - 11:30 | The Reconstruction Theorem and its consequences |

Jan 25 | 10:15 - 11:30 | Schauder estimates |

Jan 27 | 10:15 - 11:30 | KPZ equation |

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