# Quasi-linear SPDEs via a rough path approach

**Lecturer:**Felix Otto**Date:**Thursday 09:15 - 11:00**Room:**MPI MiS, A3 01**First lecture**: on Oct. 20**No lectures**: on Jan. 12, 26, Feb. 2**Language:**English**Target audience:**MSc students, PhD students, Postdocs**Keywords:**nonlinear SPDEs, Schauder theory

## Abstract

Culminating in Hairer's regularity structures, there has been much recent progress in developing a robust solution theory for nonlinear stochastic partial differential equations (SPDEs). This progress is inspired by Lyons' treatment of stochastic ordinary differential equations, which is much more deterministic than Ito's approach. The main deterministic ingredient can be seen as an extended Schauder theory, so a maximal regularity theory for constant-coefficient parabolic equations in Hölder spaces, where polynomials are supplemented by more general, "rough" functions. The sole stochastic ingredient is to give an "off-line" sense to a finite number of singular products of rough functions and their distributional derivatives.

So far, this approach has been limited to SPDEs where the leading-order part is the constant-coefficient diffusion operator. In this course, we will present a treatment of the *quasi-linear* SPDE $$ \partial_tu+a(u)\partial_x^2u=\sigma(u)f $$ with a noise \(f\in C^{\alpha-2}\) on the parabolic Hölder scale. Provided \(\alpha>\frac{2}{3}\) (which includes white noise in time) and giving an "off-line" sense to products of the form \(v(\cdot,a_0)\partial_x^2v(\cdot,a_0')\) with \(v(\cdot,a_0)\) solving the constant-coefficient SPDE \(\partial_tv-a_0\partial_x^2v=f\), we obtain a (small-data) solution theory \(C^\alpha\) for *u*. Loosely speaking, we extend the treatment of the singular product \(\sigma(u)f\), in the spirit of Gubinelli, to the product \(a(u)\partial_x^2u\), which has the same degree of singularity but is more nonlinear since the solution *u* appears in both factors.

Next to treating a wider class of non-linear equations, the merit is that we introduce some simpler tools. More specifically, we treat the singular product \(a(u)\partial_1^2u\) by controlling the commutator \([a(u),(\cdot)_\epsilon]\partial_x^2u\) of multiplication with the first factor *a(u)* and convolution \((\cdot)_\epsilon\), where the family of symmetric convolution operators \(\{(\cdot)_\epsilon\}_\epsilon\) satisfies a semi-group property and respects the parabolic scaling. Controlling such commutators is reminiscent of the DiPerna-Lions theory for rough transport equations. The PDE ingredient mimics the (kernel-free) Krylov-Safanov approach to ordinary Schauder theory.

This is joint work with Hendrik Weber.

## Regular Lectures (Winter 2016/2017)

**Information Theory**- Nihat Ay
- Date: Tuesday 11:00 - 12:30, MPI MiS, A3 02

**Hamilton-Jacobi Equations: Viscosity Solutions, Optimal Control and Periodic Homogenization**- Benjamin Fehrman
- Date: Thursday 16:00 - 17:30, MPI MiS A3 02

**Introduction to stochastic PDE**- Benjamin Gess
- Date: Wednesday 11:00 - 12:30, MPI MiS A3 02

**Mathematical models of neuronal and synaptic dynamics**- Jürgen Jost
- Date: Friday 13:30 - 15:00, MPI MiS A3 01

**Introduction to Real Algebraic Geometry**- Mario Kummer
- Date: Friday 11:15 - 12:45, MPI MiS A3 02

**Quasi-linear SPDEs via a rough path approach**- Felix Otto
- Date: Thursday 09:15 - 11:00, MPI MiS A3 01

**Selected Topics in Geometry**- Hans-Bert Rademacher
- Date: Thursday 13:15 - 14:45, room Uni Leipzig, SG 3-13

**IMPRS Ringvorlesung**- Rainer Verch, Klaus Kroy, Bernd Rosenow
- Lectures 2016: Wednesday 09:15 - 10:45, MPI MiS E1 05
- Lectures 2017: Tuesday 13:30 - 15:00, MPI MiS E1 05
- Tutorials: 5.12., 12.12., 19.12. and 9.1.2017, 11:00 - 12:30, MPI MiS G2 01