Schedule
| Thu. 10 | Fri. 11 | ||
|---|---|---|---|
| 10:00–10:45 | Catellier Room 734 |
Djurdjevac Room 734 | |
| 11:00–11:45 | Galeati Room 734 |
Kastberg Nilssen Room 734 | |
| 12:00–13:00 | Lunch | ||
| 13:00–13:45 | Perkowski Room 734 |
Schmeding Room 734 | |
| 14:00–14:45 | Skre Fjordholm Room 734 |
Tapia Room 734 | |
| 15:00–15:45 | Bruned Room 734 |
Round table Room 734 | |
Talks
Renormalisation from non-geometric to geometric rough paths
(Slides)Yvain Bruned
The Hairer-Kelly map has been introduced for establishing a correspondence between geometric and non-geometric rough paths. Recently, a new renormalisation on rough paths has been proposed by Tapia et Zambotti, built on this map and the Lyons-Victoir extension theorem. We will compare this renormalisation with the existing ones such as BPHZ and the local products renormalisations. We will also explore how the renormalisation behaves in another approach for moving from non-geometric to geometric rough paths.
Regularization by (very irregular) noise for the gPAM. The Young case
(Slides)Rémi Catellier
We study the generalized Parabolic Anderson Model equation with an additive space noise (possibly càdlàg). We show that when the noise is sufficiently irregular, on can give some well-posedness result for the equation for quite non-smooth field $g$. In order to define the equation in this irregular setting, two tools are introduced : non-linear Volterra Young calculus in Banach spaces and an extension of averaging operators to cover irregular càdlàg fields. This talk is based on a joint work with Fabian Harang.
Towards Finite Element Discretization of the Dean-Kawasaki equation
(Slides)Ana Djurdjevac
Dean-Kawasaki equation describes the evolution of the density function of finitely many particles obeying Langevin dynamics. Applications of the Dean-Kawasaki equation are ranging from complex fluids to the humanities. We will first briefly motivate our studies by presenting the applications in humanities. The Dean-Kawasaki equation is a nonlinear SPDE with a non-Lipschitz multiplicative noise in divergence form, which makes its solution theory different from the standard settings. We will first comment on the well-posedness of the mollified equation. The main topic of the talk is to discuss weak finite element approximation results on the Dean-Kawasaki equation that are built on the duality argument based on the work from Von Renesse and collaborators: Dean-Kawasaki Dynamics: Ill-Posedness Vs. Triviality, 2018. This is the joint work with H. Kremp and N. Perkowski.
Regularization by noise and nonlinear Young integrals
(Slides)Lucio Galeati
Regularization by noise refers to a class of phenomena in which the introduction of noise restores existence and/or uniqueness in otherwise ill-posed ODEs and PDEs. Catellier and Gubinelli have introduced a general methodology, based on the concept of nonlinear Young differential equations, to obtain such results; this approach is mostly pathwise and presents several advantages, allowing non-Markovian, non-martingale noise and yielding a straightforward construction of the associated flow. In this talk I will review these general concepts and focus on some results of regularization for SDEs recently obtained together with F. Harang.
Rough path variational principles for fluid equations
(Slides)Torstein Kastberg Nilssen
In this presentation we will first recall the insight of Arnold that Euler's equation can be understood as a geodesic equation on an infinite dimensional manifold. Using Lagrange multipliers, this yields a natural framework for understanding the structure of random/irregular perturbations of fluid equations. In this presentation we will consider rough path perturbations and the corresponding variational principles.
Two results on singular SDEs
(Slides)Nicolas Perkowski
Singular SDEs are equations with distributional drift and uniformly elliptic (often additive) noise. By now we understand very well how to construct solutions to such equations for Brownian noise via the associated martingale problem. If the drift is not too irregular, then there are also results for \alpha-stable Levy noise. I will show how to extend those results to more singular drifts, going from the Young regime to the rough path regime. As an application, we construct a Brox diffusion with Levy noise. And I will also discuss Gaussian heat kernel estimates for the transition density in the case of Brownian noise. The talk is based on joint works with Helena Kremp and Willem van Zuijlen.
Geometric vs. weakly geometric rough paths in infinite dimensions
(Slides)Alexander Schmeding
Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. It is expected that the general theory carries over to this infinite-dimensional setting, yet a number of results which are elementary cornerstones of rough path theory are still unknown in the Banach setting. In this talk we shall present criteria for when we can approximate Banach space-valued weakly geometric rough paths by signatures of curves of bounded variation, given some tuning of the Hölder parameter. Using infinite-dimensional sub-Riemannian geometry and Lie group techniques, we show that the familiar characterisation from finite dimensions carries over to rough paths on Hilbert spaces. This is joint work with E. Grong and T. Nilssen: Geometric rough paths on infinite dimensional spaces, arXiv:2006.06362
The zero noise limit for ordinary differential equations with irregular velocities
(Slides)Ulrik Skre Fjordholm
It is well-known that ordinary differential equations (as well as PDEs such as transport or continuity equations) may be ill-posed in the presence of irregular velocity fields (say, non-Lipschitz). On the other hand, their stochastic counterparts enjoy both (strong) existence, uniqueness and stability. This leads to the natural question of whether deterministic equations may be approximated by stochastic ones, and which (if any) solutions can appear as the limit of SDEs with vanishing noise. In this talk I will talk about a modest contribution to this (quite difficult) problem by considering one-dimensional and autonomous drifts which are merely $L^\infty$-bounded. This is joint work with Andrey Pilipenko and Markus Musch.
Transport and continuity equations with (very) rough noise
(Slides)Nikolas Tapia
We study the solution theory of linear transport equations driven with rough multiplicative noise. We show existence and uniqueness for rough flows driven by an arbitrary geometric rough path, and obtain a rough version of the classical method of characteristics, under a boundedness condition for the vector fields. We also obtain an adjoint RDE for the derivatives of the induced flow. Dually, we show existence and uniqueness for the associated continuity equation.