Stochastic Analysis BRATS

• Carlo Bellingeri (TU Berlin)

Title: Singular Hölder spaces and applications

Abstract: We introduce the class of singular Hölder paths and singular controlled rough paths. These spaces arise naturally from the context of singular modelled distributions, one of the main technical notions in regularity structures. However, the simplified setting allow to describe most of their properties, showing some interesting connections with other branches of rough analysis. In particular, we will apply them to study the SLE trace and the rough-volatility regularity structure.

• Martin Brückerhoff (Münster)

Shadow martingales - A stochastic mass transport approach to

the peacock problem

Given a family of real probability measures $(\mu_t)_{t\geq 0}$ increasing in convex order (a so-called peacock) we describe a new systematic method to create a martingale exactly fitting the marginals at any time. This method is based on a stochastic mass transport

perspective on continuous time martingales. More precisely, the distribution of a shadow martingale is uniquely determined by the property that certain parts of the initial distribution $\mu_0$ evolve as concentrated as possible through the given profile $(\mu_t)_{t\geq 0}$ of marginals. As a special instance of a shadow martingale, we

obtain the unique solution to a continuous-time version of the martingale optimal transport problem. Furthermore, our method enriches the knowledge on the Predictable Representation Property (PRP) since any shadow martingale comes with a canonical Choquet representation in extremal Markov martingales. (j.w. with Martin Huesmann and Nicolas Juillet)

• Giuseppe Cannizzaro

A new Universality Class in (1+1)-dimensions: the Brownian Castle

In the context of randomly fluctuating interfaces in (1+1)-dimensions two Universality Classes have generally been considered, the Kardar-Parisi-Zhang and the Edwards-Wilkinson. Models within these classes exhibit universal fluctuations under 1:2:3 and 1:2:4 scaling respectively. Starting from a modification of the classical Ballistic Deposition model we will show that this picture is not exhaustive and another Universality Class, whose scaling exponents are 1:1:2, has to be taken into account.

We will describe how it arises, briefly discuss its connections to KPZ and introduce a new stochastic process, the Brownian Castle, deeply connected to the Brownian Web, which should capture the large-scale behaviour of models within this Class. This talk is based on a joint ongoing work with Martin Hairer.

• Alberto Chiarini (Eindhoven)

Entropic repulsion for the occupation-time field of random interlacements by disconnection.

The model of random interlacements was introduced in 2007 by A.-S. Sznitman, motivated by questions about the disconnection of discrete cylinders or tori by the trace of simple random walk. Since then, it has gained popularity among probabilists due to its percolative properties and also because of its connections to the free field. Random interlacements on transient graphs can be constructed as a Poisson point process of doubly infinite trajectories. After reviewing this model, we will focus on the rare event that these trajectories disconnect a macroscopic body from infinity, in the strongly percolative regime. We will ask the following question: What is the most efficient way for random interlacements to enforce such disconnection? In other words, how do the trajectories of random interlacements look like conditionally on disconnection?

• Paolo Grazieschi (Bath)

Convergence of the three-dimensional Ising-Kac model to \phi^4

We introduce the Ising-Kac model, which is a modification of the classical Ising model with long range interaction. Motivated by previous works, we conjecture convergence of this interacting particle system to the solution to the \phi^4 equation in dimension 3. It is well known that this equation is ill-posed in dimension 3, so that we rely on the Theory of Regularity Structures to give a meaning to it: our plan is therefore to build an expansion of the solution of the Ising-Kac model in the spirit of Regularity Structures and prove tightness of the discrete trees that make up this expansion. We will then show how we aim to get the conjectured convergence.

• Marta Leocata (Roma LUISS)

A global existence result for a quadratic system of stochastic reaction diffusion equations

We study the existence, and the regularity, of solutions to systems of stochastic reaction diffusion equations under some specific structure hypotheses (which imply preservation of positivity and control of an entropy in particular). Joint work with Julien Vovelle.

• Pierre Perruchaud (Notre Dame)

Geometric perturbations of the Euler equations for perfect fluids

It is known since the 1960s that the Euler equations for incompressible inviscid fluids can be rephrased as geodesic motions on suitable infinite-dimensional manifolds. Points in these spaces represent configurations of the fluid, and the Riemannian metric corresponds to the physical notion of the total energy. In this talk, I will describe random perturbations of these equations, focussing particularly on the so-called kinetic Brownian motion. It will provide an occasion to discuss infinite-dimensional geometry and convergence of stochastic processes using rough paths, two important ingredients for constructing a sturdy infinite-dimensional Cartan development.

• Leonardo Tolomeo (Bonn HCM)

Quasi invariance and growth of Sobolev norms for Hamiltonian PDEs

In this talk, we consider a Hamiltonian PDE with a suitable gaussian initial data, with the goal of studying the growth in time of the solution to this equation.

The study of this kind of problems was (arguably) started by Bourgain in ’96, who considered the Schrödinger equation with cubic nonlinearity posed on the 2-dimensional torus (2d-NLS). In his work, Bourgain exploited the formal invariance of the Gibbs measure in order to construct solutions to 2d-NLS in negative Sobolev regularity. These solutions satisfy a logarithmic bound on the growth of their norm.

A crucial observation is that Bourgain’s argument relies only on quasi-invariance (the existence of a density for the law of the solution with respect to the initial gaussian measure), together with some assumptions on the density.

In 2015, N. Tzvetkov developed a strategy to prove quasi-invariance that depends only on the structure of the transport measure associated with the flow. This approach was expanded in 2018 by Planchon, Tzvetkov and Visciglia, in order to incorporate some information about the solution that come from the deterministic study of the PDE. This allows to obtain quasi-invariance as a consequence of deterministic global well posedness in a plethora of situations.

In this talk, we suggest a further improvement to the previous techniques, that relies on finer space-time properties of the density of the transported measure. As an application, we obtain quasi-invariance and polynomial growth of solutions for the fourth-order Schrödinger equation with initial data in negative Sobolev regularity.

This is a joint work with J. Forlano (University of Edinburgh).