This is the third workshop of the Bernoulli Center Program "New developments and challenges in Stochastic Partial Differential Equations”. 

The workshop includes five mini-courses plus several research talks.

The workshop will be streamed via Zoom with Meeting ID: 635 5674 9098 (link). You are wekcome to download and import the iCalendar (.ics) files to your calendar system.

Minicourses

• Lubomir Banas (Bielefeld University) - Robust numerical approximation of the stochastic Cahn-Hilliard equation
  
  The Cahn-Hilliard equation is a prototype model for describing the motion of interfaces in binary mixtures. The equation involves a (small) parameter $\varepsilon$ which imposes restrictions on the discretization parameters in numerical approximation schemes. We will cover two topics related to the influence of the $\varepsilon$ parameter on the numerical approximation of the stochastic Cahn-Hilliard equation.

In the first part, we will discuss the numerical approximation of the problem near the sharp-interface limit, i.e., for $\varepsilon \rightarrow 0$, with asymptotically vanishing noise. We will derive error estimates for the numerical approximation that are robust with respect to $\varepsilon$ and show the convergence of the numerical approximation to its sharp-interface limit, which is the deterministic Hele-Shaw problem.

The theory in the first part relies on the use of a spectral estimate for the linearized (deterministic) Cahn-Hilliard operator, which restricts its applicability to the case of small noise. In the second part, we will circumvent this restriction by employing a different approach. We will derive a posteriori estimates for the numerical approximation that remain valid in the case of non-vanishing noise. We will also address the approximation in the case of space-time white noise.

• Benjamin Fehrman (Louisiana State University) - The kinetic formulation of the skeleton equation and SPDEs of fluctuating hydrodynamics type
  
  Stochastic PDEs of fluctuating hydrodynamics type, like the Dean--Kawasaki equation and related models, have attracted a large amount of attention in both mathematics and statistical physics due to their success describing non-equilibrium phenomenon in physical systems.  However, their application has until recently lacked a precise mathematical meaning for several reasons.  The stochastic PDEs are formally supercritical in the language of singular SPDEs and are therefore not renormalizable, they exhibit non-Lipschitz noise coefficients including the square root, and attempts at establishing certain weak solution theories have been shown to be either ill-posed or trivial.  At first glance, this rather negative evidence might suggest that the SPDEs are merely formal rewritings of the underlying microscopic dynamics, and that they themselves offer little in the way of understanding and simulation.

A primary purpose of these lectures will be to show that this is not the case.  After introducing a suitable spatial regularization of the noise---a step motivated, for example, by the grid-length of the particle system, coarse-graining, and numerical approximations---we will establish a robust well-posedness theory for a general class of SPDEs of fluctuating hydrodynamics type based on the equation's kinetic formulation.  The kinetic form will first be introduced in the context of the skeleton equation, for which we will establish the well-posedness and stability of solutions.  We will then discuss how these ideas can be used to treat the related SPDEs.  In the final part of these lectures, we will show that in the case of the generalized Dean--Kawasaki equation, along appropriate scaling limits, the SPDEs accurately describe the non-equilibrium behavior of the zero range process, which makes rigorous in this context the link between macroscopic fluctuation theory and fluctuating hydrodynamics.

• Julian Fischer (IST Austria) - Mathematical approaches to the rigorous justification of fluctuating hydrodynamics
  
  The theory of fluctuating hydrodynamics attempts to describe density fluctuations in systems of interacting particles in the regime of large particle numbers by means of suitable SPDEs.

The Dean-Kawasaki equation - a strongly singular SPDE - is one of the most basic equations of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N weakly interacting diffusing particles in the regime of large particle numbers N. The singular nature of the Dean-Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification: Besides being non-renormalizable by regularity structures or paracontrolled calculus, it has recently been shown to not even admit nontrivial martingale solutions.

In this lecture series, we discuss approaches to the rigorous justification of the Dean-Kawasaki equation as a quantitative model for density fluctuations. We show that structure-preserving discretizations of the Dean-Kawasaki equation may approximate the density fluctuations of N weakly interacting diffusing particles to arbitrary order in 1/N (in suitable weak metrics), the accuracy being only limited by the numerical scheme. We subsequently discuss how the situation differs in the case of the continuum (non-discretized) Dean-Kawasaki equation with regularized noise.

• Benjamin Gess (Bielefeld University) - From large deviations about porous media to PDEs with irregular coefficients, gradient flow structures, and SPDEs

In this series of lectures, we will outline how the analysis of large deviations of the rescaled zero-range process about its hydrodynamic limit, the porous medium equation, leads to the analysis of the skeleton equation, an energy-critical, degenerate parabolic-hyperbolic PDE with irregular drift. Subsequently, we will demonstrate that such large deviations are intimately related to a gradient flow interpretation of the porous medium equation. Furthermore, this gradient flow interpretation reveals a canonical fluctuation correction to the porous medium equation—a stochastic PDE with noise resembling a stochastic scalar conservation law. This will motivate the introduction of a kinetic solution approach to such stochastic PDEs. Based on this, in the final part of the lecture, we will investigate regularisation by noise for (stochastic) scalar conservation laws.

• Hendrik Weber (University of Münster) - Global existence for stochastic PDEs using regularity structures

Initiated in groundbreaking works by Hairer, Gubinelli and others, the theory of Parabolic Stochastic PDEs has made spectacular progress over the last few years. A systematic solution theory for various interesting and previously intractable equations from Mathematical Physics is now available. Examples include the KPZ equation and the stochastic quantisation equations for the 3D $\Phi^4$ and Yang Mills measures. These solutions have been shown to arise as scaling limits of discrete models of Statistical Mechanics and to display interesting phenomena, such as phase transitions, in their own right.

In these lectures I will review some aspects of these developments. The main focus will be the question of global existence / a priori bounds for solutions: The first works on regularity structures and paracontrolled distributions were mostly concerned with constructing local-in-time solutions using a fixed point argument in a suitable space of distributions. The focus was primarily on the description of solutions on small scales and dealing with  “infinite terms” which arise when applying non-linear functions to distributions. The matching a priori bounds were only developed more recently, among others in a series of papers by Mourrat, Moinat, Chandra, Feltes and me. I will explain these results, the main challenges and the arguments that permitted to overcome them.

Invited speakers

• Yvain Bruned (Université de Lorraine) - Chain rule symmetry for singular SPDEs

In this talk, we will present the characterisation of the chain rule symmetry for the geometric stochastic heat equations in the full subcritical regime for Gaussian and non-Gaussian noises. Such a result is due to a change of perspective on several levels and the use of ideas coming from operad theory and homological algebra. One consequence is local well-posedness for quasi-linear SPDEs. This is a joint work with Vladimir Dotsenko.

• Seiichiro Kusuoka (Kyoto University) - Construction of the three-dimensional polymer measure with self-interactions and associated Dirichlet form by smoothing approximation

I present a new construction of the three-dimensional polymer measure with self-interactions (called Edward model). This construction is not by removal of the diagonal part, but by smoothing. Also we construct the associate Dirichlet form of the constructed polymer measure. In this talk, I will explain the strategy and techniques for the construction. This is a joint work with Sergio Albeverio, Song Liang and Makoto Nakashima.

• Khoa Le (University of Leeds) - Regularization by noise for the stochastic heat equation
  
  We consider the stochastic heat equation in one spatial dimension driven by a space-time white noise with a distributional drift. It has been recently established that the noise induces a regularization effect which makes the equation well-posed when the drift has a Besov regularity index larger than -1. In this talk, I will present these results and a recent progress in the case of multiplicative non-degenerate noise.
  
  

• Harprit Singh (University of Edinburgh) - Renormalisation and homogenisation of (certain) non-translation invariant SPDEs

As is well understood, renormalisation of singular SPDEs involving non-constant coefficient differential operators requires the use of (diverging) renormalisation functions, thus a-priori resulting in a notion of solution depending on uncountably many parameters. In the first part of the talk I shall discuss, using as examples the φ4d and g-PAM equation, that these renormalisation functions can be chosen to be local functionals of the coefficient field, resulting in a finite dimensional notion of solution with several desirable properties. In the second part, which is based on joint work with M. Hairer, I shall discuss homogenisation results for these equations. In particular, we observe that the (divergent) counter-terms split into a local ‘small scale’ part as above and an additional ‘large scale’ part involving familiar objects from homogenisation theory.

• Lutz Weis (Karlsruher Institut für Technologie) - A stochastic maximal function and regularity estimates for parabolic stochastic evolution equations

There is a well developped regularity theory for deterministic evolution equations given in terms of an analytic generator on UMD Banach spaces, including maximal regularity estimates. We propose a corresponding theory for stochastic parabolic evolution equation on UMD spaces. This requires however modification of the necessary tools from harmonic analysis such as stochastic maximal functions. So far such a theory only exists on L_p spaces for p>2.

• Deng Zhang (Shanghai Jiao Tong University) - The three dimensional stochastic Zakharov system
  
  This talk is concerned with recent results for the three dimensional stochastic Zakharov system in the energy space,  which couples a stochastic Schrödinger equation driven by linear multiplicative noise and a stochastic wave equation with additive noise. We will show the well-posedness of this system up to the maximal existence time and provide a blow-up alternative. The global existence of solutions below the ground state is also derived. Furthermore, we present a noise regularization result on finite time blowup before any given time. Two main ingredients of the proof are the refined rescaling approach and the normal form method. In contrast to the deterministic setting, our functional framework also incorporates local smoothing estimates,  which can control Schrödinger equations with derivative perturbations arising from the noise. Another key point for the noise regularization effect is a Strichartz estimate for the Schrödinger equation with a potential solving the free wave equation. This work is in joint with Sebastian Herr, Michael Röckner and Martin Spitz. 

Schedule