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

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

The workshop will be streamed via Zoom. If you are interested in attending online please contact M. Gubinelli via this form. 

Minicourses

• Mario Maurelli (University of Pisa) - Regularisation by noise
  
  We say that an ordinary or partial differential equation is regularized by noise if the addition of a suitable noise term restores well-posedness or improves regularity of the equation. Regularization by noise is by now well understood for ODEs and linear transport-type PDEs, but it is less understood for nonlinear PDEs like Euler and Navier-Stokes equations, with many open questions.
  In this minicourse, after a short introduction to the topic, we focus our attention on the effect of a particular transport-type noise, which is divergence-free, Gaussian, white in time and poorly correlated in space (nonsmooth Kraichnan noise). The associated linear transport model, introduced by Kraichnan, has been widely studied due to its peculiar features, like spontaneous stochasticity. In the case of incompressible 2D Euler equations, we show that the addition of nonsmoooth Kraichnan noise improves significantly the well-posedness theory, for example bringing uniqueness among finite enstrophy solutions. In the last part of the minicourse, we show further regularization properties of the Kraichnan noise, both in the linear case and for generalized surface quasi-geostrophic equation.
  Based on joint works with Marco Bagnara, Michele Coghi, Lucio Galeati, Francesco Grotto.

• Felix Otto (MPI Leipzig) - A continuity argument in regularity structures
  
  Recently, an approach to regularity structures that is more intrinsic in the sense that the model parameterizes (only) the solution manifold has been proposed, and its stochastic estimate established. In this talk, we address the actual solution theory for a boundary value problem, beyond a priori estimates. As a side effect of the sparser model, the usual “Duhamel” fixed point formulation cannot be encoded. We in- stead resort to a classical continuity method in the nonlinearity. We implement this in the simplest case of periodic boundary conditions and a $\phi^4$-type model. This is joint work with Lucas Broux and Rhys Steele.

• Hao Shen (University of Wisconsin - Madison) - Stochastic quantization of Yang-Mills

In these lectures we will discuss our recent work with Ilya Chevyrev, which shows that the Yang-Mills measure on 2D torus is the unique invariant measure for the Yang-Mills Langevin dynamic constructed by Chandra-Chevyrev-Hairer-S. To implement an argument of Bourgain we prove the continuum limit of lattice gauge theories using discrete regularity structures. We also develop a novel way to identify the gauge invariant limit. The tentative plan for the lectures is:

Lecture 1: 2D Yang-Mills models and main results;

Lecture 2: From lattice to continuum;

Lecture 3: Invariant measure.

• Nikolaos Zygouras (University of Warwick) - Critical SPDEs

Thanks to the theories of Paracontrolled Distributions,  Regularity structures and Renormalisation we now have a complete theory of singular SPDEs, which are “sub-critical” . I will discuss recent efforts to approach the situation  of “critical” SPDEs and related statistical mechanics models. We will start with the two-dimensional Stochastic Heat Equation and the construction of the Critical 2d Stochastic Heat Flow (derived in joint works with Caravenna and Sun) and then describe the treatment of some non-linear examples such Allen-Cahn, KPZ and non-linear SHE derived in various works by various  groups. We will aim to sketch some of the methods which include multi-scale analysis, LIndeberg principles, martingale methods, chaos and Butcher expansions.

Invited Speakers

• Leonie Canet (Université Joseph Fourier Grenoble Alpes)  Space-time dependence of correlation functions in turbulence
  
  I will consider the stochastic Navier-Stokes and the Burgers-KPZ equations. I will show how (extended) symmetries can be exploited within the framework of the functional renormalisation group to obtain the spatio-temporal dependence of generic n-point velocity correlation functions in the limit of large wave-numbers. I will then compare these results with data from direct numerical simulations and experiments.
  
  

• Sandra Cerrai (University of Maryland) - Asymptotic problems for SPDEs with constraints

I will first review some results obtained in collaboration with Z. Brzezniak about the small mass limit for stochastic wave equations subject to suitable functional constraints. Then, I will describe some more recent results obtained in collaboration with M. Xie in the case of systems of constrained SPDEs.

• Martin Hairer (EPFL) - Invariant measure for infinite-energy stochastic Navier-Stokes equations
  
  We consider the 2D Navier-Stokes equations driven by noise that is “essentially” (but not exactly) space-time white. Despite solutions having infinite energy, we show that they admit an invariant measure with stretched exponential tail bounds.
  
  

• Dejun Luo (Chinese Academy of Sciences) -  Some recent results on the Boussinesq hypothesis

The Boussinesq hypothesis states that turbulent small scales are dissipative on the mean part of fluid. In the vorticity formulation of 3D fluid dynamics equations, small-scale turbulent fluctuations can be modeled by suitable space-time noise involving both transport and stretching parts. Our purpose is to study the effects of such noises on the fluid equations. We first recall a previous result which shows that the Leray-projected transport noise can suppress the blow-up of vorticity with large probability. Then we introduce more recent attempts in covering the stretching part of the noise. The talk is mainly based on joint works with Professor Flandoli.

• Tadahiro Oh (University of Edinburgh) -  Fourier restriction norm method adapted to controlled paths
  
  Over the last decade, there has been a significant development in the study of stochastic dispersive PDEs, broadly interpreted with random initial data and/or additive stochastic forcing, where the difficulty comes from roughness in spatial regularity.  In this talk, I consider well-posedness of stochastic dispersive PDEs with multilpicative noises, whose Ito solutions were contructed in 80's for the wave case and in 90's for the Schrödinger case, and present the first results on pathwise well-posedness for stochastic nonlinear wave equations (SNLW) and stochastic nonlinear Schrödinger equations (SNLS).
  
  The main challenge of this problem comes from the deficiency of temporal regularities.  We overcome this issue by considering operator-valued rough paths adapted to the wave/Schrödinger flow in the spirit of Gubinelli-Tindel (2010) and implementing the Fourier restriction norm method (with the V^p-spaces and their preduals called U^p-spaces due to Koch-Tataru (2007)) for the setting of controlled paths.
  
  

• Szymon Peszat (Jagiellonian University) - Differentiability of transition semigroup of generalized Ornstein-Uhlenbeck proces; probabilistic approach 

Let $(P_t)$ be the transition semigroup on the space $B_b(E)$ of bounded measurable functions on a Banach space $E$, of the process defined by the linear equation with additive noise
$$
d X= \left(AX + a\right)d t + Bd W, \qquad X(0)=x\in E.
$$
Our goal is to establish gradient and hessian  formulae
\begin{align*}
\nabla P_t\phi (x)[y] &=\mathbb{E}\,\phi(X(t))Y(t,y), \\
\nabla ^2 P_t\phi (x)[y_1,y_2]&=\mathbb{E}\,\phi(X(t))Y(t,y_1,y_2),
\end{align*}
as well as estimates
\begin{align*}
\|\nabla P_t\phi(x)\|_{L(E)}&\le \rho(t)\|\phi\|_{B_b(E)},\\
\|\nabla ^2P_t\phi(x)\|_{L(E\times E)}&\le \tilde \rho(t)\|\phi\|_{B_b(E)}, 
\end{align*}
with properly chosen rates $\rho$ and $\tilde \rho$. Our next goal is to study Liouville type theorem for the corresponding generator.

• Fabio Toninelli (TU Vienna) - The 2-dimensional Stochastic Burgers equation: from weak to strong coupling

The stochastic Burgers equation in dimension d=2 is singular and critical. While being formally diffusively scale invariant, its  diffusion coefficient diverges as (log t)^2/3 for t->infty. To tame the divergence, one can either send the coupling constant logarithmically to zero with the regularization cutoff (weak coupling limit), or correct logarithmicaly the diffusive rescaling (strong coupling limit), or a combination of the two. In the whole crossover window ranging from weak to strong coupling, we prove a Gaussian large-scale limit for the equation. Based on ongoing work with G. Cannizzaro, and Q. Moulard

• Rongchan Zhu (Beijing Institute of Technology) - Langevin dynamics of lattice Yang-Mills-Higgs and applications
  
  In this talk I will talk about the Langevin dynamics of various lattice formulations of the Yang-Mills-Higgs model,  where the Higgs component takes values in $\mathbb{R}^N$, $\mathbb{S}^{N-1}$ or a Lie group.  We prove the exponential ergodicity of the dynamics on the whole lattice via functional inequalities. As an application, we establish that correlations for a broad range of observables decay exponentially. Specifically, the infinite volume measure exhibits a strictly positive mass gap under strong coupling conditions. Moreover, appropriately rescaled observables exhibit factorized correlations in the large $N$ limit when the state space is compact.
   

• Xiangchan Zhu (Chinese Academy of Sciences) - Global solutions to a class of singular SPDE
  
  In this talk I will recall recent progress in global solution theory for singular SPDE. In particular I will give two examples. First we will revisit the problem of global well-posedness for the generalized parabolic Anderson model within the framework of paracontrolled calculus [GIP15]. Second, we establish the existence of infinitely many non-Gaussian probabilistically strong solutions as well as ergodic stationary solutions to surface quasi-geostrophic equation perturbed by derivatives of space-time white noise. 

Schedule