New developments and challenges in
Stochastic Partial Differential Equations

Workshop 1


17-21 June 2024

This is the first 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 at the following coordinates:

https://york-ac-uk.zoom.us/j/92358382806 - Meeting ID: 923 5838 2806  - Please contact zb500@york.ac.uk for the passcode.

Minicourses

First, the basics of stochastic Navier-Stokes with spatially regular additive forcing will be recalled. Higher order drift conditions capable of controlling exponential time integrals will then be derived. With these hand, conjectures related to chaos and turbulence will be discussed. Finally, results on Lagrangian chaos and the exponential mixing of passive scalars will be discussed. 


We present a construction of the Gibbs measure of the fractional $\Phi^4$ model of Euclidean quantum field theory in three-dimensions. The measure is obtained as a perturbation of the Gaussian measure with covariance given by the inverse of a fractional Laplacian. Since the Gaussian measure is supported in the space of Schwartz distributions and the quartic interaction potential of the model involves pointwise products, to construct the measure it is necessary to solve the so-called renormalization problem. To this end, we study the stochastic quantization equation, which is a nonlinear parabolic PDE driven by the white noise. We prove a certain a priori estimate for solutions of this equation using the flow equation approach to singular stochastic PDEs and the maximum principle. We consider the entire range of powers of the fractional Laplacian for which the model is subcritical (i.e. super-renormalizable). Based on a joint work with M. Gubinelli and P. Rinaldi.

How predictable are turbulent flows? We discuss the fluctuating hydrodynamics of Landau-Lifschitz, which describes the effect of thermal noise in fluid flows by including stochastic terms into the Navier-Stokes equations. Developed turbulence can be seen as a limit where both the viscous and noise terms disappear, leading to deterministic Euler equations. However, numerical evidence suggests that the solutions remain stochastic in this limit, even though they solve a formally deterministic system. This phenomenon is called spontaneous stochasticity. We discuss mathematical models and methods by which spontaneous stochasticity and its properties can be approached.


Invited speakers

We first discuss the separation of particles into sparse and dense phases in Glauber-Kawasaki dynamics of non-gradient type and show that the phase-separating interface evolves according to the anisotropic curvature flow.  Then, we study the fluctuation of the interface in a simple situation and derive a linear SPDE via the Boltzmann-Gibbs principle.  We also discuss heuristically the derivation of nonlinear SPDEs.  The talk is partly based on joint work with Chenlin Gu, Han Wang (Tsinghua University), Hyunjoon Park (Meiji University), Claudio Landim (IMPA), Sunder Sethuraman (University of Arizona).  The results for the first part can be found in arXiv:2404.18364, arXiv:2404.12234, arXiv:2403.01732.

We revisit a result due to Robinson and Sadowski (2009), who first showed a.e. uniqueness of Lagrangian trajectories for admissible weak solutions to $3$D Navier-Stokes, for sufficiently regular $u_0$. We give an alternative proof, based on a newly established asymmetric Lusin-Lipschitz property of Leray solutions, exploited crucially in the arguments from Caravenna-Crippa (2021) and Brué-Colombo-De Lellis (2021). This approach is more robust, requiring no assumptions on $u_0$ and being applicable also to the stochastic characteristics of the system. Finally, if $u_0$ is regular (say $u_0\in H^{1/2}$), then we are able to exploit the diffusive behaviour of stochastic trajectories to further prove that, for any fixed $x_0\in\mathbb{R}^d$, path-by-path uniqueness for the SDE $d X_t = u(t,X_t) d t + d B_t, X|t=0 = x_0$.

Short-range correlation in noise generally dissipates over time, leading to dynamics that exhibit characteristics of independent noise. Here we focus on long-range dependence (LRD) noise, as LRD has been observed in data and is generally considered significant. The distinction between short-range and long-range dependence can be seen in the central limit theorems for a sequence of random variables. In the context of the stochastic heat equation and the KPZ equation, the correlation in the model with short-range correlation does not manifest in the fluctuations from their means. This study will discuss how this phenomenon differs when dealing with long-range dependent noise.


In this work we extend the paradifferential approach to quasilinear wave equations of Bahouri and Chemin to study quasilinear wave equations with distributional forcing. To achieve this, we combine their methods with the paracontrolled calculus of Gubinelli, Imkeller, and Perkowski. In two dimensions, we prove a well-posedness result for forcing with Hölder regularity better than -1/8 for an arbitrary initial data and for forcing with regularity better than -1 with restrictive initial data. This is a joint work with Nicolas Perkowski and Immanuel Zachhuber.

Inspired by problems from fluid dynamics, we introduce an approach to obtain lower bounds to Lyapunov exponents of stochastic PDEs. Our proof relies on the introduction of a Lyapunov functional for the projective process associated to the equation, based on the study of dynamics of energy level sets and on a notion of non-degeneracy of the noise that leads to high-frequency stochastic instability. We address some nonlinear problems. Joint works with A. Blessing , M. Hairer, and (in progress) with M. Hairer, S. Punshon-Smith and J. Yi.

The notion of weak Dirichlet process is the natural extension of the concept of semimartingale. Among the examples we find the following:
 1.Irregular Markov processes solutions of SDEs with distributional drift with jumps;
 2.Solutions of (even continuous) path-dependent SDEs with distributional drift;
 3.(Path-dependent) Bessel processes. Identification problem in BSDEs driven by random measure.
The talk puts the emphasis on a BSDE with distributional driver. The presentation covers joint work with E. Issoglio (Torino) and E. Bandini (Bologna).

Schedule