Tag - Stochastic DEs

Ilya Chevyrev: Pre-Lie algebras in stochastic PDEs

21st October, 2024

In this talk, I will discuss a general method to renormalize singular stochastic partial differential equations (SPDEs) using the theory of regularity structures. It turns out that, to derive the renormalized equation, one can employ a convenient multi-pre-Lie algebra. The pre-Lie products in this algebra are reminiscent of the pre-Lie product on the Grossman-Larson algebra of trees, but come with several important twists. For the renormalization of SPDEs, the important feature of this multi-pre-Lie algebra is that it is free in a certain sense.

Ilya Chevyrev: Invariant measure and universality of the 2D Yang-Mills Langevin dynamic, II

13th June, 2023

In this talk, I will present a recent work on the invariance of the 2D Yang-Mills measure for its Langevin dynamic. The Langevin dynamic both in 2D and 3D had previously been constructed in joint work with Chandra-Hairer-Shen, but it was an open problem to show the existence of an invariant measure even in 2D. In establishing this invariance, we follow Bourgain’s invariant measure argument by taking lattice approximations, but with several twists. An important one, which I will focus on, is that the approximating invariant measures require gauge-fixing, which we achieve by developing a rough version of Uhlenbeck compactness combined with rough path estimates of random walks. I will also present several corollaries of our main result, including a representation of the YM measure as a perturbation of the Gaussian free field, and a new universality result for its discrete approximations.

Nicolas Perkowski: Energy solutions and generators for singular SPDEs

13th June, 2023

I will discuss how to use tools from Gaussian analysis and operator semigroups together with some commutator estimates to construct Markov semigroups for some singular SPDEs. This yields in particular uniqueness for Goncalves-Jara-Gubinelli type energy solutions. The method applies to some critical equations and, in finite dimensions, even for some supercritical equations. In infinite dimensions we get Markov semigroups for supercritical equations but we lack a uniqueness result for supercritical energy solutions in infinite dimensions. The main SPDE examples where this works are of Burgers type: quadratic, divergence-free nonlinearity and Gaussian quasi-invariant measure.

Hao Shen: Invariant measure and universality of the 2D Yang-Mills Langevin dynamic, I

13th June, 2023

In an earlier work with Chandra, Chevyrev and Hairer, we constructed the local solution to the stochastic Yang-Mills equation on 2D torus, which was shown to have gauge covariance property and thus induces a Markov process on a singular space of gauge equivalent classes. In this talk, we discuss a more recent work with Chevyrev, where we consider the Langevin dynamics of a large class of lattice gauge theories on 2D torus, and prove that these discrete dynamics all converge to the same limiting dynamic. A novel step in the argument is a geometric way to identify the limit using Wilson loops. This universality of the dynamics is crucial for obtaining a sequence of important results for 2D Yang-Mills, including for instance the invariance of the 2D Yang-Mills measure for its Langevin dynamic, which will be discussed by Ilya Chevyrev.

Martin Hairer: The role of symmetry in renormalisation

13th June, 2023

There are several interesting situations where the solutions to singular SPDEs exhibit a symmetry at a formal level that could in principle be broken by the renormalization procedure required to define them. We’ll discuss a relatively simple argument showing that, in many cases, the renormalization can be chosen in such a way that the symmetry does indeed hold and we’ll apply it to the stochastic quantization of the 3D Yang-Mills theory.

Nikos Zygouras: SPDEs at the critical dimension

13th June, 2023

I will make an overview of the progress on treating SPDEs at the critical dimension, the current status and further challenges. Examples will include stochastic heat equations and a more recent Allen-Cahn.

Luca Fresta: The forward-backward SDE for subcritical Euclidean fermionic field theories

12th June, 2023

In this talk, I will describe a synergy between the renormalization group (RG) in the form of Polchinski's equation and the stochastic quantisation in the form of a forward-backward stochastic differential equation (FBSDE). This approach can be used for constructing subcritical Grassmann Gibbsian measures and is based on controlling the solution of the FBSDE by means of a flow equation with respect to a scale parameter. However, unlike the standard RG approach, we only need to solve Polchinski’s equation in an approximate way, resulting in a great simplification of the analysis.

Xiangchan Zhu: A class of singular SPDEs via convex integration

12th June, 2023

In this talk I will talk about our recent work on a class of singular SPDEs via convex integration method. In particular, we establish global-in-time existence and non-uniqueness of probabilistically strong solutions to the three dimensional Navier–Stokes system driven by space-time white noise. In this setting, solutions are expected to have space regularity at most −1/2 − κ for any κ > 0. Consequently, the convective term is ill-defined analytically and probabilistic renormalization is required. Up to now, only local well-posedness has been known. With the help of paracontrolled calculus we decompose the system in a way which makes it amenable to convex integration. By a careful analysis of the regularity of each term, we develop an iterative procedure which yields global non-unique probabilistically strong paracontrolled solutions. Our result applies to any divergence free initial condition in L2 ∪ B−1+κ∞,∞, κ > 0, and implies also non-uniqueness in law. Finally I will show the existence, non-uniqueness, non-Gaussianity and non-unique ergodicity for singular quasi geostrophic equation in the critical and supercritical regime.

Jocelyne Bion Nadal: Approximation and calibration of laws of solutions to stochastic differential equations

4th September, 2018

In many situations where stochastic modeling is used, one desires to choose the coefficients of a stochastic differential equation which represents the reality as simply as possible. For example one desires to approximate a diffusion model with high complexity coefficients by a model within a class of simple diffusion models. To achieve this goal, we introduce a new Wasserstein type distance on the set of laws of solutions to d-dimensional stochastic differential equations.

This new distance W̃2 is defined similarly to the classical Wasserstein distance W̃2 but the set of couplings is restricted to the set of laws of solutions of 2d-dimensional stochastic differential equations. We prove that this new distance W̃2 metrizes the weak topology. Furthermore this distance W̃2 is characterized in terms of a stochastic control problem. In the case d = 1 we can construct an explicit solution. The multi-dimensional case, is more tricky and classical results do not apply to solve the HJB equation because of the degeneracy of the differential operator. Nevertheless, we prove that this HJB equation admits a regular solution.