Stochastic Analysis Afternoon 2021

### Program

#### May 5, 2021 (Wednesday)

14:00-15:00 Rongchan Zhu (Beijing Institute of Technology, China)Large N limit of the O(N) linear sigma model via stochastic quantization
15:15-16:15 Ilya Chevyrev (University of Edinburgh, UK)Gauge fixing the Yang-Mills measure
16:30-17:30 Nicolas Perkowski (Freie Universität Berlin, Germany)Martingale problems for some singular SPDEs

#### May 6, 2021 (Thursday)

15:30-16:30 Andreas Eberle (University of Bonn, Germany) Hamiltonian Monte Carlo in high dimensions

16:45-17:45 Mateusz Majka (Heriot-Watt University, UK)Approximation of heavy-tailed distributions via stable-driven SDEs

Please note that all the times are indicated in the local time in Helsinki, Finland, that is, Eastern European summertime (EEST/UTC+3).

The talks are online in zoom and will be recorded. Click on the title to access to the recordings.

Organizers

Gerardo Barrera (gerardo.barreravargas@helsinki.fi)
Jonas Tölle (jonas.tolle@helsinki.fi)

#### Title and abstracts

Rongchan ZhuTitle: Large N limit of the O(N) linear sigma model via stochastic quantization

Abstract: In this talk we discuss large $N$ limits of a coupled system of $N$ interacting $\Phi^4$ equations posed over $\mathbb{T}^{d}$ for $d=1,2,3$, known as the $O(N)$ linear sigma model. Uniform in $N$ bounds on the dynamics are established, allowing us to show convergence to a mean-field singular SPDE, also proved to be globally well-posed. Moreover, we show tightness of the invariant measures in the large $N$ limit.

For large enough mass, they converge to the (massive) Gaussian free field, the unique invariant measure of the mean-field dynamics, at a rate of order $1/\sqrt{N}$ with respect to the Wasserstein distance. We also consider fluctuations and obtain tightness results for certain $O(N)$ invariant observables, along with an exact description of the limiting correlations in $d=1,2$.

This talk is based on joint work with Hao Shen, Scott Smith, and Xiangchan Zhu.

Ilya ChevyrevTitle: Gauge fixing the Yang-Mills measure

Abstract: The Yang-Mills measure, formally a probability measure on principal connections, is a quantum field theory which models particles carrying the fundamental forces. In this talk, I will present two recent approaches based on stochastic analysis to gauge-fixing the Yang-Mills measure in dimension 2, which is the only dimension in which a rigorous construction of the measure is currently known. The first approach is based on lattice approximations and an Uhlenbeck-type theorem which is applicable to distributional 1-forms. The second is based on stochastic quantisation, and relies on the theory of parabolic singular stochastic PDEs. I will discuss the link between these approaches, as well as the link to other constructions in the literature, and how a novel state space helps to make sense of the space of gauge orbits on which the measure is supported. Finally, I will discuss the difficulties in extending the construction to 3 dimensions.

Based partly on a joint work with A. Chandra, M. Hairer, and H. Shen.

Nicolas Perkowski Title: Martingale problems for some singular SPDEs
Abstract: Most techniques for solving singular SPDEs, such as regularity structures, are based on pathwise calculus. It would be interesting to study singular SPDEs from a more probabilistic perspective, for example via the martingale problem. In general, that is a too difficult problem at the moment, but there are some equations for which we can do this. I will explain the ideas on the example of the conservative stochastic Burgers equation and indicate how to extend the results to a larger class of equations that share a similar structure. A novelty of this approach is that it allows to prove (weak) well-posedness for some scaling critical singular SPDEs.

Based on works with Massimiliano Gubinelli and Lukas Gräfner.

Andreas EberleTitle: Hamiltonian Monte Carlo in high dimensions

Abstract: Hamiltonian Monte Carlo (HMC) is a Markov Chain Monte Carlo method that is widely used in applications. It is based on a combination of Hamiltonian dynamics and momentum randomizations. The Hamiltonian dynamics is discretized, and the discretization bias can either be taken into account (unadjusted HMC) or corrected by a Metropolis-Hastings accept-reject step (Metropolis adjusted HMC). Despite its empirical success, until a few years ago there have been almost no convergence bounds for the algorithm. This has changed in the last years where approaches to quantify convergence to equilibrium based on coupling, conductance and hypocoercivity have been developed. In this talk, I will present the coupling approach, and show how it can be used to obtain a rather precise understanding of the dimension dependence for unadjusted HMC in several high dimensional model classes. I will also discuss some open questions.

Mateusz MajkaTitle: Approximation of heavy-tailed distributions via stable-driven SDEs

Abstract: Constructions of numerous Markov Chain Monte Carlo algorithms for approximate sampling from Gibbs probability measures are based on the well-known fact that such measures are stationary distributions of ergodic stochastic differential equations (SDEs) driven by the Brownian motion. However, for some heavy-tailed distributions, it can be shown that the associated SDE is not exponentially ergodic and that related sampling algorithms may perform poorly. A natural idea that has recently been explored in the machine learning literature in this context is to make use of stochastic processes with heavy tails instead of the Brownian motion. In this talk, I will discuss how to study the problem of approximating heavy-tailed distributions via ergodic SDEs driven by symmetric (rotationally invariant) alpha-stable processes.