# SPDEs & friends

## May 31, June 1 and 2, 2021

Sessions will be hosted via Zoom, with Gather.town breakout rooms. We suggest that you download the Gather.town app (Windows, macOS).

Sessions will be hosted via Zoom, with Gather.town breakout rooms. We suggest that you download the Gather.town app (Windows, macOS).

Participation in the conference is free of charge, but we ask everyone to fill out a registration form for organizational purposes.

Links to recordings of the talks and the corresponding slides will apear below next to the abstracts. Make sure to also check out our YouTube Channel

Mon. 31/05 | Tue. 01/06 | Wed. 02/06 | |
---|---|---|---|

09:00–09:55 | Shirikyan / Tolomeo | Rhodes / Viklund | Mytnik / Derrida |

10:00–10:55 | Zambotti / Bellingeri | Holden / Yuan | Powell / Hernandez-Torres |

11:00–11:55 | Chandra / Chevyrev | Crisan / Coghi | Perkowski / Friz |

12:00–14:00 | Break | ||

14:00–14:55 | Khanin / Dunlap | — | Mattingly / Barashkov |

15:00–15:55 | Quastel / Orenshtein | Remenik / Knizel | Cerrai / Tapia |

16:00–16:55 | Khoshnevisan / Butkovsky | Staffilani / Fan | Krylov / Gerencsér |

17:00–17:55 | — | Bakhtin / Bates | — |

**Title:** Dynamic polymers: invariant measures and ordering by noise.

**Abstract:** Gibbs measures describing directed polymers in random potential are tightly related to the stochastic Burgers/KPZ/heat equations. One of the basic questions is: do the local interactions of the polymer chain with the random environment and with itself define the macroscopic state uniquely for these models? We establish and explore the connection of this problem with ergodic properties of an infinite-dimensional stochastic gradient flow. Joint work with Hong-Bin Chen and Liying Li.

**Title:** On the small mass limit for infinite-dimensional systems with state-dependent damping

**Abstract:** I will present a series of results on the asymptotic behavior, with respect to the small mass, of infinite-dimensional stochastic systems described by a damped waves equation perturbed by a Gaussian noise. I will consider in particular the case when the friction coefficient depends on the position of the particles and I will show how things change drastically, compared to what happens in the case of constant friction.

**Title:** Stochastic Quantization of Yang Mills

**Abstract:** I will result some recent work along with work in progress on the construction of a stochastic dynamic and a corresponding state space that in principle should have the Yang Mills Euclidean quantum field theory as its invariant measure. This is joint work with Ilya Chevyrev, Martin Hairer, and Hao Shen.

**Title:** Well-posedness Properties for a Stochastic Rotating Shallow Water Model

**Abstract:** The rotating shallow water (RSW) equations describe the evolution of a compressible rotating fluid below a free surface. The typical vertical length scale is assumed to be much smaller than the horizontal one, hence the shallow aspect. The RSW equations are a simplification of the primitive equations which are the equations of choice for modelling atmospheric and oceanic dynamics. In this talk, I will present some well-posedness properties of a viscous rotating shallow water system. The system is stochastically perturbed in such a way that two key properties of its deterministic counterpart are preserved. First, it retains the characterisation of its dynamics as the critical path of a variational problem. In this case, the corresponding action function is stochastically perturbed. Secondly, it satisfies the classical Kelvin circulation theorem. The introduction of stochasticity replaces the effects of the unresolved scales. The stochastic RSW equations are shown to admit a unique maximal strong solution in a suitably chosen Sobolev space which depends continuously on the initial datum. The maximal stopping time up to which the solution exist is shown to be strictly positive and, for sufficiently small initial datum, the solution is shown global in time with positive probability. This is joint work with Dr Oana Lang (Imperial College London) and forms part of the ERC Synergy project "Stochastic transport in upper ocean dynamics" (https://www.imperial.ac.uk/ocean-dynamics-synergy/)

**Title:** Integrability of the Schramm-Loewner evolution via conformal welding of random surfaces

**Abstract:** The Schramm-Loewner evolution is a one-parameter family of random fractal curves which describe the scaling limit of statistical physics models. We derive an explicit formula for the moments of the derivative of a particular uniformizing conformal map associated with an SLE. The problems is hard to approach via classical Ito calculus methods, and our proof relies instead on conformal welding of Liouville quantum gravity surfaces along with integrability results from Liouville conformal field theory. Joint work with Morris Ang and Xin Sun.

**Title:** On Stationary Solutions to the Stochastic Heat Equation

**Abstract:** I plan to discuss the problem of uniqueness of global solutions to the random Hamilton-Jacobi equation. I will formulate several conjectures and present results supporting them. Then I will discuss a new uniqueness result for the Stochastic Heat equation in the regime of weak disorder.

**Title:** Phase Analysis of a Family of Stochastic Reaction-Diffusion Equations

**Abstract:** We consider a wide family of reaction-diffusion equations that are forced with multiplicative space-time white noise, and show that if the level of the noise is sufficiently high then the resulting SPDE has a unique invariant measure. By contrast, we prove also that when the level of the noise is sufficiently low, then there are infinitely many invariant measures. In that case, we prove that the collection of all invariant measures is a line segment; that is, there are two extreme points. Time permitting, we will say a few thing about the two extremal invariant measures as well in the low-noise case. The phase picture that is described here was predicted in an earlier work of Zimmerman et al (2000).

This is based on joint work with Carl Mueller (University of Rochester, USA), Kunwoo Kim (POSTECH, S Korea), and Shang-Yuan Shiu (National Central University, Taiwan).

**Title:** On diffusion processes with drift in a Morrey class containing \(L_{d+2}\).

**Abstract:** We present new conditions on the drift of the Morrey type with mixed norms allowing us to obtain Aleksandrov type estimates of potentials of time inhomogeneous diffusion processes in spaces with mixed norms and, for instance, in \(L_{d_0+1}\) with \(d_0 < d\).

**Title:** The Gaussian Structure of the Stochastically Forced Burgers Equation and related problems

**Abstract:** I will explain some recent results which show that the law of the stochastic burgers equation at a fixed time t is absolutely continuous with respect to the natural Gaussian measure on the spatial domain. The results will apply to forcing just up to the point where the roughness of the forcing corresponds to the classical KPZ equation in the Burgers setting. As one approaches this level of roughness, the equations must be understood in as a Singular SPDEs (in the sense of Hairer ). As such the construction helps illuminate the structure of the equation and makes clear in what sense we might call these equations “truly” elliptic in this infinite dimensional setting. I will also make comments connecting back to previous results on the 2D Naiver Stokes equation. This work is joint with Marco Romito and Langxuan Su and builds on work with Andrea Watkins Hairston.

**Title:** On the speed of a front for stochastic reaction-diffusion equations with non-Lipschitz drift

**Abstract:** We study the asymptotic speed of a random front for solutions to stochastic reaction-diffusion equations with non-Lipschitz drift and Wright-Fisher noise proportional to \(\sigma\). Under some conditions on the drift, we show the existence of the speed of the front and derive its asymptotics depending on \(\sigma\).

This talk is based on joint works with C. Mueller, L. Ryzhik, C. Barnes and Z. Sun

**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.

**Title:** Brownian half-plane excursions, CLE_{4} and critical Liouville quantum gravity

**Abstract:** I will discuss a coupling between a Brownian excursion in the upper half plane and an exploration of nested CLE_{4} loops in the unit disk. In this coupling, the CLE_{4} is drawn on top of an independent “critical Liouville quantum gravity surface” known as a quantum disk. This is based on a forthcoming joint work with Juhan Aru, Nina Holden and Xin Sun, and describes the analogue of Duplantier-Miller-Sheffield’s “mating-of-trees correspondence” in the critical regime (κ=4).

**Title:** The KPZ fixed point Part I

**Abstract:** The 1-d KPZ universality class contains random interface growth models as well as random polymer free energies and driven diffusive systems. Various exact asymptotic distributions have been computed over the last two decades, some of them coming from random matrix theory. These are special cases of the strong coupling fixed point, which turns out to be a completely integrable Markov process: its transition probabilities are described by classical integrable PDE’s.

**Title:** The KPZ fixed point Part II

**Abstract:** The KPZ fixed point, the universal limit of all models in the KPZ universality class, is obtained as the scaling limit of the totally asymmetric simple exclusion process (TASEP). The main ingredient in the construction is an explicit formula for the distribution of TASEP in terms of the Fredholm determinant of a kernel which involves certain random walk hitting times. The formula has a natural scaling limit which defines the KPZ fixed point and can be used to show that its transition probabilities are integrable.

**Title:** Gluing random surfaces: conformal bootstrap in Liouville theory via Segal’s axioms

**Abstract:** The law of Markov processes indexed by time (the real line) take a simple form when expressed in terms of the action of the associated semigroup. The generalization of this question to the case when the process is indexed by higher dimensional manifolds is more intricate and this question is particularly relevant in the study of quantum field theories.
A general proposal was formalized by G. Segal in the eighties in this direction. Yet, concrete examples of QFTs where the Segal axioms are indeed valid are extremely limited (beyond trivial cases). We treat here the case of a specific Conformal Field Theory (CFT), called the Liouville theory which is a probabilistic model of 2D random surfaces. The outcome is the validity of the conformal bootstrap, i.e. a bridge between probability and representation theory: correlation functions are expressed in terms of (universal) holomorphic functions of the moduli parameters of the Riemann surface, called conformal blocks which have a strong representation theoretical content, and the structure constants of the CFT, here the DOZZ formula.
Conformal bootstrap was conjectured in physics in the eighties to be the universal structure of CFTs and Liouville theory is perhaps the first non trivial example where it can be shown to hold mathematically. This talk will be introductory to these topics.

**Title:** On almost sure well-posedness for certain dispersive PDE

**Abstract:** In this talk we summarize some of the many almost sure well-posedness results proved in recent years for dispersive equations. This study goes back to the work of Bourgain on invariant Gibbs measures and continued with the applications and evolution of his original ideas to address the question of local and global well-posedness for equations that are in a sense “supercritical”. If time permits we will also present some results for stochastic NLS equations, a direction of research started by de Bouard and Debussche.

**Title:** Mixing for PDEs with degenerate noise: an overview and open problems

**Abstract:** We discuss some results about uniqueness and stability of a stationary measure for randomly forced PDEs arising in fluid dynamics. Two different scenarios expressed in terms of controllability properties of the associated deterministic problem will be presented. We show, in particular, how they influence the choice of admissible random forces. We also formulate some open problems in the field.

**Title:** Weighted norms and a priori estimates for rough paths

**Abstract:** We introduce weighted norms on controlled paths and prove some a priori estimates for solutions to rough differential equations. This method seems particularly effective to handle equations with Lipschitz but unbounded non-linearities. Joint work with F. Caravenna and M. Gubinelli.