# 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).

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

# Speakers & Discussants

• Yuri BAKHTIN
• Sandra CERRAI
• Ajay CHANDRA
• Dan CRISAN
• Nina HOLDEN
• Konstantin KHANIN
• Davar KHOSHNEVISAN
• Nicolai KRYLOV
• Jonathan MATTINGLY
• Leonid MYTNIK
• Nicolas PERKOWSKI
• Ellen POWELL
• Jeremy QUASTEL
• Daniel REMENIK
• Armen SHIRIKYAN
• Gigliola STAFFILANI
• Lorenzo ZAMBOTTI

## Organizing Committee

• Oleg BUTKOVSKY (WIAS)
• Peter K. FRIZ (WIAS/TU Berlin)
• Nikolas TAPIA (WIAS/TU Berlin)

# Schedule

All times CET (GMT+2)
Mon. 31/05Tue. 01/06Wed. 02/06
09:00–09:55TalkTalkTalk
10:00–10:55ZambottiHoldenPowell
11:00–11:55TalkCrisanTalk
12:00–14:00Break
14:00–14:55TalkTalkTalk
15:00–15:55TalkTalkTalk
16:00–16:55TalkStaffilaniKrylov

# Abstracts

## Yuri BAKHTIN

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.

## Dan CRISAN

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/)

## Nicolai KRYLOV

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 $$d_0 < d$$.

## Ellen POWELL

Title: Brownian half-plane excursions, CLE4 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 CLE4 loops in the unit disk. In this coupling, the CLE4 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).

## Gigliola STAFFILANI

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.