TITLES AND ABSTRACTS ------------------------------------------------------------------------------- Nils Berglund Renormalisation and sharp asymptotics for metatable transition times in stochastic Allen-Cahn equations We are interested in the Allen-Cahn-Chaffee-Infante equation on the d-dimensional torus, with weak additive space-time white noise. This equation provides a description of phase separation in an alloy. Its solution theory is equivalent to the solution theory of the Phi^4_d model from Euclidean quantum field theory, but its qualitative dynamics is quite different, as it shows metastability. In dimension d = 1, we obtain sharp asymptotics of Eyring-Kramers time for the transition time between the two deterministically stable states, using methods from potential theory. A similar result holds in dimension 2, but there a renormalisation procedure is needed to make the equation well-posed. Interestingly, the prefactor of the metastable transition time depends on the choice of renormalisation counterterm. In dimension 3, showing well-posedness of the renormalised equation requires sophisticated methods such as regularity structures, but the question of sharp asymptotics of transition times is still open. I will present some ongoing work aiming at settling that question, using algebraic methods to take track the combinatorics in renormalising Feynman diagrams. Based on joint works with Barbara Gentz, Giacomo Di Gesu', Hendrik Weber, and Tom Klose. ------------------------------------------------------------------------------- Federico Cornalba Numerical aspects for stochastic PDEs of Fluctuating Hydrodynamics Interacting particle systems can be often be viewed as (exact) solutions to (singular) stochastic PDEs. Despite these SPDEs being singular, their numerical approximations offer computational efficiency (over of the particle systems) when it comes to estimating fluctuations in relevant scaling regimes. We will detail this in the case of the Dean-Kawasaki model (based on joint works with Julian Fischer, Jonas Ingmanns, Claudia Raithel). We will then discuss some open problems and aspects of interest concerning numerics for this class of SPDEs. ------------------------------------------------------------------------------- Benoit Dagallier Uniqueness of the invariant measure of the phi42 dynamics in infinite volume I will discuss the phi42 dynamics, a singular stochastic partial differential formally corresponding to the Langevin dynamics for the phi42 field theory model. The goal is to characterise infinite volume invariant measures for these dynamics. The associated phi42 theory is known to undergo phase transitions as parameters in the model are varied, with the field transitioning from short-distance correlations (in the sense that the susceptibility of the measure defined on a finite box is bounded uniformly on the box size) to long-distance correlations. One expects this behaviour to be reflected in the dynamics, with a unique infinite volume invariant measure in absence of phase transition and possibly more than one in the strongly correlated regime. We prove that this picture is indeed correct in the sense that there is a unique invariant measure for the dynamics whenever the susceptibility is finite. This is done by adapting to the field-theory setting the Holley-Stroock-Zegarlinski approach to uniqueness for statistical mechanics models. This approach is based on a volume-independent bound on the log-Sobolev constant of the associated dynamics, together with crude, model-independent bounds. In the phi42 case the log-Sobolev has recently been established, but substantial work is required to adapt the other parts of the argument. The talk is based on joint work in progress with R. Bauerschmidt and H. Weber. ------------------------------------------------------------------------------- Arnaud Debussche From correlated to white transport noise in fluid models Stochastic fluid model with transport noise are popular, the transport noise models unresolved small scales. The main assumption in these models is a very strong separation of scales allowing this representation of small scales by white - ie fully decorrelated - noise. It is therefore natural to investigate whether these models are limits of models with correlated noises. Also, an advantage of correlated noises is that they allow classical calculus. In particular, it allows to revisit the derivation of stochastic models from variational principle and allows to derive equation for the evolution of the noise components. The advantage of having such equation is that in most works, the noise components are considered as given and stationary with respect to time which is non realistic. Coupling stochastic fluid models with these gives a more realistic systems. -------------------------------------------------------------------------------- Davide Gabrielli Solvable Stationary Non Equilibrium States Boundary-driven stochastic lattice gases are simple but effective models for non-equilibrium statistical mechanics. Apart from special cases, such as the zero-range model where the invariant measure is always of product type, they generally exhibit long-range correlations in the steady state. I will discuss some examples of models for which it is possible to provide simple representations of the invariant measures by enlarging the state space. I will focus on the boundary-driven Kipnis Marchioro Presutti model, also showing the relation with the structure of rate functionals of large deviations. Joint work with Pablo Ferrari and Anna de Masi. ------------------------------------------------------------------------------- Benjamin Gess Gradient flow structures and large deviations for porous media equations While the derivation of nonlinear but uniformly parabolic equations from microscopic dynamics, fluctuations around these limits, and the corresponding canonical choice of a gradient flow structure are now well-understood, less is known for equations with either degenerate, or unbounded diffusivity. Specifically, for the model case of the porous medium equation (PME), multiple gradient flow structures have been identified since the works of Brézis and Otto; however, it remains unclear which, if any, are thermodynamic in nature, meaning that they arise through the large deviations of a microscopic model. In this talk, to demonstrate that the (formal) geometric picture we obtain is thermodynamic, we examine a rescaling of the zero-range process (ZRP) that converges to the PME and prove a full large deviations principle. The proof of this result is complicated by the degeneracy and unboundedness of the diffusivity. We then discuss how the large deviations rigorously identify a gradient flow structure for the PME. ------------------------------------------------------------------------------- Patricia Goncalves TBA ------------------------------------------------------------------------------- Francesco Grotto Stochastic transport models of turbulence The advection of passive scalar and vector fields by random velocity fields can be used as a phenomenological model of turbulent flows. When the velocity field is delta-correlated these are exactly solvable models for which physically relevant quantities can be computed analytically, and that are able to reproduce phenomena such as anomalous dissipation of energy if an appropriate power law spectrum is prescribed for the space covariance (Kraichnan's model). Nevertheless, a rigorous pathwise description of solutions in those cases is not trivial. I will present recent results concerning the anomalous regularization of solutions with distribution-valued initial data, due to the irregular and stochastic nature of the advecting velocity field. Based on joint works with M. Bagnara, L. Galeati, M. Maurelli. ------------------------------------------------------------------------------- Xue-Mei Li Fluctuations of Stochastic Heat Equation and KPZ equation In this talk, we explore the stochastic heat equation and the KPZ equation, subject to space time Gaussian noise with long-range spatial dependence. Our focus is on the fluctuation problem associated with diffusively scaled solutions from their average. While the behaviour of compactly supported correlations - typically known to dissipate at large scales - is well-documented, our research shifts to examining long-range dependent noise. We investigate whether this dependence is maintained in the large-scale scaling limit and identify the limit. ------------------------------------------------------------------------------- Kirone Mallick An exact solution of the macroscopic fluctuation theory for symmetric exclusion Gianni Jona-Lasinio and his collaborators have proposed in the early 2000’s a non-linear action functional that encodes the macroscopic fluctu- ations and the large deviations for a wide class of diffusive systems out of equilibrium. In this Macroscopic Fluctuation Theory (MFT), optimal fluc- tuations far from equilibrium can be found, at a coarse-grained scale, by solving two coupled non-linear hydrodynamic equations. In this talk, we shall show that the MFT equations for the symmetric exclusion process are classically integrable and can be solved with the help of the inverse scattering method, originally developed to study solitons in the KdV or the NLS equations. This exact solution will allow us to calculate the large deviations of the current and the optimal profiles that generates a given fluctuation, both at initial and final times. This macroscopic solution matches previous results that were derived, by applying the Bethe Ansatz, at the microscopic level. ------------------------------------------------------------------------------- Vlad Margarint A bridge between Random Matrix Theory and Schramm-Loewner Evolutions Theory I will describe a newly introduced toolbox that connects two areas of Probability Theory: Schramm-Loewner Evolutions and Random Matrix Theory. This machinery opens new avenues of research that allow the use of techniques from one field to another. One aspect of this research direction is centered in an interacting particle systems model, namely the Dyson Brownian motion. I will first describe the toolbox and then I will describe one of the recent applications. Then, I am to describe some of the open problems that emerge using this newly introduced toolbox. This is a joint work with A. Campbell and K. Luh. ------------------------------------------------------------------------------ Neill O'Connell Discrete Whittaker processes I will discuss a Markov chain on reverse plane partitions (of a given shape) which is closely related to the Toda lattice. This process has non-trivial Markovian projections and a unique entrance law starting from the reverse plane partition with all entries equal to plus infinity. I will also outline some connections with imaginary exponential functionals of Brownian motion, a random polymer model with purely imaginary disorder, interacting corner growth processes and discrete delta-Bose gas, and hitting probabilities for some low rank examples. ------------------------------------------------------------------------------- Max von Renesse TBA ------------------------------------------------------------------------------- Marco Romito About some notions of irregularity with application to the enhanced dissipation of rough shear flows In the first part of the talk we will discuss some notions of irregularity of functions and the associated problems of regularization by (noiseless) noise, starting from the seminal work of Catellier, Gubinelli (2016). In particular we will discuss a new notion, based on small ball estimates, directly connected to Besov regularity. The second part of the talk will focus on enhanced dissipation of passive scalars driven by very rough shear flows. Joint works with Leonardo Tolomeo (Edinburgh) and Leonardo Roveri (Pisa) ------------------------------------------------------------------------------- Karl-Theodor Sturm Conformally invariant random geometry in dimensions n>2 In this talk, I will present an approach to conformally invariant Liouville geometry in dimensions n>2. In particular, I will discuss * conformally invariant log-correlated Gaussian fields based on Paneitz and GJMS operators, * Liouville quantum gravity measures, Liouville Brownian motion on n-dim manifolds, * Polyakov-Liouville measure and conformal anomaly involving Branson’s Q-curvature. ------------------------------------------------------------------------------- Lorenzo Zambotti Reflected, Skew or Bessel SPDEs and stochastic sewing In this talk I would like to review old results on stochastic(partial) differential equations with coefficients dependent on the local times of the solution, and relate them with more recent results based on stochastic sewing techniques. ------------------------------------------------------------------------------