Short talks

Two sessions of flash talks will take place during the summer school. The first will take place Monday 7 July, 16:00-18:00 and the second will take place Thursday 10 July, 18:00-18:30. The talks will be around 5-7 minutes long, with a couple of minutes for questions. See below for details.

Monday 7 July

Time: 16:00
Speaker: Quentin Moulard
Title: The Stochastic Burgers Equation in the critical dimension 2
Abstract: The Stochastic Burgers Equation (SBE) is a singular SPDE which was introduced to model the fluctuations of stochastic driven diffusive systems with a conserved scalar quantity, such as the ASEP. In space dimension 2, the SBE is a critical SPDE and it was conjectured by physicists that the diffusivity diverges as $log t to the power two thirds$ for large time. I will present a recent result, in joint work with Giuseppe Cannizzaro and Fabio Toninelli, which not only proves this conjecture but also shows that, under a super-diffusive space-time rescaling, the SBE converges to a stochastic heat equation (with additive noise).

Time: 16:10
Speaker: Lorenzo Facciaroni
Title: Para-Markov chains and related non-local equations
Abstract: In the last decades Markov chains and semi-Markov chains have found applications in several fields. We here go beyond the semi-Markov setting, by defining some non-Markovian chains whose waiting times are assumed to be stochastically dependent. This creates a long memory tail in the evolution, unlike what happens for semi-Markov processes. As a special case, we study a particular counting process which extends the fractional Poisson process.

Time: 16:20
Speaker: Davide Papapicco
Title: A numerical estimate of escape rates
Abstract: Given an overdamped particle in a metastable scalar potential, whose evolution obeys Langevin dynamics, there will be a non-zero probability of it to cross critical thresholds and visits multiple basins of stabilities. These escapes are, in the low-noise limit, rare and quantified by Kramer's approximation. A novel adaptation of this setup is recently being involved in the theory of tipping points in dynamical systems. We will present a numerical method that can approximate these tipping points based on timeseries data alone by leveraging the relationship between the escape rate and the scalar potential.

Time: 16:30
Speaker: Fabio Bugini
Title: (Non)linear rough Fokker-Planck equations
Abstract: (Non)linear rough Fokker–Planck equations are measure-valued PDEs driven by a rough path. They model, for example, the evolution of probability distributions in (McKean–Vlasov) stochastic dynamics under rough common noise. We present some existence and uniqueness results, combining rough path theory, stochastic sewing techniques (and Lions' differential calculus on Wasserstein spaces). This is ongoing work with Peter K. Friz and Wilhelm Stannat.

Time: 16:40

Speaker: Juan Pablo Chávez Ochoa

Title: Brownian motion with asymptotically oblique reflection in unbounded domains: from subexponential to uniform ergodicity.

Abstract: We study the behaviour of a continuous diffusion on a generalised muti-dimensional parabolic domain with oblique asymptotic reflection towards the origin at the boundary. Based on the shape of the domain, we prove phase transitions between subexponential, exponential and uniform ergodicity, and we give upper and lower decay bounds on the tails of the return times to compact sets and the invariant distribution. Furthermore, we prove that on the uniformly ergodic case, the diffusion SDE admits a weak solution entering from infinity, and we estimate its speed in terms of the shape of the domain.

Time: 16:50
Speaker: Anna Donadini
Title: Noise sensitivity
Abstract: In this talk, we will describe the concept of noise sensitivity, a notion that has deep implications in statistical physics and was first introduced in 1998 by Benjamini, Kalai, and Schramm in the context of Boolean functions.

Time: 17:00
Speaker: Ethan Baker
Title: White Noise and Newtonian Limits for the Generalised Relativistic Langevin Equation
Abstract: The Langevin equation (LE) describes the motion of a particle under the influence of an external potential according to Newton’s second law with a random noise, whereas the generalised Langevin equation (GLE) is a non-Markovian extension of LE. Recently, relativistic counterparts to these equations, which describe such motion of a particle with relativistic kinetic energy, have been considered in the literature. We will introduce these equations, as well as state the Markovian formulation of the GLE and generalised relativistic Langevin equation, and describe the derivations of the Langevin equations from classical and relativistic Newtonian systems.

Time: 17:10
Speaker: Harry Giles
Title: Diffusivity of Glauber dynamics for dimers
Abstract: In this talk, I will introduce a model of discrete surface growth in 2+1 dimensions. We can show that information spreads through the system at a diffusive rate, which leads one to conjecture that the model lies in the Edwards-Wilkinson class.

Thursday 10 July

Time: 18:00

Speaker: Da Li

Title: Weak coupling limit for polynomial stochastic Burgers equations in 2d.
Abstract: We explore the weak coupling limit for stochastic Burgers type equation in critical dimension, and show that it is given by a Gaussian stochastic heat equation, with renormalised coefficient depending only on the second order Hermite polynomial of the nonlinearity. We use the approach of Cannizzaro, Gubinelli and Toninelli (2024), who treat the case of quadratic nonlinearities, and we extend it to polynomial nonlinearities. In that sense, we extend the weak universality of the KPZ equation shown by Hairer and Quastel (2018) to the two dimensional generalized stochastic Burgers equation. A key new ingredient is the graph notation for the generator. This enables us to obtain uniform estimates for the generator. This is joint work with Nicolas Perkowski.

Time: 18:10
Speaker: Jack Piazza
Title: Stochastic conformal flows in even dimensions
Abstract: We define two stochastic analogs of a geometric flow on even-dimensional manifolds called Q-curvature flow, and use the theory of Dirichlet forms to construct weak solutions to both. This generalizes the stochastic Ricci flow previously studied by Dubédat and Shen. We also discuss the connection between these flows and the even-dimensional Polyakov-Liouville measures recently defined by Dello Schiavo, Herry, Kopfer, and Sturm.

Time: 18:20
Speaker: Rohan Shiatis
Title: The integrable snake model
Abstract: On the integer lattice $Z squared$ , a "pure snake configuration" is a permutation on the vertices containing no two-cycles and such that every vertex is mapped to either itself, the vertex to the right, the vertex above or the vertex below. As such, a pure snake configuration admits an interpretation as a collection of snaking, non-intersecting paths on $Z squared$ . Pure snake configurations are a generalisation of lozenge tilings, which are in natural correspondence with paths that only travel right or up. We consider a partition function on a finite version of this model and see the probabilistic properties of random pure snake configurations chosen according to their contribution to this partition function. Under a suitable weighting, the model is integrable in the sense that we have access to explicit formulas for its partition function and correlation function. We will note the integrable and determinantal structure of this model, even through its various scaling limits, and realise a traffic representation of ASEP on the ring.