Sarah-Jean Meyer
The Forward Backward Approach to sine-Gordon up to 6π
30th September 2024
I will present a novel approach for a rigorous construction of the sine-Gordon quantum field theory on the full-space in the first few regions. The approach relies on a scale decomposition of the QFT and the observation that the dynamics of moving between scales is described by a forward-backward stochastic differential equation (FBSDE). Using a simplified version of the problem, I will explain the general set-up and issues that arise in the analysis of the FBSDE. Once the FBSDE can be controlled, it serves as a powerful tool to show several properties of the resulting QFT. I will give an overview over results such as the verification of the Osterwalder-Schrader axiom, decay of correlations, as well as a variational problem and a resulting large deviations principle for the semi-classical limit.
Tommaso Rosati
Lower bounds to energy dissipation in passive scalar advection
7th October 2024
We consider a passive scalar advected by a stochastic velocity field. Under some non-degeneracy assumptions on the noise, we prove a lower bound on the energy dissipation that is quantitative in the diffusivity of the scalar. This partially addresses a conjecture by Miles and Doering and proves a first lower bound to the so-called Batchelor scale. The proof is based on dynamics of energy level sets, and a refined short- time and high-frequency expansion. Joint work with M. Hairer, S. Punshon-Smith and J. Yi.
Sophie Laruelle
Urn Models and Stochastic Approximation : From Pólya Urns to Nonlinear Randomized Urns
14th October 2024
This talk will begin with the presentation of the classic model of Pólya urns and the results obtained to introduce the notations used subsequently. We will then look at variants where we modify the deterministic replacement matrix (without replacement, adding a different color or several colors).
We will then study the case of stochastic replacement matrix with uniform drawing and balanced urn. This has been motivated by applications in clinical trials and finance, and can be solved more efficiently using stochastic algorithms on the dynamics of the proportions of balls in the urn.
We will end this talk with a final generalisation: the drawing rule will no longer be uniform, but will be modified by a convex or concave function. With 2 colors, this will allow us to exhibit a phase transition (in the convex case) passing from one equilibrium to several (the number of which depends on the function). For the number of balls greater than 2, work is in progress, but we have some new results for 3 colors and some ideas we will discuss for going further.
James Foster and Dáire O'Kane
On the convergence of adaptive approximations for SDEs
and
Simulating underdamped Langevin dynamics: A third order convergent method
21st October 2024 at 14:15
On the convergence of adaptive approximations for SDEs
When using ordinary differential equations (ODEs), numerical solutions are often approximated and propagated in time via discrete step sizes. For a large variety of ODE problems, performance can be improved by making these step sizes “adaptive” – that is, adaptively changed based on the state of system. However, for stochastic differential equations (SDEs), adaptive numerical methods can be difficult to study and even fail to converge due to the rough nature of Brownian motion.
In this talk, we will show that convergence does indeed occur, provided the underlying Brownian motion is discretized in an adaptive but “martingale-like” fashion. Whilst this prevents adaptive steps from skipping over time points (which we show can prevent convergence), we believe our convergence theory is the first that is applicable to standard SDE solvers. We will discuss the key ingredients in this analysis – including martingale convergence, rough path theory and the approximation of Brownian motion by polynomials.
Based on our theory, we also modify an adaptive “Proportional-Integral” (PI) step size controller for use in the SDE setting. Unlike those used for ODEs, this new PI controller is designed to revisit time points where the Brownian motion was previously sampled. Finally, we conclude with a numerical experiment showing that SDE solvers can achieve an order of magnitude more accuracy with adaptive step sizes than with constant step sizes.
Joint work with Andraž Jelinčič.
Simulating underdamped Langevin dynamics: A third order convergent method
Underdamped Langevin dynamics (ULD) is a stochastic differential equation of great interest to those in the molecular dynamics and machine learning communities. Interest from the latter is due to ergodic properties of ULD, meaning that under assumptions on the potential function, the process can be used as proxy for generating samples from an unnormalized (log-concave) target density. Due to the presence of a non-linear term, one cannot simulate ULD exactly and must apply a numerical discretisation.
In this talk, we introduce the numerical method “QUICSORT”, which is a third order convergent method for simulating ULD. Our method is based off discretising the “Shifted ODE” method of Foster, Lyons and Oberhauser, where Brownian motion is replaced by a piecewise linear path matching higher order stochastic integrals. During the talk we will motivate the construction of QUICSORT, discuss the key ideas in the proof of third order convergence (in the 2-Wasserstein metric) and highlight the key role played by numerical contractivity. To the best of our knowledge, this is the first method to achieve third order convergence over the infinite-time horizon for strongly convex MCMC problems. We conclude with a numerical experiment (courtesy of Andraž Jelinčič) illustrating the performance of our method against the current state of the art sampler for high dimensional problems.
Joint work with James Foster.
Rivka Mitchell
Discrete Snakes with Globally Centered Displacements
28th October 2024
We consider certain size-conditioned critical Bienaymé trees, in which each vertex is endowed with a spatial location that is a random displacement away from their parent’s location. By construction, the positions along each vertex’s lineage form a random walk. It is convenient to encode the genealogy and spatial locations using a path-valued process called the discrete snake. We prove that under a global finite variance and a tail behaviour assumption on the displacements, any globally centered discrete snake on a Bienaymé tree whose offspring distribution is critical and admits a finite third moment has the Brownian snake driven by a normalised Brownian excursion as its scaling limit. Our proof relies on two perspectives of Bienaymé trees. To prove convergence of finite dimensional distributions we rely on a line-breaking construction from a recent paper of Addario-Berry, Blanc-Renaudie, Donderwinkel, Maazoun and Martin. To prove tightness, we adapt a method used by Haas and Miermont in the context of height functions of Markov branching trees.
This is based on joint work in progress with Louigi Addario-Berry, Serte Donderwinkel, and Christina Goldschmidt.
Gady Kozma
A reduction of the theta(p_c) problem to a conjectured inequality
4th November 2024
A famous open question asks if critical percolation has an infinite cluster or not. This question has been resolved in dimension 2 and in sufficiently high dimensions, but is open in intermediate dimensions. We will discuss a reduction of this problem to a conjectured FKG-like inequality. Joint work with Shahaf Nitzan.
Nikos Zygouras
The Critical 2d Stochastic Heat Flow and Critical SPDEs
11th November 2024
Thanks to the theories of Regularity Structures and Paracontrolled Distributions we now have a complete theory of singular SPDEs, which are “sub-critical” in the sense of renormalisation. Recently, there have been efforts to approach the situation of “critical” SPDEs and statistical mechanics models. A first such treatment has been through the study of the two-dimensional stochastic heat equation, which has revealed a certain phase transition and has led to the construction of the novel object called the Critical 2d Stochastic Heat Flow. In this talk we will present some aspects of this model and its construction and some first steps towards an approach to critical SPDEs.
Most of the talk will be based on joint works with Caravenna and Sun.
Giuseppe Cannizzaro
Superdiffusive Central Limit Theorem for the critical Stochastic Burgers Equation
11th November 2024
The Stochastic Burgers Equation (SBE) was introduced in the eighties by van Beijren, Kutner and Spohn as a way to encode the fluctuations of driven diffusive systems with one conserved quantity (e.g. ASEP). In the subcritical dimension d=1, it coincides with the derivative of the KPZ equation whose large-scale behaviour is polynomially superdiffusive and given by the KPZ Fixed Point, and in the super-critical dimensions d>2, it was recently shown to be diffusive and rescale to a Stochastic Heat equation. At the critical dimension d=2, the SBE was conjectured to be logarithmically superdiffusive with a precise exponent but this has only been shown up to lower order corrections. In the present talk, we pin down the logarithmic superdiffusivity exactly by identifying the limit of the so-called diffusion coefficient and show that, once the logarithmic corrections to the scaling are taken into account, the solution of the SBE satisfies a central limit theorem.
Joint work with Q. Moulard and F. Toninelli.
Matthew Jenssen
Sphere packing in high dimensions
2nd December 2024
The classical sphere packing problem asks: what is the densest possible arrangement of identical, non-overlapping spheres in Rd? Over the past century, this question has been intensely studied by mathematicians and physicists alike. In this talk I will discuss some different perspectives on this problem along with some recent progress. In particular, I will sketch a proof that there exists a sphere packing in Rd with density at least
(1+o(1)) d log d / 2{d+1}
This improves upon previous bounds by a factor of order log d and is the first improvement by more than a constant factor to Rogers’ bound from 1947.
This is joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe.
Shuo Qin
Recurrence and transience of multidimensional elephant random walks
4th December 2024 (Note: Wednesday!)
We prove a conjecture by Bertoin that the multidimensional elephant random walk on Zd is transient in dimensions d ≥ 3. We show that it undergoes a phase transition in dimensions d = 1, 2 between recurrence and transience at p = (2d + 1)/(4d). We also generalize these results to the step-reinforced random walk under mild moments conditions.
Thomas Powell
Quantitative results for stochastic processes
9th December 2024
It is an elementary fact from real analysis that any monotone bounded sequence of real numbers converges. It turns out that the monotone convergence theorem can be given an equivalent finitary formulation: roughly that any sufficiently long monotone bounded sequence experiences long regions where the sequence is metastable. This so-called "finite convergence principle" is carefully motivated and discussed by Terence Tao in a 2007 blog post ('Soft analysis, hard analysis, and the finite convergence principle'), but was already known to proof theorists, where the use of logical methods to both finitize infinitary statements and provide uniform quantitative information for the finitary versions plays a central role in the so-called proof mining program.
The goal of my talk is to discuss how methods from proof theory can be applied to the theory of stochastic processes, an area of mathematics hitherto unexplored from this perspective. I will begin by discussing the simple monotone convergence principle, and will then focus on how everything I mentioned above can be generalised to the analogous result in the stochastic setting: Doob's martingale convergence theorems. This then sets the groundwork for new applications of proof theory in stochastic optimization, where almost-supermartingales play a crucial role, and I will give a high level overview of some work in progress in this direction.
Avi Mayorcas
Large deviations for the Φ43 measure via Stochastic Quantisation
6th January 2025
The Φ43 measure is one of the easiest non-trivial examples of a Euclidean quantum field theory (EQFT) whose rigorous construction in the 1970’s has been one of the celebrated achievements of the constructive QFT community. In recent years, progress in the field of singular stochastic PDEs, initially by the theory of regularity structures, has allowed to construct the Φ43 EQFT as the invariant measure of a previously ill-posed Langevin dynamics—a strategy originally proposed by Parisi and Wu (’81) under the name stochastic quantisation. In this talk, I will demonstrate that the same idea also allows for the transference of large deviation principles for the Φ43 dynamics, obtained by Hairer and Weber (’15), to the corresponding EQFT. Our strategy is inspired by earlier work of Sowers (’92) and Cerrai and Röckner (’05) for non-singular dynamics and potentially also applies to other EQFT measures. The talk is based on joint work with Tom Klose (University of Oxford).
Sefika Kuzgun
Time-dependent averages of a critical long-range stochastic heat equation
13th January 2025
In this talk we will introduce a critical long-range linear stochastic heat equation. We will discuss the solution theory which dates back to 2004 work of Mueller and Tribe. We will then talk about the time-dependent spatial averages and the limiting behavior on different scales. The latter part of this talk is based on a joint work with Ran Tao.
Ajay Chandra
Non-commutative singular SPDE
3rd February 2025
In this talk, I will describe some recent progress on singular stochastic partial differential equations in the setting of non-commutative probability theory - examples will include the stochastic quantization of Fermionic quantum field theories and also the setting of free probability. This is based on joint work with Martin Hairer and Martin Peev.
Daniel Heydecker
The Porous Medium Equation: large deviations and gradient flow with degenerate and unbounded diffusion
10th February 2025
We consider a rescaling of the zero-range process, which converges in the limit to the porous medium equation ∂_t u = Δuα , and study the dynamical large deviations. The degeneracy of the diffusion close to u=0, as well as the unboundedness as u → ∞, mean that the usual proof of the key superexponential estimate cannot be applied; instead, we give a new argument based on pathwise regularity. As a consequence, we exhibit a formulation of the porous medium equation as the gradient flow of the entropy in a degenerate Wasserstein geometry through the entropy-dissipation equality.
Léonie Papon
TBA
17th February 2025
Abstract to appear.
Yoan Tardy
TBA
24th February 2025
Abstract to appear.
Alexandre Stauffer
Non-monotone phase transition in interacting particle systems
10th March 2025
In this talk we will discuss a reaction-diffusion particle system which has a non-monotone phase transition.
I will explain the techniques used to analyze monotone models and how they can be refined to
analyze non-monotone particle systems.
Based on a joint work with Leandro Chiarini.
Marco Seiler
TBA
17th March 2025
Abstract to appear.
Vittoria Silvestri
TBA
24th March 2025
Abstract to appear.
Sebastien Martineau
TBA
31st March 2025
Abstract to appear.
Hugo Vanneuville
TBA
22nd April 2025
Abstract to appear.
Franco Severo
TBA
28th April 2025
Abstract to appear.
Tom Klose
TBA
5th May 2025
Abstract to appear.
Serte Donderwinkel
TBA
19th May 2025
Abstract to appear.