Past Seminars and Recordings
Tuesday, February 20, 2024
Zhenfu Wang (BICMR-Peking University) [Online]
We derive the quantitative estimates of propagation of chaos for the large interacting particle systems in terms of the relative entropy between the joint law of the particles and the tensorized law of the mean field PDE. We resolve this problem for the first time for the viscous vortex model that approximating 2D Navier-Stokes equation in the vorticity formulation on the whole space. We obtain as key tools the Li-Yau-type estimates and Hamilton-type heat kernel estimates for 2D Navier-Stokes on the whole space. This is based on a joint work with Xuanrui Feng from Peking University.
Renyuan Xu (University of Southern California) [Online]
Generative diffusion models, which transform noise into new data instances by reversing a Markov diffusion process in time, have become a cornerstone in modern generative models. A key component of these models is to learn the associated Stein’s score function. While the practical power of diffusion models has now been widely recognized, the theoretical developments remain far from mature. Notably, it remains unclear whether gradient-based algorithms can learn the score function with a provable accuracy. In this talk, we develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models and the accuracy of score estimation. Our analysis covers both the optimization and the generalization aspects of the learning procedure, which also builds a novel connection to supervised learning and neural tangent kernels.
This is based on joint work with Yinbin Han and Meisam Razaviyayn (USC).
Tuesday, February 13, 2024
Oana Lang (Imperial College London)
This talk will be focused on local and global properties for a class of stochastic partial differential equations (SPDEs) which includes equations coming from geophysical fluid dynamics. There are many SPDEs in the literature for which global existence of solutions does not hold, with several such examples coming from fluid dynamics. This has important consequences not just analytically, but also in applications. I will discuss conditions under which such equations can admit existence of a global solution and implications at their structural level. This is joint work with Dan Crisan.
Tuesday, February 6, 2024
Johannes Muhle-Karbe (Imperial College London)
Tuesday, January 30, 2024
Ying Zhang (The Hong Kong University of Science and Technology (Guangzhou))
Title: Langevin dynamics-based algorithm e-THeO POULA for stochastic optimization with discontinuous stochastic gradient
We introduce a new Langevin dynamics-based algorithm, called e-THO POULA, to solve optimization problems with discontinuous stochastic gradients which naturally appear in real-world applications such as quantile estimation, vector quantization, CVaR minimization, and regularized optimization problems involving ReLU neural networks. We demonstrate both theoretically and numerically the applicability of the e-THO POULA algorithm. More precisely, under the conditions that the stochastic gradient is locally Lipschitz in average and satisfies a certain convexity at infinity condition, we establish non-asymptotic error bounds for e-THeO POULA in Wasserstein distances and provide a non-asymptotic estimate for the expected excess risk, which can be controlled to be arbitrarily small. Two key applications in finance are provided, namely, multi-period portfolio optimization and transfer learning in multi-period portfolio optimization, which involve neural networks with (Leaky)-ReLU activation functions. Numerical experiments conducted using real-world datasets illustrate the superior empirical performance of e-THeO POULA compared to SGLD, ADAM, and AMSGrad in terms of model accuracy.
Tuesday, January 23, 2024
Xell Brunet Guasch (University of Edinburgh)
Title: Multi-Type Critical Birth-Death Process: A Model of Error-Induced Extinction
Tuesday, January 16, 2024
Catherine Wolfram (MIT) [Online]
Title: The dimer model in 3D
A dimer tiling of Z^d is a collection of edges such that every vertex is covered exactly once. In 2000, Cohn, Kenyon, and Propp showed that 2D dimer tilings satisfy a large deviations principle. In joint work with Nishant Chandgotia and Scott Sheffield, we prove an analogous large deviations principle for dimers in 3D. A lot of the results for dimers in two dimensions use tools and exact formulas (e.g. the height function representation of a tiling or the Kasteleyn determinant formula) that are specific to dimension 2. I will explain how to formulate the large deviations principle in 3D, show simulations, try to give some intuition for why three dimensions is different from two. Time permitting, I will explain some of the ways that we use a smaller set of tools (e.g. Hall’s matching theorem or a double dimer swapping operation) in our arguments.
Tuesday, December 5, 2023
Maxim Raginsky (University of Illinois at Urbana-Champaign) [Online]
Recently, diffusion models have achieved remarkable breakthroughs in generative modeling, with practical advances outpacing theoretical understanding. This has led to a surge of interest in the theory of diffusion processes, as well as in the fundamental links between stochastic control, nonequilibrium thermodynamics, and learning. In this talk, I will present a unified perspective on sampling, inference, and path estimation (or smoothing) in diffusion processes through the lens of free energy minimization. This variational perspective ties together the Gibbs variational principle with the Cameron-Martin-Girsanov change of measure and the Hamilton-Jacobi-Bellman optimal control formulation. Several existing results emerge as special cases, including the variational formulation of nonlinear estimation due to Newton and Mitter and the recent control-theoretic derivation of diffusion-based generative modeling due to Berner, Richter, and Ullrich.
Corrected Slides: Raginsky.pdf
Tuesday, November 28, 2023
Sarah-Jean Meyer(University of Oxford)
We present a forward-backward approach to stochastic quantisation for the sine-Gordon Euclidean quantum field on the full space up to the second threshold, i.e. for . The basis of our method is a forward-backward stochastic differential equation (FBSDE) for a decomposition of the interacting Euclidean field along a scale parameter . This FBSDE describes the optimiser of a stochastic control representation of the Euclidean QFT introduced by Barashkov and Gubinelli and provides a scale-by-scale description of the interacting field without cut-offs. I will present the general idea of the approach, and illustrate how the FBSDE can be used effectively to study the sine-Gordon measure to obtain results such as large deviations, integrability, decay of correlations for local observables, Osterwalder–Schrader axioms and other properties.
This talk is based on joint work with Massimiliano Gubinelli.
Tuesday, November 21, 2023
Gal Kronenberg (University of Oxford)
Independent sets in bipartite regular graphs have been studied extensively in combinatorics, probability, computer science and more. The problem of counting independent sets is particularly interesting in the d-dimensional hypercube , motivated by the lattice gas hardcore model from statistical physics. Independent sets also turn out to be very interesting in the context of random graphs.
The number of independent sets in the hypercube was estimated precisely by Korshunov and Sapozhenko in the 1980s and recently refined by Jenssen and Perkins.
In this talk we will discuss new results on the number of independent sets in a random subgraph of the hypercube. The results extend to the hardcore model and rely on an analysis of the antiferromagnetic Ising model on the hypercube.
This talk is based on joint work with Yinon Spinka.
Tuesday, November 14, 2023
Nikolas Nüsken (King’s College London)
Connecting optimal transport and variational inference, this talk is about a principled and systematic framework for score-based sampling and generative modelling centred around forward and reverse-time stochastic differential equations. In particular, I will discuss the novel Controlled Monte Carlo Diffusion (CMCD) methodology for Bayesian computation, a score-based annealing technique that crucially adapts both forward and backward dynamics in a diffusion model. Time permitting, we will discuss relationships to optimal control and Schroedinger bridges, as well as connections to the Jarzinsky and Crooks identities from statistical physics. This is joint work with F. Vargas, S. Padhy and D. Blessing.
Tuesday, November 7, 2023
Harprit Singh (University of Edinburgh)
We shall discuss the solution theory to a large class of singular SPDEs in various nontranslation invariant settings. In particular, I shall present results on Riemannian manifolds and homogeneous Lie groups as well as related results for non-translation invariant parabolic and hypo-elliptic equations on Euclidean space. Based on joint work with M. Hairer, respectively A. Mayorcas.
Tuesday, October 31, 2023
Guillaume Barraquand (ENS)
The Kardar-Parisi-Zhang (KPZ) equation is a nonlinear stochastic PDE introduced in Physics to model interface growth. In one dimension, the KPZ equation on the line admits a remarkably simple stationary measure: the Brownian motion. For the KPZ equation on an interval or a half-line, however, stationary measures are non-Gaussian and depend on the boundary conditions imposed. Their explicit description has been obtained only very recently, through the detailed study of integrable probabilistic models, such as ASEP or directed polymer models, that can be viewed as discretizations of the KPZ equation. We will first review these results, which are based on a series of works by several groups of authors. I will then outline a new method to compute the stationary distributions of stochastic integrable models on a lattice such as last passage percolation, or directed polymer models.
Tuesday, October 24, 2023
Roger Van Peski (KTH)
Title: New local limits in discrete random matrix theory
Random matrices over finite fields, integers, and p-adic integers have been studied since the late 1980s as natural models for random groups appearing in number theory and combinatorics. Recently it has also become clear that the theory has close structural parallels with singular values of complex random matrices, bringing new techniques from integrable probability and motivating new questions. In particular, in this talk I will explain recent results on the local limits of products of such matrices, which yield a new object that may be viewed as a discrete and non-determinantal analogue of (certain deformations of) the extended sine and Airy processes in classical random matrix theory.
Tuesday, October 17, 2023
Ioannis Gasteratos (Imperial College London)
We consider a multiscale system of stochastic differential equations in which the slow component is perturbed by a small fractional Brownian motion with Hurst index and the fast component is driven by an independent Brownian motion. Working in the framework of Young integration, we use tools from fractional calculus and weak convergence arguments to establish a Large Deviation Principle (LDP) in the homogenized limit, as the noise intensity and time-scale separation parameters vanish at an appropriate rate. Our approach is based in the study of the limiting behavior of an associated controlled system. We show that, in certain cases, the non-local rate function admits an explicit non-variational form. The latter allows us to draw comparisons to the case which corresponds to the classical Freidlin-Wentzell theory. Moreover, we study the asymptotics of the rate function as and show that it is discontinuous at .
Tuesday, October 10, 2023
Raphaël Maillet (CEREMADE) [Online]
In this talk, we explore the long-term behaviour of solutions to a nonlinear McKean-Vlasov equation with common noise. This equation arises naturally when studying the collective behaviour of interacting particles driven by both individual and common noise. Our main focus is to understand how the presence of common noise affects the stability of the system. We will first present the results for the case without common noise and then discuss how the introduction of common noise can enhance stability in certain respects.
Tuesday, October 3, 2023
Kohei Suzuki (Durham University)
The Dyson Brownian Motion (DBM) is an eigenvalue process of a particular Hermitian matrix-valued Brownian motion introduced by Freeman Dyson in 1962, which has been one of the central subjects in the random matrix theory. In this talk, we study the DBM from a geometric perspective. We show that the infinite particle DBM possesses a lower bound of the Ricci curvature à la Bakry-Émery. As a consequence, we obtain various quantitative estimates of the transition probability of the DBM (e.g., the local spectral gap, the local log-Sobolev, and the dimension-free Harnack inequalities) as well as the characterisation of the DBM as the gradient flow of the Boltzmann entropy in a particular Wasserstein-type space.
September 26, 2023
Ofer Busani (University of Edinburgh)
Title: Scaling limit of multi-type stationary measures in the KPZ class
The KPZ class is a very large set of models that are meant to describe random growth interfaces. It is believed that upon scaling, the long time behavior of members in this class is universal and is described by a limiting random object, a Markov process called the KPZ fixed-point. The (one-type) stationary measures for the KPZ fixed-point as well as many models in the KPZ class are known – it is a family of distributions parametrized by some set that depends on the model. For the k-type stationary distribution with intensities is a coupling of one-type stationary measures of indices that is stationary with respect to the model dynamics. In this talk we will present recent progress in our understanding of the multi-type stationary measures of the KPZ fixed-point as well as the scaling limit of multi-type stationary measures of two families of models in the KPZ class: metric-like models (e.g. last passage percolation) and particle systems (e.g. exclusion process). Based on joint work with Timo Seppalainen and Evan Sorensen.
June 14, 2023
Yufei Zhang (LSE)
Title: Exploration-exploitation trade-off for continuous-time reinforcement learning
Recently, reinforcement learning (RL) has attracted substantial research interests. Much of the attention and success, however, has been for the discrete-time setting. Continuous-time RL, despite its natural analytical connection to stochastic controls, has been largely unexplored and with limited progress. In particular, characterising sample efficiency for continuous-time RL algorithms remains a challenging and open problem.
In this talk, we develop a framework to analyse model-based reinforcement learning in the episodic setting. We then apply it to optimise exploration-exploitation trade-off for linear-convex RL problems, and report sublinear (or even logarithmic) regret bounds for a class of learning algorithms inspired by filtering theory. The approach is probabilistic, involving analysing learning efficiency using concentration inequalities for correlated continuous-time observations, and applying stochastic control theory to quantify the performance gap between applying greedy policies derived from estimated and true models.
May 3, 2023: North British Probability Workshop
Neil O’Connell (University College Dublin) [Online]
I will discuss a natural Markov chain on reverse plane partitions
which is closely related to the Toda lattice.
Athanasios Vasileiadis (University of Nice)
In this talk we are going to use common noise as a form of exploration to learn the Nash equilibrium of a Linear Quadratic Mean Field Game. We develop a scheme based on Fictitious Play and study its convergence to the analytical solution of the limit system. We illustrate the analysis by numerical implementations using regression on a basis of Hermit polynomials and also artificial neural networks.
Stefano Bruno (University of Edinburgh)
April 26, 2023:
Benjamin Fehrman (University of Oxford)
Non-equilibrium behavior in physical systems is widespread. A statistical description of these events is provided by macroscopic fluctuation theory, a framework for non-equilibrium statistical mechanics that postulates a formula for the probability of a space-time fluctuation based on the constitutive equations of the system. This formula is formally obtained via a zero noise large deviations principle for the associated fluctuating hydrodynamics, which postulates a conservative, singular stochastic PDE to describe the system far-from-equilibrium. In this talk, we will focus particularly on the fluctuations of the zero range process about its hydrodynamic limit. We will show how the associated MFT and fluctuating hydrodynamics lead to a class of conservative SPDEs with irregular coefficients, and how the study of large deviations principles for the particles processes and SPDEs leads to the analysis of parabolic-hyperbolic PDEs in energy critical spaces. The analysis makes rigorous the connection between MFT and fluctuating hydrodynamics in this setting, and provides a positive answer to a long-standing open problem for the large deviations of the zero range process.
April 19, 2023:
Patrik Ferrari (University of Bonn)
March 29, 2023:
Mao Fabrice Djete (École Polytechnique) [Online]
Title: Non–regular McKean–Vlasov equations and calibration problem in local stochastic volatility models
Christian Bayer (WIAS Berlin) [Online]
We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process X.
We consider classic and randomized stopping times represented by linear and non-linear functionals of the rough path signature associated to X, and prove that maximizing over these classes of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a(deterministic) optimization problem depending only on the (truncated) expected signature. By applying a deep neural network approach to approximate the non-linear signature functionals, we can efficiently solve the optimal stopping problem numerically.
The only assumption on the process X is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion, which fail to be either semi-martingales or Markov processes, and can be used, in particular, for American-type option pricing in fractional models, e.g. of financial or electricity markets. (Based on joint work with Paul Hager, Sebastian Riedel, and John Schoenmakers)
March 15, 2023:
Guangqu Zheng (University of Liverpool)
Title: Pathwise local well-posedness of stochastic nonlinear Schrödinger equations with multiplicative noises.
In this talk, I will present a recent work on the stochastic nonlinear Schrödinger equation (SNLS) with multiplicative noises. The equation was first proposed in physics literature, and has attracted much attention in the mathematical community. It has been studied by various researchers such as de Bouard, Debussche, Barbu, Röckner, either in Ito framework or via specific transform. As a result, it is not pathwise. The major difficulty towards a pathwise theory lies at how to give a meaning to the product of the noise and an unknown in presence of deficiency of temporal regularities. Despite some development in the parabolic setting by Gubinelli-Tindel (2010) and by Hairer-Pardoux (2015), the dispersive problem remains unsolved.
In this talk, I will present a joint work with T. Oh (Edinburgh), which established the first pathwise local well-posedness result for SNLS with multiplicative noises in the spatially smooth setting. This work exploits Gubinelli-Tindel’s framework of algebraic integration, in combination with random matrix/tensor estimates from a recent work by Deng-Nahmod-Yue (2022), and in particular, the notion of operator-valued controlled rough paths adapted to the Schrödinger flow is introduced.
March 8, 2023:
Emanuela Gussetti (Universität Bielefeld)
Title: An application of rough paths theory to the study of the stochastic Landau-Lifschitz-Gilbert equation
The Landau-Lifschitz-Gilbert equation is a model describing the magnetisation of a ferromagnetic material. The stochastic model is studied to observe the role of thermal fluctuations. We interpret the linear multiplicative noise appearing by means of rough paths theory and we study existence and uniqueness of the solution to the equation on a one dimensional domain . As a consequence of the rough paths formulation, the map that to the noise associates the unique solution to the equation is locally Lipschitz continuous in the strong norm . This implies a Wong-Zakai convergence result, a large deviation principle, a support theorem.
We prove also continuity with respect to the initial datum initial datum in , which allows to conclude the Feller property for the associated semigroup. We also discuss briefly a Path-wise central limit theorem and a moderate deviations principle for the stochastic LLG: in this case the rough paths formulation leads to a path-wise convergence, not easily reachable in the classical It‘s calculus setting.
The talk is based on a joint work with A. Hocquet [https://arxiv.org/abs/2103.00926] and on [https://arxiv.org/abs/2208.02136].
February 15, 2023:
Laure Dumaz (ENS)
Title: Some aspects of the Anderson Hamiltonian in 1D.
In this talk, I will present several results on the Anderson Hamiltonian with white noise potential in dimension 1. This operator formally writes « – Laplacian + white noise ». It arises as the scaling limit of various discrete models and its explicit potential allows for a detailed description of its spectrum. We will discuss localization of its eigenfunctions as well as the behavior of the local statistics of its eigenvalues. Around large energies, we will see that the eigenfunctions are delocalized and follow a universal shape given by the exponential of a Brownian motion plus a drift, a behavior already observed by Rifkind and Virag in tridiagonal matrix models. Based on joint works with Cyril Labbé.
February 8, 2023:
Fabian Germ (University of Edinburgh)
Abstract: I will present the filtering problem for a partially observed jump diffusion system Z=(X,Y) driven by Wiener processes and Poisson martingale measures. We will derive the filtering equations, describing the time evolution of the normalised conditional distribution and the unnormalised conditional distribution of the unobservable signal X t given the observations Y. Recent results on L_p-calculus for systems with jumps have made it possible to prove existence and regularity properties of (unnormalised) conditional densities in certain Sobolev spaces under fairly general assumptions. We will see such conditions and derive existence and regularity results based on SPDEs for the (unnormalised) conditional density processes.
This is joint work with Istvan Gyongy.
November 24, 2022
Mo Dick Wong (Durham University)
Title: What can we hear about the geometry of an LQG surface?
Abstract: The Liouville quantum gravity (LQG) surface, formally defined as a 2-dimensional Riemannian manifold with conformal factor being the exponentiation of a Gaussian free field, is closely related to random planar geometry as well as scaling limits of models from statistical mechanics. In this talk we consider the bounded, simply connected setting, and explain the Weyl’s law for the asymptotics of the eigenvalues of the (random) Laplace-Beltrami operator. Our approach is purely probabilistic and relies on the study of the so-called Liouville Brownian motion, the canonical diffusion process on an LQG surface. This is a joint work with Nathanael Berestycki.
James Foster (University of Bath) -online talk
Title: High order splitting methods for stochastic differential equations
Abstract: In this talk, we will discuss how ideas from rough path theory can be leveraged to develop high order numerical methods for SDEs. To motivate our approach, we consider what happens when the Brownian motion driving an SDE is replaced by a piecewise linear path. We show that this procedure transforms the SDE into a sequence of ODEs – which can then be discretized using an appropriate ODE solver. Moreover, to achieve a high accuracy, we construct these piecewise linear paths to match certain “iterated” integrals of the Brownian motion. At the same time, the ODE sequences obtained from this path-based approach can be interpreted as a splitting method, which neatly connects our work to the existing literature. For example, we show that the well-known Strang splitting falls under this framework and can be modified to give an improved convergence rate. We will conclude the talk with a couple of examples, demonstrating the flexibility and convergence properties of our methodology.
(joint work with Gonçalo dos Reis and Calum Strange)
November 17, 2022
Jiawei Li (University of Edinburgh)
Title: On distributions of random vorticity and velocity fields in turbulent flows
November 10, 2022
Mustazee Rahman (Durham University)
Title: Scaling limit of the second class particle
Abstract: The totally asymmetric simple exclusion process (TASEP) is a microscopic analogue of Burgers equation. The characteristics of Burgers equation play an important role in its solution. The second class particle in TASEP mirrors the role of a microscopic characteristic. Although much is known about the macroscopic behaviour of the second class particle, it is of interest to understand its fluctuations. I will discuss the behaviour of the second class particle when the initial conditions of TASEP converge under KPZ scaling. The second class particle has a scaling limit, which can be described in terms of geometric concepts arising in the KPZ universality class. Joint work with Balint Virag.
November 3, 2022
Alexey Korepanov (University of Paris)
Title: Almost sure approximation by Brownian motion for Bunimovich flowersAbstract: I will talk about approximation of processes which appear on Bunimovich flowers (deterministic dynamical systems where a point particle moves inside a table made of several concave arcs, well known for slow memory loss), and about their approximation by a Brownian motion by reconstructing both on the same probability space so that they are close. This is a rather special result about a rather particular system, but in a way it completes a long line of research. I’ll attempt to introduce the topic and explain the kind of problems we are dealing with. This is a joint work in progress with C.Cuny, J.Dedecker and F.Merlevede.
October 27, 2022
Samuel Cohen (University of Oxford)
Title: Neural-SDEs and Market Models of Options
Abstract: Measuring market risk for option portfolios is challenging; it requires modelling joint dynamics of liquid options under the real world measure, simulating realistic trajectories of risk factors, and evaluating option prices efficiently under a large of amount of risk scenarios. In this talk we will construct an arbitrage-free neural-SDE market model for European call options. This involves building neural networks satisfying certain structural assumptions. Through backtesting analysis, we show that our models are more computationally efficient and accurate for evaluating Value-at-Risk (VaR) of option portfolios, with better coverage performance and less procyclicality than standard filtered historical simulation approaches.
Karen Habermann (University of Warwick)
Title: A polynomial expansion for Brownian motion and the associated fluctuation process
Abstract: We start by deriving a polynomial expansion for Brownian motion expressed in terms of shifted Legendre polynomials by considering Brownian motion conditioned to have vanishing iterated time integrals of all orders. We further discuss the fluctuations for this expansion and show that they converge in finite dimensional distributions to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle. We then link the asymptotic convergence rates of approximations for Brownian Lévy area which are based on the Fourier series expansion and the polynomial expansion of the Brownian bridge to these limit fluctuations. We close with a general study of the asymptotic error arising when approximating the Green’s function of a Sturm-Liouville problem through a truncation of its eigenfunction expansion, both for the Green’s function of a regular Sturm-Liouville problem and for the Green’s function associated with the classical orthogonal polynomials.