15:30
Forthcoming events in this series
15:30
18:00
q-Gaussian Rough Paths and the q-Itô Formula
Abstract
TBA
15:30
Kinetic Optimal Transport
Abstract
We present a kinetic version of the optimal transport problem for probability measures on phase space. The central object is a second-order discrepancy between probability measures, analogous to the 2-Wasserstein distance, but based on the minimisation of the squared acceleration. We discuss the equivalence of static and dynamical formulations and characterise absolutely continuous curves of measures in terms of reparametrised solutions to the Vlasov continuity equation. This is based on joint work with Giovanni Brigati (ISTA) and Filippo Quattrocchi (ISTA).
15:30
Well-Posedness and Regularity of SDEs in the Plane with Non-Smooth Drift
Abstract
Keywords: SDE on the plane, Brownian sheet, path by path uniqueness, space time local time integral, Malliavin calculus
In this talk, we discuss the existence, uniqueness, and regularisation by noise for stochastic differential equations (SDEs) on the plane. These equations can also be interpreted as quasi-linear hyperbolic stochastic partial differential equations (HSPDEs). More specifically, we address path-by-path uniqueness for multidimensional SDEs on the plane, under the assumption that the drift coefficient satisfies a spatial linear growth condition and is componentwise non-decreasing. In the case where the drift is only measurable and uniformly bounded, we show that the corresponding additive HSPDE on the plane admits a unique strong solution that is Malliavin differentiable. Our approach combines tools from Malliavin calculus with variational techniques originally introduced by Davie (2007), which we non-trivially extend to the setting of SDEs on the plane.
This talk is based on a joint works with A. M. Bogso, M. Dieye and F. Proske.
15:30
Variance renormalisation of singular SPDEs
Abstract
Scaling arguments give a natural guess at the regularity condition on the noise in a stochastic PDE for a local solution theory to be possible, using the machinery of regularity structures or paracontrolled distributions. This guess of ``subcriticality'' is often, but not always, correct. In cases when it is not, a the blowup of the variance of certain nonlinear functionals of the noise necessitates a different, multiplicative renormalisation. This led to a general prediction and the first results in the case of the KPZ equation in [Hairer '24]. We discuss recent developments towards confirming this prediction. Based on joint works with Fabio Toninelli and Yueh-Sheng Hsu.
15:30
Transport of Gaussian measures under the flow of semilinear (S)PDEs: quasi-invariance and singularity.
Abstract
In this talk, we consider the Cauchy problem for a number of semilinear PDEs, subject to initial data distributed according to a family of Gaussian measures.
We first discuss how the flow of Hamiltonian equations transports these Gaussian measures. When the transported measure is absolutely continuous with respect to the initial measure, we say that the initial measure is quasi-invariant.
In the high-dispersion regime, we exploit quasi-invariance to build a (unique) global flow for initial data with negative regularity, in a regime that cannot be replicated by the deterministic (pathwise) theory.
In the 0-dispersion regime, we discuss the limits of this approach, and exhibit a sharp transition from quasi-invariance to singularity, depending on the regularity of the initial measure.
We will also discuss how the same techniques can be used in the context of stochastic PDEs, and how they provide information on the invariant measures for their flow.
This is based on joint works with J. Coe (University of Edinburgh), J. Forlano (Monash University), and M. Hairer (EPFL).
15:30
Quantitative Convergence of Deep Neural Networks to Gaussian Processes
Abstract
In this seminar, we explore the quantitative convergence of wide deep neural networks with Gaussian weights to Gaussian processes, establishing novel rates for their Gaussian approximation. We show that the Wasserstein distance between the network output and its Gaussian counterpart scales inversely with network width, with bounds apply for any finite input set under specific non-degeneracy conditions of the covariances. Additionally, we extend our analysis to the Bayesian framework, by studying exact posteriors for neural networks, when endowed with Gaussian priors and regular Likelihood functions, but we also provide recent advancements in quantitative approximation of trained networks via gradient descent in the NTK regime. Based on joint works with A. Basteri, and A. Agazzi and E. Mosig.
15:30
Weak Error of Dean-Kawasaki Equation with Smooth Mean-Field Interactions
Abstract
We consider the weak-error rate of the SPDE approximation by regularized Dean-Kawasaki equation with Itô noise, for particle systems exhibiting mean-field interactions both in the drift and the noise terms. Global existence and uniqueness of solutions to the corresponding SPDEs are established via the variational approach to SPDEs. To estimate the weak error, we employ the Kolmogorov equation technique on the space of probability measures. This work generalizes previous results for independent Brownian particles — where Laplace duality was used. In particular, we recover the same weak error rate as in that setting. This paper builds on joint work with X. Ji., H. Kremp and N. Perkowski.
15:30
Uniqueness of Dirichlet operators related to stochastic quantisation for the \(exp(φ)_{2}\)-model
Abstract
In this talk, we consider Dirichlet forms related to stochastic quantisation for the \(exp(φ)_{2}\)-model on the torus. We show strong uniqueness of the corresponding Dirichlet operators by applying an idea of (singular) SPDEs. This talk is based on ongoing joint work with Hirotatsu Nagoji (Kyoto University).
15:30
Recent progress on quantitative propagation of chaos
Abstract
When and how well can a high-dimensional system of stochastic differential equations (SDEs) be approximated by one with independent coordinates? This fundamental question is at the heart of the theory of mean field limits and the propagation of chaos phenomenon, which arise in the study of large (many-body) systems of interacting particles. This talk will present recent sharp quantitative answers to this question, both for classical mean field models and for more recently studied non-exchangeable models. Two high-level ideas underlie these answers. The first is a simple non-asymptotic construction, called the independent projection, which is a natural way to approximate a general SDE system by one with independent coordinates. The second is a "local" perspective, in which low-dimensional marginals are estimated iteratively by adding one coordinate at a time, leading to surprising improvements on prior results obtained by "global" arguments such as subadditivity inequalities. In the non-exchangeable setting, we exploit a surprising connection with first-passage percolation.
15:30
Spin glasses with multiple types
Abstract
Spin glasses are models of statistical mechanics in which a large number of elementary units interact with each other in a disordered manner. In the simplest case, there are direct interactions between any two units in the system, and I will start by reviewing some of the key mathematical results in this context. For modelling purposes, it is also desirable to consider models with more structure, such as when the units are split into two groups, and the interactions only go from one group to the other one. I will then discuss some of the technical challenges that arise in this case, as well as recent progress.
15:30
Sharp bounds for parameter-dependent stochastic integrals
Abstract
We provide sharp bounds in the supremum- and Hölder norm for parameter-dependent stochastic integrals. As an application we obtain novel long-term bounds for stochastic partial differential equations as well as novel bounds on the space-time modulus of continuity of the stochastic heat equation. This concerns joint work with Joris van Winden (TU Delft).
15:30
Stochastic wave equations with constraints: well-posedness and Smoluchowski-Kramers diffusion approximation
Abstract
I will discuss the well-posedness of a class of stochastic second-order in time-damped evolution equations in Hilbert spaces, subject to the constraint that the solution lies on the unit sphere. A specific example is provided by the stochastic damped wave equation in a bounded domain of a $d$-dimensional Euclidean space, endowed with the Dirichlet boundary conditions, with the added constraint that the $L^2$-norm of the solution is equal to one. We introduce a small mass $\mu>0$ in front of the second-order derivative in time and examine the validity of the Smoluchowski-Kramers diffusion approximation. We demonstrate that, in the small mass limit, the solution converges to the solution of a stochastic parabolic equation subject to the same constraint. We further show that an extra noise-induced drift emerges, which in fact does not account for the Stratonovich-to-It\^{o} correction term. This talk is based on joint research with S. Cerrai (Maryland), hopefully to be published in Comm Maths Phys.
15:30
Analyzing the Error in Score-Based Generative Models: A Stochastic Control Approach
Abstract
Score-based generative models (SGMs), which include diffusion models and flow matching, have had a transformative impact on the field of generative modeling. In a nutshell, the key idea is that by taking the time-reversal of a forward ergodic diffusion process initiated at the data distribution, one can "generate data from noise." In practice, SGMs learn an approximation of the score function of the forward process and employ it to construct an Euler scheme for its time reversal.
In this talk, I will present the main ideas of a general strategy that combines insights from stochastic control and entropic optimal transport to bound the error in SGMs. That is, to bound the distance between the algorithm's output and the target distribution. A nice feature of this approach is its robustness: indeed, it can be used to analyse SGMs built upon noising dynamics that are different from the Ornstein-Uhlenbeck process . As an example, I will illustrate how to obtain error bounds for SGMs on the hypercube.
15:30
Adapted optimal transport for stochastic processes
Abstract
15:30
Heat kernel for critical percolation clusters on the binary tree.
Abstract
15:30
Chasing regularization by noise of 3D Navier-Stokes equations
Abstract
Global well-posedness of 3D Navier-Stokes equations (NSEs) is one of the biggest open problems in modern mathematics. A long-standing conjecture in stochastic fluid dynamics suggests that physically motivated noise can prevent (potential) blow-up of solutions of the 3D NSEs. This phenomenon is often referred to as `regularization by noise'. In this talk, I will review recent developments on the topic and discuss the solution to this problem in the case of the 3D NSEs with small hyperviscosity, for which the global well-posedness in the deterministic setting remains as open as for the 3D NSEs. An extension of our techniques to the case without hyperviscosity poses new challenges at the intersection of harmonic and stochastic analysis, which, if time permits, will be discussed at the end of the talk.
15:30
Stochastic quantization of fractional $\Phi^4_3$ model of Euclidean quantum field theory
Abstract
The construction of the measure of the $\Phi^4_3$ model in the 1970s has been one of the major achievements of constructive quantum field theory. In the 1980s Parisi and Wu suggested an alternative way of constructing quantum field theory measures by viewing them as invariant measures of certain stochastic PDEs. However, the highly singular nature of these equations prevented their application in rigorous constructions until the breakthroughs in the area of singular stochastic PDEs in the past decade. After explaining the basic idea behind stochastic quantization proposed by Parisi and Wu I will show how to apply this technique to construct the measure of a certain quantum field theory model generalizing the $\Phi^4_3$ model called the fractional $\Phi^4$ model. The measure of this model is obtained as a perturbation of the Gaussian measure with covariance given by the inverse of a fractional Laplacian. Since the Gaussian measure is supported in the space of Schwartz distributions and the quartic interaction potential of the model involves pointwise products, to construct the measure it is necessary to solve the so-called renormalization problem. Based on joint work with M. Gubinelli and P. Rinaldi.
15:30
Critical phenomena in intermediate dimensions
Abstract
The talk will focus on recent developments regarding the (near-)critical behaviour of certain statistical physics models with long-range dependence in dimensions larger than 2, but smaller than 6, above which mean-field behaviour is known to set in. This “intermediate” regime remains a great challenge for mathematicians. The models revolve around a certain percolation phase transition that brings into play very natural probabilistic objects, such as random walk traces and the Gaussian free field.
17:00
The Brooke Benjamin Lecture in Fluid Dynamics: The Elusive Singularity
Abstract
The Seventeenth Brooke Benjamin Lecture 2024
The Elusive Singularity
I will describe the open problems of singularity formation in incompressible fluids. I will discuss a list of related models, some results, and some more open problems.
Date: Monday, 11 November 2024
Time: 5pm GMT
Location: Lecture Theatre 1, Mathematical Institute
Speaker: Professor Peter Constantin
15:30
Statistical Inference for weakly interacting diffusions and their mean field limit
Abstract
We consider the problem of parametric and non-parametric statistical inference for systems of weakly interacting diffusions and of their mean field limit. We present several parametric inference methodologies, based on stochastic gradient descent in continuous time, spectral methods and the method of moments. We also show how one can perform fully nonparametric Bayesian inference for the mean field McKean-Vlasov PDE. The effect of non-uniqueness of stationary states of the mean field dynamics on the inference problem is elucidated.
15:30
Higher Order Lipschitz Functions in Data Science
Abstract
The notion of Lip(gamma) Functions, for a parameter gamma > 0, introduced by Stein in the 1970s (building on earlier work of Whitney) is a notion of smoothness that is well-defined on arbitrary closed subsets (including, in particular, finite subsets) that is instrumental in the area of Rough Path Theory initiated by Lyons and central in recent works of Fefferman. Lip(gamma) functions provide a higher order notion of Lipschitz regularity that is well-defined on arbitrary closed subsets, and interacts well with the more classical notion of smoothness on open subsets. In this talk we will survey the historical development of Lip(gamma) functions and illustrate some fundamental properties that make them an attractive class of function to work with from a machine learning perspective. In particular, models learnt within the class of Lip(gamma) functions are well-suited for both inference on new unseen input data, and for allowing cost-effective inference via the use of sparse approximations found via interpolation-based reduction techniques. Parts of this talk will be based upon the works https://arxiv.org/abs/2404.06849 and https://arxiv.org/abs/2406.03232.
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