Forthcoming events in this series


Tue, 10 Oct 2023
11:00
Lecture Room 4, Mathematical Institute

DPhil Presentations

DPhil Students
Abstract

As part of the internal seminar schedule for Stochastic Analysis for this coming term, DPhil students have been invited to present on their works to date. Student talks are 20 minutes, which includes question and answer time.

Mon, 17 Jun 2019

15:45 - 16:45
L3

Mathematical and computational challenges in interdisciplinary bioscience: efficient approaches for stochastic models of biological processes.

RUTH BAKER
(University of Oxford)
Abstract

Simple mathematical models have had remarkable successes in biology, framing how we understand a host of mechanisms and processes. However, with the advent of a host of new experimental technologies, the last ten years has seen an explosion in the amount and types of data now being generated. Increasingly larger and more complicated processes are now being explored, including large signalling or gene regulatory networks, and the development, dynamics and disease of entire cells and tissues. As such, the mechanistic, mathematical models developed to interrogate these processes are also necessarily growing in size and complexity. These detailed models have the potential to provide vital insights where data alone cannot, but to achieve this goal requires meeting significant mathematical challenges. In this talk, I will outline some of these challenges, and recent steps we have taken in addressing them.

Mon, 17 Jun 2019

14:15 - 15:15
L3

Path Developments and Tail Asymptotics of Signature

XI GENG
(University of Melbourne)
Abstract

It is well known that a rough path is uniquely determined by its signature (the collection of global iterated path integrals) up to tree-like pieces. However, the proof the uniqueness theorem is non-constructive and does not give us information about how quantitative properties of the path can be explicitly recovered from its signature. In this talk, we examine the quantitative relationship between the local p-variation of a rough path and the tail asymptotics of its signature for the simplest type of rough paths ("line segments"). What lies at the core of the work a novel technique based on the representation theory of complex semisimple Lie algebras. 

This talk is based on joint work with Horatio Boedihardjo and Nikolaos Souris

Mon, 10 Jun 2019

15:45 - 16:45
L3

Towards Geometric Integration of Rough Differential Forms

DARIO TREVISAN
(University of Pisa Italy)
Abstract

We discuss some results on integration of ``rough differential forms'', which are generalizations of classical (smooth) differential forms to similar objects involving Hölder continuous functions that may be nowhere differentiable. Motivations arise mainly from geometric problems related to irregular surfaces, and the techniques are naturally related to those of Rough Paths theory. We show in particular that such a geometric integration can be constructed substituting appropriately differentials with more general asymptotic expansions (of Stratonovich or Ito type) and by summing over a refining sequence of partitions, leading to a two-dimensional extension of the classical Young integral, that coincides with the integral introduced recently by R. Züst. We further show that Stratonovich sums gives an advantage allowing to weaken the requirements on Hölder exponents, and discuss some work in progress in the stochastic case. Based on joint works with E. Stepanov, G. Alberti and I. Ballieul.

 

Mon, 10 Jun 2019

14:15 - 15:15
L3

Gibbs measures of nonlinear Schrodinger equations as limits of many-body quantum states

VEDRAN SOHINGER
(University of Warwick)
Abstract

Gibbs measures of nonlinear Schrödinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. We study the problem of the derivation of Gibbs measures as high-temperature limits of thermal states in many-body quantum mechanics.

In our work, we apply a perturbative expansion in the interaction. This expansion is then analysed by means of Borel resummation techniques. In two and three dimensions, we need to apply a Wick-ordering renormalisation procedure. Moreover, in one dimension, our methods allow us to obtain a microscopic derivation of the time-dependent correlation functions for the cubic nonlinear Schrödinger equation. This is based partly on joint work with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein.

Mon, 03 Jun 2019

14:15 - 15:15
L3

Mean Field Langevin Dynamics and Its Applications to Neural Networks

DAVID SISKA
(University of Edinburgh)
Abstract

 

Neural networks are undoubtedly successful in practical applications. However complete mathematical theory of why and when machine learning algorithms based on neural networks work has been elusive. Although various representation theorems ensures the existence of the ``perfect’’ parameters of the network, it has not been proved that these perfect parameters can be (efficiently) approximated by conventional algorithms, such as the stochastic gradient descent. This problem is well known, since the arising optimisation problem is non-convex. In this talk we show how the optimization problem becomes convex in the mean field limit for one-hidden layer networks and certain deep neural networks. Moreover we present optimality criteria for the distribution of the network parameters and show that the nonlinear Langevin dynamics converges to this optimal distribution. This is joint work with Kaitong Hu, Zhenjie Ren and Lukasz Szpruch. 

 

Mon, 20 May 2019

15:45 - 16:45
L3

Low degree approximation of real singularities

ANTONIO LERARIO
(SISSA ITALY)
Abstract

In this talk I will discuss some recent results that allow to approximate a real singularity given by polynomial equations of degree d (e.g. the zero set of a polynomial, or the number of its critical points of a given Morse index) with a singularity which is diffeomorphic to the original one, but it is given by polynomials of degree O(d^(1/2)log d).
The approximation procedure is constructive (in the sense that one can read the approximating polynomial from a linear projection of the given one) and quantitative (in the sense that the approximating procedure will hold for a subset of the space of polynomials with measure increasing very quickly to full measure as the degree goes toinfinity).

The talk is based on joint works with P. Breiding, D. N. Diatta and H. Keneshlou      

Mon, 20 May 2019

14:15 - 15:15
L3

The renormalized wave equation in 3d with quadratic nonlinearity and additive white noise

HERBERT KOCH
(University of Bonn)
Abstract

Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a periodic domain. There are two new ingredients as compared to the parabolic setting. (i) In constructing stochastic objects, we have to carefully exploit dispersion at a multilinear level. (ii) We introduce novel random operators and leverage their regularity to overcome the lack of smoothing of usual paradifferential commutators

Mon, 13 May 2019

15:45 - 16:45
L3

Weak universality for the KPZ equation (and also others)

WEIJUN XU
(University of Oxford)
Abstract

Many singular stochastic PDEs are expected to be universal objects that govern a wide range of microscopic models in different universality classes. Two notable examples are KPZ and \Phi^4_3. In these cases, one usually finds a parameter in the system, and tunes according to the space-time scale in such a way that the system rescales to the SPDE in the large-scale limit. We justify this belief for a large class of continuous microscopic growth models (for KPZ) and phase co-existence models (for Phi^4_3), allowing microscopic nonlinear mechanisms far beyond polynomials. Aside from the framework of regularity structures, the main new ingredient is a moment bound for general nonlinear functionals of Gaussians. This essentially allows one to reduce the problem of a general function to that of a polynomial. Based on a joint work with Martin Hairer, and another joint work in progress with Chenjie Fan and Jiawei Li. 

Mon, 13 May 2019

14:15 - 15:45
L3

Solving nonlinear PDE's in the presence of singular randomness.

NIKOLAY TZETKOB
(University of Clergy France)
Abstract

We will start by presenting two basic probabilistic effects for questions concerning the regularity of functions and nonlinear operations on functions. We will then overview well-posedenss results for the nonlinear wave equation, the nonlinear Schr\"odinger equation and the nonlinear heat equation, in the presence of singular randomness.

Mon, 29 Apr 2019

15:45 - 16:45
L3

Inference of a large rank-one matrix and Hamilton-Jacobi equations

JEAN-CHRISTOPHE MOURRAT
(ENS FRANCE)
Abstract

We observe a noisy version of a large rank-one matrix. Depending on the strength of the noise, can we recover non-trivial information on the matrix? This problem, interesting on its own, will be motivated by its link with a "spin glass" model, which is a model of statistical mechanics where a large number of variables interact with one another, with random interactions that can be positive or negative. The resolution of the initial question will involve a Hamilton-Jacobi equation

Mon, 29 Apr 2019

14:15 - 15:15
L3

Scaling limits and surface tension for gradient Gibbs measure

WEI WU
(Warwick University)
Abstract

I will discuss new results for the gradient field models with uniformly convex potential (also known as the Ginzburg-Landau field). A connection between the scaling limits of the field and elliptic homogenization was introduced by Naddaf and Spencer in 1997. We quantify the existing central limit theorems in light of recent advances in quantitative homogenization; and positively settle a conjecture of Funaki and Spohn about the surface tension. Joint work with Scott Armstrong. 

 

Mon, 04 Mar 2019

15:45 - 16:45
L3

Numerical approximation of BSDEs with polynomial growth driver

ARNAUD LIONNET
(Birmingham University)
Abstract

Backward Stochastic Differential Equations (BSDEs) provide a systematic way to obtain Feynman-Kac formulas for linear as well as nonlinear partial differential equations (PDEs) of parabolic and elliptic type, and the numerical approximation of their solutions thus provide Monte-Carlo methods for PDEs. BSDEs are also used to describe the solution of path-dependent stochastic control problems, and they further arise in many areas of mathematical finance. 

In this talk, I will discuss the numerical approximation of BSDEs when the nonlinear driver is not Lipschitz, but instead has polynomial growth and satisfies a monotonicity condition. The time-discretization is a crucial step, as it determines whether the full numerical scheme is stable or not. Unlike for Lipschitz driver, while the implicit Bouchard-Touzi-Zhang scheme is stable, the explicit one is not and explodes in general. I will then present a number of remedies that allow to recover a stable scheme, while benefiting from the reduced computational cost of an explicit scheme. I will also discuss the issue of numerical stability and the qualitative correctness which is enjoyed by both the implicit scheme and the modified explicit schemes. Finally, I will discuss the approximation of the expectations involved in the full numerical scheme, and their analysis when using a quasi-Monte Carlo method.

Mon, 04 Mar 2019

14:15 - 15:15
L3

Support characterisation for path-dependent SDEs

ALEXANDER KALININ
(Imperial College)
Abstract

By viewing a stochastic process as a random variable taking values in a path space, the support of its law describes the set of all attainable paths. In this talk, we show that the support of the law of a solution to a path-dependent stochastic differential equation is given by the image of the Cameron-Martin space under the flow of mild solutions to path-dependent ordinary differential equations, constructed by means of the vertical derivative of the diffusion coefficient. This result is based on joint work with Rama Cont and extends the Stroock-Varadhan support theorem for diffusion processes to the path-dependent case.

Mon, 25 Feb 2019

15:45 - 16:45
L3

Reinforcement and random media

XIAOLIN ZENG
(University of Strasbourg)
Abstract

Abstract: The edge reinforced random walk is a self-interacting process, in which the random walker prefer visited edges with a bias proportional to the number of times the edges were visited. We will gently introduce this model and talk about some of its histories and recent progresses.

 

Mon, 25 Feb 2019

14:15 - 15:15
L3

Angles of Random Polytopes

DMITRY ZAPOROZHETS
(St. Petersburg University)
Abstract

We will consider some problems on calculating  the average  angles of random polytopes. Some of them are open.

Mon, 18 Feb 2019

15:45 - 16:45
L3

The branching-ruin number, the once-reinforced random walk, and other results

DANIEL KIOUS
(University of Bath)
Abstract

In a joint-work with Andrea Collevecchio and Vladas Sidoravicius,  we study  phase transitions in the recurrence/transience of a class of self-interacting random walks on trees, which includes the once-reinforced random walk. For this purpose, we define the branching-ruin number of a tree, which is  a natural way to measure trees with polynomial growth and therefore provides a polynomial version of the branching number defined by Furstenberg (1970) and studied by R. Lyons (1990). We prove that the branching-ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once-reinforced random walk on this tree. We will also mention two other results where the branching-ruin number arises as critical parameter: first, in the context of random walks on heavy-tailed random conductances on trees and, second, in the case of Volkov's M-digging random walk.

Mon, 18 Feb 2019

14:15 - 15:15
L3

Cut off phenomenon for the weakly asymmetric simple exclusion process

CYRIL LABBE
(Ceremade Dauphin)
Abstract

Consider the asymmetric simple exclusion process with k particles on a linear lattice of N sites. I will present results on the asymptotic of the time needed for the system to reach its equilibrium distribution starting from the worst initial configuration (also called mixing time). Two main regimes appear according to the strength of the asymmetry (in terms of k and N), and in both regimes, the system displays a cutoff phenomenon: the distance to equilibrium falls abruptly from 1 to 0. This is a joint work with Hubert Lacoin (IMPA).

 

 

Mon, 11 Feb 2019

15:45 - 16:45
L3

Small time asymptotics for Brownian motion with singular drift

TUSHENG ZHANG
(Manchester University)
Abstract

We consider Brownian motion with Kato class measure-valued drift.   A small time large deviation principle and a Varadhan type asymptotics for the Brownian motion with singular drift are established. We also study the existence and uniqueness of the associated Dirichlet boundary value problems.

Mon, 11 Feb 2019

14:15 - 15:15
L3

'Semilinear PDE and hydrodynamic limits of particle systems on fractals'

MICHAEL HINZ
(University Bielefeld)
Abstract

We first give a short introduction to analysis and stochastic processes on fractal state spaces and the typical difficulties involved.

We then discuss gradient operators and semilinear PDE. They are used to formulate the main result which establishes the hydrodynamic limit of the weakly asymmetric exclusion process on the Sierpinski gasket in the form of a law of large numbers for the particle density. We will explain some details and, if time permits, also sketch a corresponding large deviations principle for the symmetric case.

Mon, 04 Feb 2019

15:45 - 16:45
L3

The parabolic Anderson model in 2 d, mass- and eigenvalue asymptotics

WILLEM VAN ZUIJLEN
(WIAS Berlin)
Abstract


In this talk I present work in progress with Wolfgang König and Nicolas Perkowski on the parabolic Anderson model (PAM) with white noise potential in 2d. We show the behavior of the total mass as the time tends to infinity. By using partial Girsanov transform and singular heat kernel estimates we can obtain the mass-asymptotics by using the eigenvalue asymptotics that have been showed in another work in progress with Khalil Chouk. 

Mon, 04 Feb 2019

14:15 - 15:15
L3

Space-time localisation for the dynamic $\Phi^4_3$ model

HENDRIK WEBER
(University of Bath)
Abstract

We prove an a priori bound for solutions of the dynamic $\Phi^4_3$ equation.

This bound provides a control on solutions on a compact space-time set only in terms of the realisation of the noise on an enlargement of this set, and it does not depend on any choice of space-time boundary conditions.

We treat the  large and small scale behaviour of solutions with completely different arguments.For small scales we use bounds akin to those presented in Hairer's theory of regularity structures. We stress immediately that our proof is fully self-contained, but we give a detailed explanation of how our arguments relate to Hairer's. For large scales we use a PDE argument based on the maximum principle. Both regimes are connected by a solution-dependent regularisation procedure.

The fact that our bounds do not depend on space-time boundary conditions makes them useful for the analysis of large scale properties of solutions. They can for example be used in a compactness argument to construct solutions on the full space and their invariant measures

Mon, 28 Jan 2019

15:45 - 16:45
L3

A geometric perspective on regularity structures

YOUNESS BOUTAIB
(BERLIN UNIVERSITY)
Abstract

Abstract: We use groupoids to describe a geometric framework which can host a generalisation of Hairer's regularity structures to manifolds. In this setup, Hairer's re-expansionmap (usually denoted \Gamma) is a (direct) connection on a gauge groupoid and can therefore be viewed as a groupoid counterpart of a (local) gauge field. This definitions enables us to make the link between re-expansion maps (direct connections), principal connections and path connections, to understand the flatness of the direct connection in terms of that of the manifold and, finally, to easily build a polynomial regularity structure which we compare to the one given by Driver, Diehl and Dahlquist. (Join work with Sara Azzali, Alessandra Frabetti and Sylvie Paycha).

Mon, 28 Jan 2019

14:15 - 15:15
L3

Recent progress in 2-dimensional quantum Yang-Mills theory

THIERRY LEVY
(Paris)
Abstract

Quantum Yang-Mills theory is an important part of the Standard model built by physicists to describe elementary particles and their interactions. One approach to the mathematical substance of this theory consists in constructing a probability measure on an infinite-dimensional space of connections on a principal bundle over space-time. However, in the physically realistic 4-dimensional situation, the construction of this measure is still an open mathematical problem. The subject of this talk will be the physically less realistic 2-dimensional situation, in which the construction of the measure is possible, and fairly well understood.

In probabilistic terms, the 2-dimensional Yang-Mills measure is the distribution of a stochastic process with values in a compact Lie group (for example the unitary group U(N)) indexed by the set of continuous closed curves with finite length on a compact surface (for example a disk, a sphere or a torus) on which one can measure areas. It can be seen as a Brownian motion (or a Brownian bridge) on the chosen compact Lie group indexed by closed curves, the role of time being played in a sense by area.

In this talk, I will describe the physical context in which the Yang-Mills measure is constructed, and describe it without assuming any prior familiarity with the subject. I will then present a set of results obtained in the last few years by Antoine Dahlqvist, Bruce Driver, Franck Gabriel, Brian Hall, Todd Kemp, James Norris and myself concerning the limit as N tends to infinity of the Yang-Mills measure constructed with the unitary group U(N). 

 

Mon, 14 Jan 2019

15:45 - 16:45
L3

Nonparametric pricing and hedging with signatures

IMANOL PEREZ
(University of Oxford)
Abstract

We address the problem of pricing and hedging general exotic derivatives. We study this problem in the scenario when one has access to limited price data of other exotic derivatives. In this presentation I explore a nonparametric approach to pricing exotic payoffs using market prices of other exotic derivatives using signatures.