Forthcoming events in this series


Mon, 09 Feb 2015

14:15 - 15:15
Oxford-Man Institute

The Renormalization Group as a tool of Rigorous Probability Theory

Ajay Chandra
(Warwick University)
Abstract

The Renormalization Group (RG) was pioneered by the physicist Kenneth Wilson in the early 70's and since then it has become a fundamental tool in physics. RG remains the most general philosophy for understanding how many models in statistical mechanics behave near their critical point but implementing RG analysis in a mathematically rigorous way remains quite challenging.

I will describe how analysis of RG flows translate into statements about continuum limits, universality, and cross-over phenomena - as a concrete example I will speak about some joint work with Abdelmalek Abdesselam and Gianluca Guadagni.

Mon, 02 Feb 2015

15:45 - 16:45
Oxford-Man Institute

Spectral volume and surface measures via the Dixmier trace for local symmetric Dirichlet spaces with Weyl type eigenvalue asymptotics

Naotaka Kajino
Abstract

Spectral volume and surface measures via the Dixmier trace for local symmetric Dirichlet spaces with Weyl type eigenvalue asymptotics

 

The purpose of this talk is to present the author's recent results of on an

operator theoretic way of looking atWeyl type Laplacian eigenvalue asymptotics

for local symmetric Dirichlet spaces.

For the Laplacian on a d-dimensional Riemannian manifoldM, Connes' trace

theorem implies that the linear functional  coincides with

(a constant multiple of) the integral with respect to the Riemannian volume

measure of M, which could be considered as an operator theoretic paraphrase

of Weyl's Laplacian eigenvalue asymptotics. Here  denotes a Dixmier trace,

which is a trace functional de_ned on a certain ideal of compact operators on

a Hilbert space and is meaningful e.g. for compact non-negative self-adjoint

operators whose n-th largest eigenvalue is comparable to 1/n.

The first main result of this talk is an extension of this fact in the framework

of a general regular symmetric Dirichlet space satisfying Weyl type asymptotics

for the trace of its associated heat semigroup, which was proved for Laplacians

on p.-c.f. self-simiar sets by Kigami and Lapidus in 2001 under a rather strong

assumption.

Moreover, as the second main result of this talk it is also shown that, given a

local regular symmetric Dirichlet space with a sub-Gaussian heat kernel upper

bound and a (sufficiently regular) closed subset S, a “spectral surface measure"

on S can be obtained through a similar linear functional involving the Lapla-

cian with Dirichlet boundary condition on S. In principle, corresponds to the

second order term for the eigenvalue asymptotics of this Dirichlet Laplacian, and

when the second order term is explicitly known it is possible to identify  For

example, in the case of the usual Laplacian on Rd and a Lipschitz hypersurface S,is a constant multiple of the usual surface measure on S.

Mon, 02 Feb 2015

14:15 - 15:15
Oxford-Man Institute

Maximal couplings and geometry

Sayan Banerjee
(Warwick University)
Abstract

Maximal couplings are couplings of Markov processes where the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian couplings are coupling strategies where neither process is allowed to look into the future of the other before making the next transition. These are easier to describe and play a fundamental role in many branches of probability and analysis. Hsu and Sturm proved that the reflection coupling of Brownian motion is the unique Markovian maximal coupling (MMC) of Brownian motions starting from two different points. Later, Kuwada proved that to have a MMC for Brownian motions on a Riemannian manifold, the manifold should have a reflection structure, and thus proved the first result connecting this purely probabilistic phenomenon (MMC) to the geometry of the underlying space.

Mon, 19 Jan 2015

15:45 - 16:45
Oxford-Man Institute

A stochastic free boundary problem

Martin Keller-Ressel
(Dresden University of Technology)
Abstract

Motivated by stochastic models for order books in stock exchanges we consider stochastic partial differential equations with a free boundary condition. Such equations can be considered generalizations of the classic (deterministic) Stefan problem of heat condition in a two-phase medium. 

Extending results by Kim, Zheng & Sowers we allow for non-linear boundary interaction, general Robin-type boundary conditions and fairly general drift and diffusion coefficients. Existence of maximal local and global solutions is established by transforming the equation to a fixed-boundary problem and solving a stochastic evolution equation in suitable interpolation spaces. Based on joint work with Marvin Mueller.

Mon, 19 Jan 2015

14:15 - 15:15
Oxford-Man Institute

'Optimal Switching in Finite Horizon under State Constraints’

Idris Kharoubbi
(Université Paris Dauphine)
Abstract

'We study an optimal switching problem with a state constraint: the controller is only allowed to choose strategies that keep the controlled diffusion in a closed domain. We prove that the value function associated to the weak formulation of this problem is the limit of the value function associated to an unconstrained switching problem with penalized coefficients, as the penalization parameter goes to infinity. This convergence allows to set a dynamic programming principle for the constrained switching problem. We then prove that the value function is a constrained viscosity solution to a system of variational inequalities (SVI for short). We finally prove that the value function is the maximal solution to this SVI. All our results are obtained without any regularity assumption on the constraint domain.’

Mon, 01 Dec 2014
14:15
Oxford-Man Institute

Conformal restriction: 3-point chordal case.

Wei Qian
(ETH Zurich)
Abstract

Lawler, Schramm and Werner studied 2-point chordal restriction measures and gave several constructions using SLE tools.

It is possible to characterize 3-point chordal restriction measures in a similar manner. Their boundaries are SLE(8/3)-like curves with a slightly different drift term.

@email

Mon, 24 Nov 2014

15:45 - 16:45
Oxford-Man Institute

Recombination, Scenario reduction, and nested high order integration with positive weights.

Terry Lyons and Maria Tchernychova
(Oxford University)
Abstract

Cubature is the business of describing a probability measure in terms of an empirical measure sharing its support with the original measure, of small support, and with identical integrals for a class of functions (eg polynomials with degree less than k). 

Applying cubature to already discrete sets of scenarios provides a powerful tool for scenario management and summarising data.  We refer to this process as recombination. It is a feasible operation in real time and has lead to high accuracy pde solvers.

The practical complexity of this operation has changed! By a factor corresponding to the dimension of the space of polynomials. 

We discuss the algorithm and give home computed examples of nested sparse grids with only positive weights in moderate dimensions (eg degree 1-8 in dimension 7).  Positive weights have significant advantage over signed ones when available.
 

Mon, 24 Nov 2014
14:15
Oxford-Man Institute

Learning in high dimension with multiscale invariants

Stephane Mallat
(CMAP ecole polytechnique)
Abstract

   Stéphane Mallat

   Ecole Normale Superieure

Learning functionals in high dimension requires to find sources of regularity and invariants, to reduce dimensionality. Stability to actions of diffeomorphisms is a strong property satisfied by many physical functionals and most signal classification problems. We introduce a scattering operator in a path space, calculated with iterated multiscale wavelet transforms, which is invariant to rigid movements and stable to diffeomorphism actions. It provides a Euclidean embedding of geometric distances and a representation of stationary random processes. Applications will be shown for image classification and to learn quantum chemistry energy functionals.

Mon, 17 Nov 2014
15:45
Oxford-Man Institute

An ergodic backward stochastic differential equation approach to large time behaviour of some parabolic semilinear PDEs

ADRIEN RICHOU
(Bordeaux University)
Abstract

In this talk we study the large time behaviour of some semilinear parabolic PDEs by a purely probabilistic approach. For that purpose, we show that the solution of a backward stochastic differential equation (BSDE) in finite horizon $T$ taken at initial time behaves like a linear term in $T$ shifted with a solution of the associated ergodic BSDE taken at inital time. Moreover we give an explicit rate of convergence: we show that the following term in the asymptotic expansion has an exponential decay. This is a Joint work with Ying Hu and Pierre-Yves Meyer from Rennes (IRMAR - France).

Mon, 17 Nov 2014
14:15
Oxford-Man Institute

LÉVY KHINTCHINE FORMULA FOR ROUGH PATHS

ATUL SHEKHAR
(Berlin University of Technology)
Abstract

In this talk, we develop rough integration with jumps, offering a pathwise view on stochastic integration against cadlag processes.  A class of Marcus-like rough paths is introduced,which contains D. Williams’ construction of stochastic area for Lévy processes. We then established a Lévy–Khintchine type formula for the expected signature, based on“Marcus(canonical)"stochastic calculus. This calculus fails for non-Marcus-like Lévy rough paths and we treat the general case with Hunt’ theory of Lie group valued Lévy processes is made.

Mon, 10 Nov 2014
15:45
Oxford-Man Institute

"Limit theorems for ambit fields"

MARK PODOLSKIJ
(Heidelberg University)
Abstract

In this talk we will present some recent developments in the theory of ambit fields with a particular focuson limit theorems.
Ambit fields is a tempo-spatial class of models, which has been originally introduced by Barndorff-Nielsen and Schmiegel in the context of turbulence,
but found applications also in biology and finance. Its purely temporal analogue, Levy semi-stationary processes, has a continuous moving average structure
with an additional multiplicative random input (volatility or intermittency). We will briefly describe the main challenges of ambit stochastics, which
include questions from stochastic analysis, statistics and numerics. We will then focus on certain type of high frequency functionals typically called power variations.
We show some surprising non-standard limit theorems, which strongly depend on the driving Levy process. The talk is based on joint work with O.E. Barndorff-Nielsen, A. Basse-O'Connor,
J.M. Corcuera and R. Lachieze-Rey. 

Mon, 10 Nov 2014
14:15
Oxford-Man Institute

A stochastic free boundary problem

Martin Keller-Ressel
(Dresden University of Technology)
Abstract

Motivated by stochastic models for order books in stock exchanges we consider stochastic partial differential equations with a free boundary condition. Such equations can be considered generalizations of the classic (deterministic) Stefan problem of heat condition in a two-phase medium. 

Extending results by Kim, Zheng & Sowers we allow for non-linear boundary interaction, general Robin-type boundary conditions and fairly general drift and diffusion coefficients. Existence of maximal local and global solutions is established by transforming the equation to a fixed-boundary problem and solving a stochastic evolution equation in suitable interpolation spaces. Based on joint work with Marvin Mueller.

@email 

Mon, 03 Nov 2014
15:45
Oxford-Man Institute

Selection and dimension

Nic Freeman
(Bristol University)
Abstract

I will describe the Spatial Lambda-Fleming-Viot process, which is a model of evolution in a spatial continuum, and discuss the time and spatial scales on which selectively advantageous genes propagate through space. The appropriate scaling depends on the dimension of space, resulting in three distinct cases; d=1, d=2 and d>=3. In d=1 the limiting genealogy is the Brownian net whereas, by contrast, in d=2 local interactions give rise to a delicate damping mechanism and result in a finite limiting branching rate. This is joint work with Alison Etheridge and Daniel Straulino.

Mon, 03 Nov 2014
14:15
Oxford-Man Institute

The Parabolic Anderson Model on R^3

Cyril Labbe
(University of Warwick)
Abstract

The theory of regularity structures allows one to give a meaning to several stochastic PDEs, including the Parabolic Anderson Model. So far, these equations have been considered on a torus. The goal of this talk is to explain how one can define the PAM on the whole space R^3. This is a joint work with Martin Hairer.

Mon, 27 Oct 2014

15:45 - 16:45
Oxford-Man Institute

Phase transitions in Achlioptas processes

Lutz Warnke
(University of Cambridge)
Abstract

In the Erdös-Rényi random graph process, starting from an empty graph, in each step a new random edge is added to the evolving graph. One of its most interesting features is the `percolation phase transition': as the ratio of the number of edges to vertices increases past a certain critical density, the global structure changes radically, from only small components to a single giant component plus small ones.

In this talk we consider Achlioptas processes, which have become a key example for random graph processes with dependencies between the edges.

Starting from an empty graph these proceed as follows: in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. We discuss why, for a large class of rules, the percolation phase transition is qualitatively comparable to the classical Erdös-Rényi process.

                                                      

Based on joint work with Oliver Riordan.

Mon, 27 Oct 2014

14:15 - 15:15
Oxford-Man Institute

Some results on maps that factor through a tree

Roger Zuest
(Institut Maths Jussieu -Paris)
Abstract

We give a necessary and sufficient condition for a map defined on a compact, quasiconvex and simply-connected space to factor through a tree. This condition can be checked using currents. In particular if the target is some Euclidean space and the map is H\"older continuous with exponent bigger than 1/2, such maps can be characterized by the vanishing of some integrals over the winding number. Moreover, this shows that if the target is the Heisenberg group equipped with the Carnot-Carath\'eodory metric and the H\"older exponent of the map is bigger than 2/3, the map factors through a tree.

Mon, 13 Oct 2014

15:45 - 16:45
Oxford-Man Institute

A-free Groups and Tree-free Groups

IAN CHISWELL
(Queen Mary University London)
Abstract

The idea of A-free group, where A is a discrete ordered abelian group, has been introduced by Myasnikov, Remeslennikov and Serbin. It generalises the construction of free groups. A proof will be outlined that a group is A-free for some A if and only if it acts freely and without inversions on a \lambda-tree, where \lambda is an arbitrary ordered abelian group.

Mon, 13 Oct 2014

14:15 - 15:15
Oxford-Man Institute

Ito map and iterated integrals

Horatio Boedihardjo
(Oxford-Man Institute)
Abstract

The Taylor expansion of a controlled differential equation suggests that the solution at time 1 depends on the driving path only through the latter's iterated integrals up to time 1, if the vector field is infinitely differentiable. Hambly and Lyons proved that this remains true for Lipschitz vector fields if the driving path has bounded total variation. We extend the Hambly-Lyons result for weakly geometric rough paths in finite dimension. Joint work with X. Geng, T. Lyons and D. Yang.    

 

 

Mon, 16 Jun 2014

15:45 - 16:45
Oxford-Man Institute

Efficient PDE methods for multivariate option pricing

OLEG REICHMANN
(ETH Zurich)
Abstract

We consider the numerical approximation of Kolmogorov equations arising in the context of option pricing under L\'evy models and beyond in a multivariate setting. The existence and uniqueness of variational solutions of the partial integro-differential equations (PIDEs) is established in Sobolev spaces of fractional or variable order.

Most discretization methods for the considered multivariate models suffer from the curse of dimension which impedes an efficient solution of the arising systems. We tackle this problem by the use of sparse discretization methods such as classical sparse grids or tensor train techniques. Numerical examples in multiple space dimensions confirm the efficiency of the described methods.

Mon, 16 Jun 2014

14:15 - 15:15
Oxford-Man Institute

Topologies of nodal sets of band limited functions

IGOR WIGMAN
(Kings College London)
Abstract

This work is joint with Peter Sarnak.

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.

Mon, 09 Jun 2014

15:45 - 16:45
Oxford-Man Institute

Phase transitions in a class of infinite particle systems.

ALEX DALETSKI
(York University)
Abstract

We study infinite (random) systems of interacting particles living in a Euclidean space X and possessing internal parameter (spin) in R¹. Such systems are described by Gibbs measures on the space Γ(X,R¹) of marked configurations in X (with marks in R¹). For a class of pair interactions, we show the occurrence of phase transition, i.e. non-uniqueness of the corresponding Gibbs measure, in both 'quenched' and 'annealed' counterparts of the model.

Mon, 09 Jun 2014

14:15 - 15:15
Oxford-Man Institute

Integral representation of martingales motivated by the problem of market completion with derivative securities.

DANIEL C SCHWARZ
(University of Oxford)
Abstract

A model of a financial market is complete if any payoff can be obtained as the terminal value of a self-financing trading strategy. It is well known that numerous models, for example stochastic volatility models, are however incomplete. We present conditions, which, in a general diffusion framework, guarantee that in such cases the market of primitive assets enlarged with an appropriate number of traded derivative contracts is complete. From a purely mathematical point of view we prove an integral representation theorem which guarantees that every local Q-martingale can be represented as a stochastic integral with respect to the vector of primitive assets and derivative contracts.

Mon, 02 Jun 2014

15:45 - 16:45
Oxford-Man Institute

Integrating slow-varying linear one-forms against rough path.

DANYU YANG
(Univeristy of Oxford)
Abstract

Abstract: We introduce a new framework for defining integration against rough path. This framework generalizes rough integral, and gives a natural explanation of some of the regularity requirements in rough path theory.

Mon, 02 Jun 2014

14:15 - 15:15
Oxford-Man Institute

We consider the short time asymptotics of the heat content $E(s)$ of a domain $D$ of $\mathbb{R}^d$, where $D$ has a random boundary.

PHILIPPE CHARMOY
(University of Oxford)
Abstract

When $\partial D$ is spatially homogeneous, we show that we can recover the lower and upper Minkowski dimensions of $\partial D$ from the sort time behaviour of $E(s)$. Furthermore, when the Minkowski dimension exists, finer geometric fluctuations can be recovered and $E(s)$ is controlled by $s^\alpha e^{f(\log(1/s))}$ for small $s$, for some $\alpha \in (0, \infty)$ and some regularly varying function $f$. The function $f$ is not constant is general and carries some geometric information.

When $\partial D$ is statistically self-similar, the Minkowski dimension and content of $\partial D$ typically exist and can be recovered from $E(s)$. Furthermore, $E(s)$ has an almost sure expansion $E(s) = c s^{\alpha} N_\infty + o(s^\alpha)$ for small $s$, for some $c$ and $\alpha \in (0, \infty)$ and some positive random variable $N_\infty$ with unit expectation arising as the limit of some martingale. In some cases, we can show that the fluctuations around this almost sure behaviour are governed by a central limit theorem, and conjecture that this is true more generally.

This is based on joint work with David Croydon and Ben Hambly.