Curs 2014-2015: Abstracts i Slides

8/07/15:  Iain Johnstone, Stanford University, Standford, USA.

Likelihood ratios for eigenvalues in spiked multivariate models.

In 1964 Alan James gave a remarkable classification of many of the eigenvalue distribution problems of multivariate statistics. We revisit the classification, now from the viewpoint of high dimensional models and low rank departures from the usual null hypotheses. For each of James' models, we describe the phase transition of the largest eigenvalue, and derive the asymptotic behavior of the likelihood ratios that correspond to null and alternative hypotheses about sub- and super-critical spikes. We find that the statistical experiment of observing the eigenvalues in the super-critical regime, converges to a simple Gaussian shift experiment. Our findings for the sub-critical regime are totally different: the experiment of observing the eigenvalues converges to a Gaussian sequence experiment, and no optimal test about a sub-critical spike is available. (Joint work with Alexei Onatski and Prathapa Dharmawansa).

1/07/15:  John Walsh, University of British Columbia, Vancouver, Canada.

Some Remarks on Stochastic Partial Differential Equations

This is a rather informal account of one person's experience with and understanding of stochastic partial differential equations. It will touch on a number of the ideas that go into the subject, both as they appeared when first encountered, and the (usually simpler) way they appear now. It begins with multi-parameter martingales and the all-important Brownian sheet. This leads inevitably to SPDEs, since the two-parameter Brownian sheet $B_{st}$ is a solution of the SPDE $\frac{\partial^2 B}{\partial s \partial t} = \dot W$. Here, $\dot W_{st}$ is a white noise on the plane. (Rotate coordinates by 45 degrees: this becomes the stochastic wave equation.) Once that was realized, many known facts about the Brownian sheet became easy to explain: they're just waves! And some new facts about solutions of the stochastic wave equation became equally obvious, since they were known for the Brownian sheet. We will discuss some physical and biological examples of SPDEs, including at least one that has yet to be (successfully) treated, and we will mention a few open problems in the process. While we will make an attempt to be as intuitive as possible, some of the most important facts of the subject are basically technical. For example, in spaces of dimensions two or greater, the solutions of the stochastic heat and wave equations driven by white noise are pure Schwartz distributions, not functions. That's Technicality One. Technicality Two is that Schwartz distributions do not form an algebra: they cannot be multiplied. The consequence of these is a simple, non-technical fact: we cannot treat general non-linear SPDE in this setting. This is why the solution of non-linear equations in higher dimensions is the holy grail of the subject.

1/07/15:  Henry Schellhorn, Institute of Mathematical Sciences Claremont Graduate University Claremont, USA.

A New Representation of Smooth Brownian Martingales.

We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented as an exponential of its value at a later time. The time-dependent generator of this exponential operator is equal to one half times the second order Malliavin derivative. This result can be seen as a generalization of the semi-group theory of parabolic partial differential equations to the parabolic path-dependent partial differential equations introduced by Dupire (2009) and Cont and Fournié (2011). The exponential operator can be calculated explicitly in a series expansion, which resembles the Dyson series of quantum field theory. Indeed, in the latter, the generator of the semi-group is the Hamiltonian (divided by the Planck constant), as opposed to (one half times) the second-order Malliavin derivative for martingales. Our continuous-time martingale representation result can be proved either in the time domain or in the frequency domain. In the time domain, it is proved by a passage to the limit of a special case of a backward Taylor expansion of an approximating discrete-time martingale. The latter expansion can also be used for numerical calculations, for instance in optimal stopping problems. We present several other applications of our representation, mostly to mathematical finance. We will also extend our results to fractional Brownian motion. The talk is based on joint work with Qidi Peng, and Sixian Jin, and frequent conversations with Josep Vives.

30/05/15: Gabriel Lord, Heriot Watt University, Edinburgh.

Stochastic travelling waves and computation.

We examine a new numerical method for solving Stratonovich SDEs. In particular we are interested in computing stochastic travelling waves. Travelling waves are often of physical interest and we have applications from models of neural tissue that are both SPDEs and large SDE systems. We introduce a technique where we move to a travelling wave frame and stop the wave from moving. This has some computational advantages as a small domain can be used but we will also discuss some potential pitfalls.

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29/05/15: Istvan Gyongy, The University of Edinburgh, Edinburgh.

On stochastic partial differential equations of parabolic type.

A brief introduction to the theory of stochastic partial differential equations will be presented. Applications to nonlinear filtering problems will be discussed. In particular, new results in the innovation problem will be given. This part of the talk is based on joint work with Nick Krylov.

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27/05/15: Istvan Gyongy, The University of Edinburgh, Edinburgh.

On numerical solutions of degenerate stochastic PDEs.

We consider second order parabolic stochastic PDEs which may degenerate and become first order PDEs. We present sharp estimates for the error of approximations obtained by accelerated finite difference schemes and localizations. The talk is based on joint works with N.V. Krylov and a recent work with Máté Gerencsér.

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13/05/15: Maria Jolis, Universitat Autònoma de Barcelona, Bellaterra.

On the norming constants for normal maxima.

Given $n$ independent standard normal random variables it is known that their maxima $M_n$ can be normalized in such a way that they converge in distribution to the Gumbel law, whose distribution function is given by $\Lambda(x)=\exp\{-e^{-x}\}$. In a remarkable paper of Peter Hall it is proved that the Kolmogorov distance $d_n$ between the normalized $M_n$ and its associated limit distribution is less than $3/\log n$. In this paper we propose a different set of norming constants that allow to decrease this upper bound with $d_n\le C(m)/\log n$ for $n\ge m\ge 5.$ Furthermore, the function $C(m)$ is explicitly computed and satisfies that $C(m)\le 1$ and $\lim_{m\to \infty} C(m)=1/3$. Moreover, some new and effective norming constants are provided by using the asymptotic expansion of a Lambert $W$ type function. This is a joint paper with Armengol Gasull and Frederic Utzet.

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25/03/15: Marco Romito, Università di Pisa, Italy.

Densities for the Navier-Stokes equations with noise .

We present a proof of existence of the density with respect to the Lebesgue measure, as well as of regularity in Besov spaces, for the solution of stochastic differential equations with non--smooth'' data. As an application we show existence of densities for the finite dimensional marginal distributions of the law of solutions of the 3D Navier-Stokes equations forced by Gaussian noise. Classical methods, such as the Malliavin calculus, do not work in this setting for reasons that are strongly related to the three dimensional case. The same method provides also regularity in time of the densities, as well as absolute continuity of the laws of some quantities (energy and dissipation rate, for instance) that depend on a infinite number of components. A stronger result, namely H\"older continuity of the densities, is available through a suitable conditioned Fokker--Planck equation. When the random forcing has no full support in Fourier space the problem is open. In this case we can prove the mere existence of a density, without any regularity property, by using the backward local smoothness of trajectories, and weak--strong uniqueness.

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4/03/15:  Joaquim Ortega-Cerdà, Universitat de Barcelona, Barcelona.

Determinantal point process: The spherical ensemble.

We will overview the properties of determinantal point processes with special emphasis on the spherical ensemble. The spherical ensemble is a rotationally invariant random point process in the sphere obtained by computing the generalized eigenvalues of a couple of random matrices. We will study quantitatively the behaviour of the process as the number of points increases.

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25/02/15: Vlad Bally, Université Paris-Est Marne-la-Vallée, Paris, France.

Convergence and regularity of probability laws by using an interpolation method.

Recentely Fournier and Printems established a methodology which allows to prove the absolute continuity of the law of the solution of some stochastic equations with Holder continuous coefficients. This is of course out of reach by using already classical probabilistic methods based on Malliavin calculus. Afterwords Debussche and Romito employ some Besov space technics in order to substantially improve the result of Fournier and Printems. In our paper we show that this kind of problem naturally fits in the framework of interpolation spaces: we prove an interpolation inequality which allows to state (and even to slightly improve) the above absolute continuity result. Moreover it turns out that the above interpolation inequality has applications in a completely different framework: we use it in order to estimate the error in total variance distance in some convergence theorems. This talk is based on a joint work with Lucia Caramellino (Università di Roma - Tor Vergata).

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18/02/15: Peter Imkeller, Institut fuer Mathematik Humboldt-Universitaet zu Berlin, Germany.

A Fourier analysis based approach of integration.

In 1961, Ciesielski established a remarkable isomorphism of spaces of Hölder continuous functions and Banach spaces of real valued se- quences. The isomorphism can be established along Fourier type expansions of (rough) Holder continuous functions by means of the Haar-Schauder wavelet. We will use Schauder representations for a pathwise approach of the integral of one rough function with respect to another one, via Ciesielski's isomorphism. In a more general and analytical setting, this approach of rough path analysis can be understood in terms of Paley-Littlewood decompositions of distributions, and Bony paraproducts in Besov spaces. It is well suited for Hairer's approach of SPDE. In a stochastic analysis context, the resulting integral is closely related to Stratonovich's or Ogawa's concepts. To recover Itô's integral for instance in Föllmer's pathwise approach requires some additional knowledge of the quadratic variation. This talk is based on work in progress with M. Gubinelli (U Paris-Dauphine) and N. Perkowski (HU Berlin).

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18/02/15: Ivan Nourdin, Université de Luxembourg.

Gaussian Phase Transitions for Conic Intrinsic Volumes.

Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, for instance in the compressed sensing theory. In this talk I will explain why, in the high-dimensional limit, most conic intrinsic volumes encountered in applications can be approximated by a suitable Gaussian distribution and lead to a phase transition. This talk is based on a joint work with Larry Goldstein (Southern California) and Giovanni Peccati (Luxembourg).

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28/01/15: Samy Tindel,Dépt. de Mathématiques - IECN Université de Lorraine, Vandoeuvre-lès-Nancy, France.

Feynman-Kac representations for the stochastic heat equation.

We focus in this talk on stochastic heat equations whose noisy part is of the form u W, where u is the solution to the equation and W a rather general Gaussian noise. This model is usually called parabolic Anderson model, and is related to many physically relevant systems such as KPZ equation. We will first motivate the model, then show how to define and solve the stochastic heat equation. We shall derive some Feyman-Kac representations for the solution, either in a pathwise way or for moments. These Feyman-Kac formulae always involve some weighted Brownian local times. Finally, we obtain some moment estimates which entail the so-called intermittency phenomenon. If time allows it, we shall also give some perspectives on future works in this direction, concerning irregular noises W.

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10/12/14: Daniel Campos, Dept. Física, Universitat Autònoma de Barcelona, Bellaterra.

The Optimal Walk to the Random Walk.

First-passage processes in random walks represent a useful paradigm for many problems in physics, chemistry, economics or psychology, among other disciplines. In the last years there has been a growing interest, in particular from the ecological literature, in predicting the existence of optimal solutions to these first–passage processes in order to verify if such optimal solutions do actually reflect in real foraging trajectories followed by animals or microorganisms. Such interest has given rise to an area of research now termed as stochastic search theory. This topic represents a beautiful way in which the theory of stochastic processes can be used as a test model to improve our understanding on the (evolutionary) nature of living organisms. In this talk, I will use basic and well-known examples of random walks (founded, e.g., on Wiener, Ornstein-Uhlenbeck or Lévy processes) to illustrate some of the results and advances achieved recently, as well as the main open challenges, in this area.

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26/11/14: Eulàlia Nualart, Universitat Pompeu Fabra, Barcelona.

Noise excitability of the stochastic heat equation.

We study the behaviour of the moments and the Lyapunov exponent of the stochastic heat equation with Dirichlet boundary conditions depending on the amount of noise. Joint work with M. Foondun.

Backward stochastic differential equations with quadratic growth and measurable generators.

In a first step, we establish the existence (and sometimes the uniqueness) of solutions for a large class of quadratic backward stochastic differential equations (QBSDEs) with continuous generator and a merely square integrable terminal condition. Our approach is different from those existing in the literature. Although we are focused on QBSDEs, our existence result also covers the BSDEs with linear growth, keeping $\xi$ square integrable in both cases. As byproduct, the existence of viscosity solutions is established for a class of quadratic partial differential equations (QPDEs) with a square integrable terminal datum. In a second step, we consider QBSDEs with measurable generator for which we establish a Krylov's type a priori estimate for the solutions. We then deduce an Itô-Krylov's change of variable formula. This allows us to establish various existence and uniqueness results for classes of QBSDEs with square integrable terminal condition and sometimes a merely measurable generator. Our results show, in particular, that neither the existence of exponential moments of the terminal datum nor the continuity of the generator are necessary to the existence and/or uniqueness of solutions for quadratic BSDEs. Some comparison theorems are also established for solutions of a class of QBSDEs.

22/10/14: Lluís Quer-Sardanyons, Universitat Autònoma de Barcelona, Bellaterra.

SPDEs with fractional noise in space with index H<1/2.

We consider the stochastic wave and heat equations on $\mathbb{R}$ with non-vanishing initial conditions, driven by a Gaussian noise which is white in time and behaves in space like a fractional Brownian motion of index H, with 1/4<H<1/2. We assume that the diffusion coefficient is given by an affine function, and the initial value functions are bounded and Hölder continuous of order H. We prove the existence and uniqueness of the mild solution for both equations. We show that the solution is $L^{2}(\Omega)$-continuous and its p-th moments are uniformly bounded, for any $p \geq 2$.

The talk is based on a joint article with Raluca Balan (University of Ottawa) and Maria Jolis (UAB).

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09/09/14: Marco Ferrante, Università di Padova, Itàlia.

On a stochastic epidemic SEIHR model and its diffusion approximation.

In this talk I will present a joint paper with Elisabetta Ferraris and Carles Rovira, where we have generalized a simple SIR type model defined by Henry Tuckwell and Ruth Williams in 2007. All the SIR type models share the following structure: the population is divided into some classes which represent the state of the contagious. Usually with S we denote the class of susceptible individuals, by I that of infectious and by R that of the removed, which become immune. If one is interested in modelling diseases where an initial incubation period is present, as well as a period when the individual is infectious, but still not aware to be ill, a simple SIR model is no more adequate. For example, the varicella disease is poorly described by a SIR model. To overcome this problem, we have introduced two additional classes, in order to define what we will call a SEIHR models, where E denotes the class of the exposed individuals, who therefore are in the incubation non infectious period, and H that of the individuals which are infectious and sick, so often are hospitalized. Following [TW], we will assume that all the individuals but at least one are initially in the class S, the time is discretized and the unit length is one day. We assume that every day any susceptible individual meet a given number of different individuals and if one of these is infectious the diseases is transmitted with a given probability. When an individual enter in the class E, will stay in this class for a given, fixed number of days, then he will pass to the following class I, stay there for a fixed, possible different, given number of days and so on. Once in the class R he will be no more susceptible and will be removed'' by the system. In this way we will have a simple model of a disease with an incubation period as well as a period when the individual is infectious, but still not aware to be ill. We will derive an explicit structure for this discrete time Markov chain and we will simulate some possible scenarios. Furthermore, we will derive a simple diffusion approximation of this model. The resulting stochastic differential equation will present multiple delays, due to the presence of the additional classes E and H. We will see, via numerical simulation, how the discrete time and the diffusion models are close and we will apply both the models to the case of the varicella disease, with a good description of the multi-peak behaviour of this disease.

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05/09/14: André Suess, Universitat de Barcelona, Catalunya.

Thesis Defense: Contributions to Stochastic Integration and Stochastic Partial Differential Equations.

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04/09/14: Francesco Russo, ENSTA ParisTech, França.

Kolmogorov equations related to frames of diffusion processes and related path dependent calculus.

First we remind the framework of Banach space valued via regularizations introduced by C. Di Girolami and the speaker and the notion of robust replication of a random variable. The second part of the talk will be devoted to the study of the Kolmogorov type equation associated with the so called window Brownian motion, called path-dependent heat equation, for which well-posedness at the level of strict solutions is established. Third, a notion of strong approximating solution, called strong-viscosity solution, is introduced is introduced which is supposed to be a substitution tool to the viscosity solution. The definition of strong-viscosity solution will be extended to semilinear PDEs associated with the path dependent heat equation. This is inspired by the notion of “good" solution, and it is based again on an approximating procedure. This talk is based on a collaboration with Andrea Cosso (Politecnico Milano and Paris VII) and Cristina Di Girolami (Pescara).

04/09/14: Annie Millet, SAMM, Université Paris 1 and LPMA, França.

LDP and the zero viscosity limit for 2D NSE.

Using a weak convergence approach, we prove a Large Deviation Principle for the solution of 2D stochastic Navier Stokes equations when the viscosity converges to 0 and the noise intensity is multiplied by the square root of the viscosity. The weak convergence is proven by tightness properties of the distribution of the solution in appropriate functional spaces. This a joint work with Hakima Bessaih.

04/09/14: Robert C. Dalang, Institut de mathématiques, EPFL, Suïssa.

Multiple points of the Brownian sheet in critical dimensions.

It is well-known that an $N$-parameter $d$-dimensional Brownian sheet has no $k$-multiple points when $(k-1)d> 2kN$, and does have such points when $(k-1)d< 2kN$. Existing results on hitting probabilities for general Gaussian processes do not provide information concerning the critical dimensions. We complete the study of the existence of $k$-multiple points by showing that in the critical cases where $(k-1) d= 2kN$, there are a.s. no $k$-multiple points. This is joint work with Carl Mueller (University of Rochester).

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