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Titles / Abstracts

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Monday, June 8

Hao Wu (MIT)
Monday, June 8 * 9:00 a.m.

Conformal restriction: the chordal and the radial

When people tried to understand two-dimensional statistical physics models, it is realized that any conformally invariant process satisfying a certain restriction property has crossing or intersection exponents.

Conformal field theory has been extremely successful in predicting the exact values of critical exponents describing the behavior of two-dimensional systems from statistical physics. The main goal of this talk is to investigate the restriction property and related critical exponents. First, we will introduce Brownian intersection exponents.

Second, we discuss Conformal Restriction—the chordal case—and the relation to halp-plane Browinian intersection exponents. Finally, we discuss Conformal Restriction—the radial case—and the relation to whole-plane Brownian intersection exponents.

Ciprian Tudor (Université Lille 1)
Monday, June 8 * 10:30 a.m.

Sample paths of the solution to the heat equation with fractional noise

The solutions to the heat equation with fractional noise represents an interesting examples of  self-similar process  without stationary increments.  We will discuss various path properties of these processes with respect to the time and space variable, such as the spectral measure, the moduli of continuity or the quadratic variations.

Yuliya Mishura (National University of Kyiv)
Monday, June 8 * 11:30 a.m.

Between two self-similarities

What can happen between two self-similarities? A lot of good things! So, we discuss the problems related to Gaussian processes whose covariance distance satisfies two-sided power inequalities. At least, three properties will be considered:

  1. Asymptotivc behavior with probability 1 and connected statistical estimators.
  2. Representative properties. It means that we discuss the possibility of the representation of any generated random variable as the integral w.r.t. the corresponding Gaussian process.
  3. Asymptotic behavior of some functionals including supremum of Gaussian process.

Jan Rosinski (University of Tennessee)
Monday, June 8 * 2:30 p.m.

An isomorphism theorem for infinitely divisible processes and related topics

We propose an isomorphism theorem for Poissonian processes, i.e., infinitely divisible processes without Gaussian component. The framework of our result involves the concept of Levy measures of general infinitely divisible processes as well as series and integral representations. We will illustrate this result showing how Dynkin’s Isomorphism Theorem follows in our setting. Recall that Dynkin Isomorphism Theorem relates local times of strongly symmetric Markov processes to Gaussian processes.  Other applications of the isomorphism theorem will also be discussed.

Yizao Wang (University of Cincinnati)
Monday, June 8 * 4:00 p.m.

Invariance principles for operator-scaling Gaussian random fields

Recently, Hammond and Sheffield (2013) introduced a model of correlated random walks that scale to fractional Brownian motions with long-range dependence. In this paper, we consider a natural generalization of this model to dimension d\geq 2. We define a Z^d-indexed random field with dependence relations governed by an underlying random graph with vertices Z^d, and we study the scaling limits of the partial sums of the random field over rectangular sets. An interesting phenomenon appears: depending on how fast the rectangular sets increase along different directions, different random fields arise in the limit. In particular, there is  a critical regime where the limit random field is operator-scaling  and inherits the full dependence structure of the discrete model, whereas in other regimes the limit random fields have at least one direction that has either invariant or independent increments, no longer reflecting the dependence structure in the discrete model. The limit random fields form a general class of operator-scaling Gaussian random fields. Their increments and path properties are investigated.

Joint work with Hermine Biermé and Olivier Durieu.

Tuesday, June 9

David Nualart (University of Kansas)
Tuesday, June 9 * 9:00 a.m.

Numerical approximation schemes for fractional diffusions

The purpose of this talk is to present  a new modified Euler scheme for stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H>1/2.   The rate of convergence of this numerical scheme with step size 1/n turns out to be n^{ 1/2-H} if H<3/4, n^{-1} \sqrt{\log n} if H=3/4 and n^{-1} if H>3/4.  These results have been obtained applying techniques of Malliavin calculus. We will also discuss the corresponding weak approximation results and central limit theorems for the fluctuations of the approximation error.

Céline Lacaux (Université Elie Cartan, Nancy)
Tuesday, June 9 * 10:30 a.m.

Modulus of continuity of operator scaling random fields and sub-Gaussian fields

The first part of this talk will focus on operator scaling random fields and on their sample path properties. Such random fields, which have been introduced in [1], satisfy an anisotropic self-similarity property, which generalizes the classical self-similarity property.  To study sample paths properties of stable operator scaling random fields, the main idea is to represent them as a shot noise series. This idea has already been used in [6] to study some self-similar stable processes (d = 1) and also applies to study some locally asymptotically operator scaling random fields (see [3]).

Such study can be seen as a particular study of the modulus of continuity of a random field defined as a conditional sub-Gaussian series. Then in the second part, we will present some results on the uniform convergence of such series and an upper bound of its modulus of continuity, stated in term of a quasi-metric ρ which takes into account its anisotropy. The particular case of LePage random series, allows to recover the results of [2, 3] and also to study multi-stable random fields (see [5, 7] for examples of such fields).

This talk is based on joint works with Hermine Biermé (Poitiers, France) and H.P. Scheffler (Siegen, Germany).

[1] H. Biermé, M. M. Meerschaert, and H. P. Scheffler. Operator scaling stable random fields. Stoch. Proc. Appl., 117(3):312–332, 2007.
[2] H. Biermé, and C. Lacaux Hölder regularity for operator scaling stable random fields.  Stoch. Proc. Appl., 119(7):2222–2248, 2009.
[3] H. Biermé,  C. Lacaux, and H. P. Scheffler. Multi-operator  scaling random fields.  Stoch. Proc. Appl., 121(11):2642–2677, 2011.
[4] H. Biermé, and C. Lacaux Modulus of continuity  of some conditionally sub-Gaussian fields. Application to stable random fields to appear in Bernoulli.
[5] Falconer, K. and Le Guével,  R. and Lévy  Vehel, J.  Localisable moving average symmetric stable and multistable processes. Stochastic  Models, 25: 648–672, 2009.
[6] Kôno, N. and Maejima, M. Hölder continuity of sample paths of some self-similar stable processes. Tokyo J. Math., 14(1),:93–100, 1991.
[7] Le Guével, R. and Lévy Vehel, J. A Ferguson - Klass - LePage series representation  of multistable multi- fractional processes and related processes to be published in Bernoulli.

Hermine Biermé (Université de Poitiers)
Tuesday, June 9 * 11:30 a.m.

Simulation of some anisotropic Gaussian random fields

In medical imaging, several authors have proposed to characterize roughness of observed textures by their fractal dimensions. These studies are based on the stochastic modeling of images using the famous fractional Brownian field for which the fractal dimension is determined by its so-called Hurst parameter. However, this stochastic model does not allow to reveal texture anisotropy which may be an important characteristic for diagnostic help. We consider two anisotropic generalizations of this model and focus on the issue of simulating realizations on a regular grid. The first model is obtained and characterized by an anisotropic deformation of the fractional Brownian field spectral density. We propose an adaptation of the turning-band method, initially suggested by Matheron, in our 2D anisotropic and non-stationary framework, based on fast and exact synthesis of 1D fractional Brownian motion. This is a joint work with Lionel Moisan (MAP5, Université Paris Descartes, France) and Frédéric Richard (LATP, Aix-Marseille Université, France). The second model is obtained by an anisotropic deformation of the fractional Brownian field variogram. We propose an adaptation of the Stein method, based on a locally stationary periodic representation, to get fast an exact synthesis of this model. This is a joint work with Céline Lacaux (IECN, Université Lorraine, Nancy, France).

Alexey Kuznetsov (York University)
Tuesday, June 9 * 2:30 p.m.

Spectral analysis of stable processes in the half-line

I will explain how the Wiener-Hopf factorization for stable processes can lead to the generalized eigenfunction expansion of the semigroup of a (general, non-symmetric) stable process killed upon the first exit from the positive half-line. This talk is based on joint work with Mateusz Kwasnicki.

Victor Rivero (CIMAT, Guanajuato)
Tuesday, June 9 * 4:00 p.m.

Exponential functionals of Lévy and Markov additive Lévy processes

Let I be the exponential functional of a real valued Lévy process  This random variable plays a key rol in several areas of probability theory, for instance in fragmentation, coalescence and branching processes, financial and insurance mathematics, Brownian motion in hyperbolic spaces, random processes in random environment, positive self-similar Markov processes, etc. Obtaining distributional properties of these r.v. has been the subject of many research articles. The paper by Bertoin and Yor [1] is a thorough review on the topic. In this talk we will show that the law of this random variable satisfies an functional equation in terms of the potential measure of the underlying Levy process. We will show how this identity can be used to provide elementary proofs of several known results for I, in particular of the striking factorisation obtained in [2]. To finish we will establish that a similar identity holds when we replace ξ by a Markov additive Lévy process, and explain how our methods allow to derive analogous results to the Lévy processes case, in particular of the main result in [2].

[1] J. Bertoin and M. Yor. Exponential functionals of L ́evy processes. Probab. Surv., 2:191–212, 2005.
[2] P. Patie and M. Savov. Exponential functional of L ́evy processes: generalized Weierstrass products and Wiener-Hopf factorization. C. R. Math. Acad. Sci. Paris, 351(9-10):393–396, 2013.

Wednesday, June 10

Mladen Savov (University of Reading)
Wednesday, June 10 * 9:00 a.m.

The spectral theory of generalized Laguerre semigroups and its applications to positive self-similar Markov processes

In this talk we present a methodology that allows us to develop the spectral theory for a class of non-self-adjoint semigroups, which we call the generalized Laguerre semigroups.  The latter can be directly linked to the semigroups of the positive self-similar Markov processes thereby allowing in some instances the explicit computation of the action of their semigroups. We will discuss the main tools behind our approach and the extent of our results. We will also consider these developments and our methodology from the perspective of general non-self-adjoint Markov processes.

Yanxia Ren (Peking University)
Wednesday, June 10 * 10:30 a.m.

Central limit theorems for supercritical superprocesses

We establish a central limit theorem for a large class of general supercritical superprocesses with spatially dependent branching mechanisms satisfying a second moment condition. We are able to characterize the limit Gaussian field. In the case of supercritical super Ornstein-Uhlenbeck processes with non-spatially dependent branching mechanisms, our central limit theorem reveals more independent structures of the limit Gaussian field.

We also establish some functional central limit theorems for the supercritical super- processes mentioned above. In the particular case when the state E is a finite set and the underline motion is an irreducible Markov chain on E, our results are superprocess analogs of the functional central limit theorems of for supercritical multi-type branching processes.

The talk is based on papers with Renming Song and Rui Zhang.

Thomas Simon (Université Lille 1)
Wednesday, June 10 * 11:30 a.m.

Computing harmonic measures for Lévy stable processes, with applications

Using classical hypergeometric identities, we compute the harmonic measure of finite intervals and their complementaries for the Lévy stable process on the line. This gives a simple and unified proof of several results by Blumenthal-Getoor-Ray, Rogozin, and Kyprianou-Pardo-Watson. We deduce several explicit computations on the related Green function and Martin kernel. In the second part of the talk, I will consider the two-dimensional Markov process constructed on the stable Lévy process and its area process, and give an explicit formula for the harmonic measure of the split complex plane. This formula allows to compute the persistence exponent of the area process and to solve a problem raised by Z. Shi. If time permits, I will display a possible connection between this persistence exponent and the Hausdorff dimension of the Lagrangian points of the inviscid Burgers equation with Lévy stable initial data. This is based on two joint works with Christophe Profeta (Evry).

Thursday, June 11

Renming Song (University of Illinois)
Thursday, June 11 * 9:00 a.m.

Stochastic flows for Lévy processes with Holder drifts

In this talk I will present some new results on the following SDE in R^d:

dX_t=b(t, X_t)dt+dZ_t, X_0=x

where Z is a Lévy process. We show that for a large class of Lévy processes Z,  Holder continuous drift b, the SDE above has a unique strong solution for every starting point x\in R^d.

Moreover, these strong solutions form a C^1-stochastic flow.

In particular, we show that, when Z is a symmetric -stable process with \alpha\in (0, 1] and b is beta-Holder continuous with \beta \in (1-\alpha/2, 1], the SDE above has a unique strong solution.

Krzysztof Bogdan (Wroclaw University of Technology)
Thursday, June 11 * 10:30 a.m.

Fractional Laplacian and cones

I will discuss results related to the behavior of harmonic functions of the fractional Laplacian in cones obtained in the last 15 years. These include the asymptotics of the Martin kernel, Green function and Dirichlet heat kernel, and more. The results have been proved by Rodrigo Bañuelos, Krzysztof Michalik, Tomasz Grzywny, Michael Ryznar, Bartlomiej Siudeja, Longmin Wang, Zbigniew Palmowski and myself.

Parthanil Roy (Indian Statistical Institute, Kolkata)
Thursday, June 11 * 11:30 a.m.

Stability for point processes, branching random walks, and a conjecture of Brunet and Derrida

Stable point processes were introduced and characterized by Davydov, Molchanov and Zuyev (2008). They showed that such point processes can always be represented as a scale mixture of iid copies of one point process with the scaling points coming from an independent Poisson random measure. We obtain such a point process as a weak limit of a sequence of point processes induced by a branching random walk with regularly varying displacements. In particular, we show that a conjecture of two physicists, Brunet and Derrida (2011), remains valid in this setup, and recover a slightly improved version of a result of Durrett (1983). This talk is based on a joint work with Ayan Bhattacharya and Rajat Subhra Hazra.

Robert Dalang (Ecole Polytechnique Fédérale de Lausanne)
Thursday, June 11 * 2:30 p.m.

Polarity of points for systems of linear spde’s in critical dimensions

We are interested in systems of d linear stochastic partial differential equations in spatial dimension k ≥ 1. The d-dimensional driving noise is space-time white noise when k = 1, and is white in time with a spatially homogeneous covariance defined as a Riesz kernel with exponent β, where 0 < β < (2 ∧ k), when k > 1. In non-critical dimensions, the issue of polarity of points for the random field solution to these systems is well-understood. In this joint paper with C. Mueller and Y. Xiao, we extend to a wide class of anisotropic Gaussian random fields an argument developed by Talagrand (1998) for fractional Brownian motion. This allows us to establish polarity of points in critical dimensions for many systems of linear spde’s, such as systems of stochastic heat and wave equations in spatial dimensions k ≥ 1.

Carl Mueller (University of Rochester)
Thursday, June 11 * 4:00 p.m.

Convergence to the stochastic heat equation and comparison of moments

We consider a system of interacting diffusions on the integer lattice. By letting the mesh size go to zero and by using a suitable scaling, we show that the system converges (in a strong sense) to a solution of the stochastic heat equation on the real line. As a consequence, we obtain comparison inequalities for product moments of the stochastic heat equation with different nonlinearities.

This is joint work with Davar Khoshnevisan and Mathew Joseph.

Friday, June 12

Serge Cohen (Université Paul Sabatier, Toulouse)
Friday, June 12 * 9:00 a.m.

Fractional fields parametrized by manifolds

Among other important applications of Self-Similarity one is quite classical for finite-dimensional Euclidean spaces: it is a convenient way to obtain generic textures used in the modeling of images. But it is obvious even for non-specialist that “textures” exist on curved spaces. The most popular model for self-similar fields is certainly the fractional Brownian field, and, maybe, even before the Lévy Brownian field. Even if the definition of a self-similar field parametrized by a Riemannian manifold is not straightforward, the definition of fractional Brownian fields parametrized by metric field is not difficult. The aim of this talk will be to review some classical facts concerning these fields and to introduce some geometrical obstructions to the existence of fractional Brownian field.

Takashi Owada (Technion—Israel Institute of Technology)
Friday, June 12 * 10:30 a.m.

Functional central limit theorem for subgraph counting processes

We investigate the limiting behavior of a subgraph counting process for understanding the formation of random geometric graphs. The subgraph counting process we shall consider counts the number of subgraphs of specific shape, which exist outside of a ball expanding as the sample size increases. As an underlying distribution, we will consider the distributions with regularly varying tail and those with exponentially decaying tail. In either case, the nature of an obtained functional central limit theorem differs depending on how rapidly the ball expands. More specifically, a proper normalization of the central limit theorem and the properties of limiting Gaussian processes are all determined by whether or not an expanding ball covers a region—called core—in which the random points are densely distributed and form a single giant geometric graph.

Davar Khoshnevisan (University of Utah)
Friday, June 12 * 11:30 a.m.

Macroscopic Hausdorff Dimension and Stochastic PDEs

We present some ongoing joint work with Kunwoo Kim and Yimin Xiao on a series of loose connections between Gaussian processes and stochastic, PDEs on one hand, and the Barlow-Taylor theory of macroscopic Hausdorff dimension that was originally developed to study the large-scale fractal properties of infinite sets in Euclidean space.