SPEAKER: Tomasz Rychlik (Istituto di Matematica dell'Accademia Polacca delle Scienze)

TITLE: Effects of prior selection in nonparametric Bayes problems

ABSTRACT: We consider an arbitrary family of stochastically ordered distribution functions dependent on parameters from an interval in the real line. We compare distribution functions and moments of random variables with distribution functions being the mixtures of the members of the above stochastically ordered family. In particular, we determine lower and upper bounds on the values of one mixture distribution function under the condition that the other mixture distribution is arbitrarily fixed. We also evaluate the differences between the expectations of mixture variables in various scale units. The results are illustrated by some examples.

SPEAKER: Giacomo di Gesù (University of Wien)

TITLE: Small noise asymptotics for stochastic Allen-Cahn equations in finite volume

ABSTRACT: We consider lattice approximations of the Allen-Cahn equation on the torus perturbed by small space-time white noise and discuss metastable transition times between the two stable phases.

SPEAKER: Frederic Patras (University of Nice)

TITLE: Revisiting chaos expansions and Wick products

ABSTRACT: Recently, the interest in the structure of cumulants and Wick products for non-Gaussian variables has been revived, since they both play important roles in M. Hairer's theory of regularity structures. We present a new approach $ (group actions, Hopf algebras) that allow to understand the theory of Wick products as a deformation, a presentation from which many of their properties follow easily. We also discuss how these ideas appear in the theory of regularity structu$ Tapia, L. Zambotti.

SPEAKER: Federico Camia (New York - Abu Dhabi University)

TITLE: Limit Theorems and Random Fractal Curves in Statistical Mechanics

ABSTRACT: Rigorous statistical mechanics deals with stochastic systems that have a large number of components and for which geometry often plays an important role. The main goal is to understand their average behavior and deviations from that. Statistical mechanics has many applications to physics and other fields, but in this talk I will only focus on some aspects of its mathematical theory which combine discrete probability, stochastic processes and complex analysis. For concreteness, I will discuss two specific (two-dimensional) models, percolation and the Ising model, both with a long history. After introducing the two models, I will present an approach to the study of two-dimensional systems that leads to a special family of random fractal curves and that has produced, in the last twenty years, deep results and major breakthroughs.

SPEAKER: Alessandra Occelli (Università di Bonn)

TITLE: On time correlations for last passage percolation models

ABSTRACT: We study time correlations of last passage percolation (LPP), a model in the Kardar-Parisi-Zhang universality class, with three different geometries: step, flat and stationary. We prove the convergence of the covariances of the LPP at two different times to a limiting expression given in terms of Airy processes. Furthermore, we prove the behaviour of the covariances when the two times are close to each other, conjectured in a work of Ferrari and Spohn.

SPEAKER: Alberto Chiarini (ETH, Zurigo)

TITLE: Invariance principle for the degenerate dynamic random conductance model.

ABSTRACT: After the brilliant result of Papanicolau and Varadhan (1979) in the case of bounded stationary and ergodic environments, there has been a recent upsurge in the research of quenched homogenization in random media. In particular, to identify the optimal conditions that a general stationary and ergodic environment must satisfy in order to obtain the convergence to a non-degenerate Brownian motion, is still an open problem. In this talk, we study a continuous-time random walk on Z^d in an environment of dynamic random conductances. We assume that the law of the conductances is ergodic and stationary with respect to space-time shifts. We prove a quenched invariance principle for the random walk under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by the celebrated Moser’s iteration scheme. This is joint work with S. Andres, J-D. Deuschel and M. Slowik.

SPEAKER: Mauro Maurelli (WIAS, Berlin)

TITLE: Enhanced Sanov theorem for Brownian rough paths and an application to interacting particles

ABSTRACT: We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations principle for the (k-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn impli$

SPEAKER: Yaakov Malinovsky (University of Maryland, Baltimore County)

TITLE: Nested Group Testing Procedures and Generalized GT Problem

ABSTRACT: Group testing has its origins in the identication of syphilis in the US army during World War II. It is a useful method that has broad applications in medicine, engineering, and even in airport security control. Consider a finite population of N units, where unit i has a probability p to be defective. A group test is a simultaneous test on an arbitrary group of units with two possible outcomes: all units are good or at least one of the units is defective. The goal is to construct a procedure which classifies all units in a given population, with as small as possible expected number of tests. In this talk I shall review previously known results in the group testing literature and present new results characterizing optimality of commonly used nested group testing procedures. In the second part of the talk, the generalized group testing problem (where unit i has a probability p_{i} to be defective) will be discussed as well.

SPEAKER: Lorenzo Dello Schiavo (Institut fur Angewandte Mathematik - University of Bonn)

TITLE: Two characterizations of Dirichlet-Ferguson measures

ABSTRACT: We consider the Dirichlet-Ferguson (DF) measure, a random probability on a locally compact Polish space X introduced by Ferguson in [1]. The measure has ever since found many applications, widely ranging from Bayesian non-parametrics to population genetics and stochastic dynamics of infinite particle systems. Firstly, we compute the characteristic functional of DF measures (addressing, if time permits, connections of these measures with Lie algebra theory and Polya Enumeration Theory). Secondly, we prove a characterization of DF measures via a Mecke-type integral identity. Profiting of connections between DF measures and Poisson measures on configuration spaces, we argue how DF measures may be regarded as 'canonical' measures on the space P(X) of Borel probability measures on X. Partly based on joint work with E. W. Lytvynov, University of Swansea, Wales, UK. [1] Ferguson, T. S., Ann. Stat. 1(2), pp. 209-230, 1973.

SPEAKER: Luca Scarpa ( Department of Mathematics University College London, UK)

TITLE: Well-posedness of semilinear SPDEs with singular drift: a variational approach

ABSTRACT: Well-posedness is proved for singular semilinear SPDEs on a smooth bounded domain D in R^n. The linear part is associated to a coercive linear maximal monotone operator on L^2(D) while the drift is represented by a multivalued maximal monotone graph everywhere defined on R, on which no growth nor smoothness conditions are required. Moreover, the noise is given by a cylindrical Wiener process on a Hilbert space U, with a stochastic integrand taking values in the Hilbert-Schmidt operators from U to L^2(D): classical Lipschitz-continuity hypotheses for the diffusion coefficient are assumed. The proof consists in approximating the equation, finding uniform estimates both pathwise and in expectation on the approximated solutions, and then passing to the limit using compactness and lower semicontinuity results. Finally, possible generalizations are discussed. This study is based on a joint work with Carlo Marinelli (University College London).

SPEAKER: Elena Di Bernardino (Parigi)

TITLE: A test of Gaussianity based on the Euler characteristic of excursion sets

ABSTRACT: We deal with a stationary isotropic random field X:R^d→R and we assume it is partially observed through some level functionals. We aim at providing a methodology for a test of Gaussianity based on this information. More precisely, the level functionals are given by the Euler characteristic of the excursion sets above a finite number of levels. On the one hand, we study the properties of these level functionals under the hypothesis that the random field X is Gaussian. In particular, we focus on the mapping that associates to any level u the expected Euler characteristic of the excursion set above level u. On the other hand, we study the same level functionals under alternative distributions of X, such as chi-square, harmonic oscillator and shot noise. In order to validate our methodology, a part of the work consists in numerical experimentations. We generate Monte-Carlo samples of Gaussian and non-Gaussian random fields and compare, from a statistical point of view, their level functionals. Goodness-of-fit p−values are displayed for both cases. Simulations are performed in one dimensional case (d=1) and in two dimensional case (d=2), using R.

SPEAKER: Vittoria Silvestri (Cambdrige)

TITLE: The reset time of Internal DLA

ABSTRACT: Consider Internal DLA on cylinder graphs of the form GxZ. How does a large cluster typically look like? How long does it take for the process to forget its initial profile? In this talk I will address these questions, explaining how the answer depends on the mixing properties of the base graph G. Joint work with Lionel Levine.

SPEAKER: Ehud Lehrer (Tel Aviv University)

TITLE: Non-additive probability, integration and decision making

ABSTRACT: Frequently a decision maker (DM) does not have a full information about the underlying uncertainty. For instance, the information about the probability of some events might be missing, or the probability distribution might fail to be additive. In this talk I will present non-additive probabilities, describe a few methods of integration with respect to these probabilities and how one may use these schemes in decision making.

SPEAKER: T.G. Kurtz, University of Wisconsin-Madison.

TITLE: Stochastic equations for processes built from bounded generators

ABSTRACT: The generator for a pure jump process with bounded jump rate is a bounded operator on the space of measurable functions. For any such process, it is simple to write a stochastic equation driven by a Poisson random measure. Uniqueness for both the stochastic equation and the corresponding martingale problem is immediate, and consequently, the martingale problem and the stochastic equation are equivalent in the sense that they uniquely characterize the same process. A variety of Markov processes, including many interacting particle models, have generators which are at least formally given by infinite sums of bounded generators. In considerable generality, we can write stochastic equations that are equivalent to these generators in the sense that every solution of the stochastic equation is a solution of the martingale problem and every solution of the martingale problem determines a weak solution of the stochastic equation. It follows that uniqueness for one approach is equivalent to uniqueness for the other.

SPEAKER: C. Landim (IMPA, CNRS Rouen)

TITLE: Static large deviations for a reaction-diffusion model

ABSTRACT: We consider the superposition of a symmetric simple exclusion dynamics, speeded-up in time, with a spin-flip dynamics in a one-dimensional interval with periodic boundary conditions. We prove the large deviations principle for the empirical measure under the stationary state. We deduce from this result that the stationary state is concentrated on the stationary solutions of the hydrodynamic equation which are stable.

SPEAKER: M. Piccioni (Sapienza)

TITLE: How it happened (to me) to meet ?

ABSTRACT: The function ? was defined by Minkowski with the purpose of mapping quadratic irrationals bijectively onto the rationals of (0,1) (in addition, it maps the rationals onto the dyadic rationals). It has remarkable self-similarity properties and it maps (increasingly) th$ sion is also obtained. This is a joint work with Gerard Letac.

SPEAKER: M. Campanino (Bologna)

TITLE: Recurrence properties for random walks on a two-dimensional random graph.

ABSTRACT: Recurrcnce properties for random walks on a two-dimensional random graph. In [1] a random walk on a bi-dimensional random graph was studied. This model had been previously introduced in the physical literature. and studied numerically. It is established in [1] that, in contrast with the corresponding periodic graph, this random walk is transient with probability one with respect to the random environment. Subsequently several papers appeared on related models ([3], [4] [5] [6]). The results of [1] have been used in [7] to study a model of corner percolation on Z2. In [2] transience was obtained on a more general random environment and a transition from recurrence to transience has been proved to occur. At the moment work is going on in collaboration with G. Bosi to study the problem on a honeycomb random graph. [1] M. Campanino, D. Petritis. Random walks on randomly oriented lattices. Markov Process. Relat. Fields 9, 391-412 (2003). [2] M. Campanino, D. Petritis. Type transition of simple random walks on randomly directed regular lattices, J. Appl. Prob. 51, 1065-1080 (2014). [3] B. De Loynes. Marche aleatoire sur un di-graphe et frontière de Martin. C. R. Acad. Sci. Paris 350, 87-90 (2012). [4] A. Devuldier, F. Pene. Random walk in random environment in a two-dimensional stratified medium with orientation. Electron. J. Prob. 18, no. 18 (2013). [5] Guillotin-Plantard, A, Le Ny. A functional limit theorem for a 2D-random walk with dependent marginals. Electron. Commun. Prob. 13, 337-351 (2008). [6] Guillotin-Plantard, A, Le Ny. Transient random walks on 2D-oriented lattice. Theory Prob. Appl. 52, 699-711 (2008). [7] G. Pete. Corner percolation on Z2 and the square root of 17. Ann. Prob. 36, 1711-1747 (2008).

SPEAKER: Yosi Rinott, The Hebrew University and LUISS.

TITLE: Differential Privacy applied to common methods of dissemination of frequency tables

ABSTRACT: When official data are to be disseminated to the public, the agency that releases the data must guarantee $ It is not clear how to define and measure privacy. I will explain a notion developed in computer science known as Differen$ issues that arise in applying it to the dissemination of frequency tables.

SPEAKER: Mauro Piccioni (Sapienza)

TITLE: Conditional iid structures in exchangeable multivariate distributions: from Marshall-Olkin to 'load sharing' $

ABSTRACT: The discovery of a conditional iid structure allows the understanding of the dependence contained in a d-v$

SPEAKER: O. Blondel (Lyon, CNRS)

TITLE: More random walks on random walks

ABSTRACT: We consider a Poissonian distribution of particles performing independent simple random walks. Simultaneously, on top of this system, a random walker evolves with a drift to the right when it is on top of (at least) a particle, to the left when it is on an empty site. We obtain a LLN, CLT and large deviation bounds in high and low density. Joint work with Marcelo Hilario, Renato dos Santos, Vladas Sidoravicius, Augusto Teixeira.

SPEAKER: W. Woess (TU Graz, Austria)

TITLE: Multidimensional reflected random walk - some results and many questions

ABSTRACT: Let (Y_n,V_n) be i.i.d. distributed, with the components r and s-dimensional, respectively. Reﬂected random walk starting at a point x of the positive r-dimensional orthant is deﬁned recursively by X_0 = x, X_n = |X_{n−1}−Y_n|, where |(a_1,...,a_r)| = (|a_1|,...,|a_r|). In R^s, consider the ordinary sum S_n = V_1 +···+V_n . We are interested in (topological) recurrence of the process (X_n,v+S_n) starting at (x,v). While this is quite well understood for refelcted random walk with r=1, in higher dimension (r \geq 2) or with some non-reﬂected coordinates (s \in {1,2}) we have a few basic results and various open questions with some partial answers. This is work with Judith Kloas, with input from Marc Peigne' and Wojciech Cygan.

SPEAKER: Lorenzo Bertini (Univ. La Sapienza, Roma)

TITLE: Grandi deviazioni rispetto al moto per curvatura media

ABSTRACT: Si consideri l'equazione di Allen-Cahn in dimensione d=2 o d=3. Effettuando un riscalamento diffusivo, per dati iniziali opportuni, la dinamica limite dell'interfaccia tra le due fasi stabili e' descritta dal moto per curvatura media. Verra' introdotta una perturbazione stocastica di tale equazione e analizzata la corrispondente asintotica di grandi deviazioni nel limite di interfacce concentrate. Il corrispondente funzionale di tasso e' analogo a quello ottenuto analizzando la convergenza variazionale dei funzionali d'azione. La dimostrazione della stima di grandi deviazioni utilizza strumenti di teoria geometrica della misura.

SPEAKER: Valentina Cammarota (Sapienza Università di Roma)

TITLE: On the critical values of random spherical harmonics.

ABSTRACT: Abstract: We study the limiting distribution, in the high energy limit, of critical points and extrema of random spherical harmonics. In particular, we first derive the density functions of extrema and saddles and then we provide analytic expressions for the variances. Our arguments require a careful investigation of the validity of the Kac-Rice formula in nonstandard circumstances, entailing degeneracies of covariance matrices for first and second derivatives of the processes being analyzed. It is well known that after proper rescaling random spherical harmonics converge to Berry's random plane waves; in the second part of the talk we focus on the spatial distribution of critical points of random plane waves. Based on joint works with Dmitry Beliaev, Domenico Marinucci and Igor Wigman.

SPEAKER: Sokol Ndreca (University Center of Belo Horizonte)

TITLE: On a Problem of Kendall

ABSTRACT: Abstract: In this talk we consider a stochastic point process $i + \xi_i$, where $i\in \mathbb{N}$ and the $\xi_i's$ are i.i.d exponential random variables with standard deviation $\sigma$. Some properties of this process are investigated. We then study a discrete time single server queueing system with this process as arrival process and deterministic unit service time. We obtain a functional equation of the bivariate probability generating function of the stationary distribution for the system. The functional equation is quite singular, does not admit simple solution. We find the solution of such equation on a subset of its set of definition. Finally we prove that the stationary distribution of the system decays super-exponentially fast in the quarter plane. The queueing model, motivated by air and railway traffic, has been proposed by Kendall and others some five decades ago, but no solution of it has been found so far. This is a joint work with Gianluca Guadagni, Carlo Lancia and Benedetto Scoppola.

SPEAKER: D. Marinucci (Tor Vergata, Roma)

TITLE: A Quantitative Central Limit Theorem for the Euler-Poincaré Characteristic of Random Spherical Eigenfunctions

ABSTRACT: We establish here a Quantitative Central Limit Theorem (in Wasserstein distance) for the Euler-Poincaré Characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler-Poincaré Characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level u is fully degenerate, i.e. the Euler-Poincaré Characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. Our results can be written as an asymptotic second-order Gaussian Kinematic Formula for the excursion sets of Gaussian spherical harmonics. Based on a joint work with Valentina Cammarota.

SPEAKER: Mauro Mariani (Università La Sapienza)

TITLE: Metastable regimes of diffusion processes

ABSTRACT: I will discuss some old and new results concerning a classical example featuring a metastable behavior: finite-dimensional diffusion processes in the vanishing noise limit. Sharp estimates have been introduced during the '60s. More recently, similar ideas appeared in the context of potential theory and calculus of variations.

SPEAKER: Antonio Lijoi (Università di Pavia)

TITLE: Bayesian nonparametric inference with heterogeneous data

ABSTRACT: The talk provides an overview of some recent work on random probability measure vectors and their role in Bayesian statistics. Indeed, dependent nonparametric priors are useful tools for drawing inferences on data that arise from different, though related, studies or experiments and for which the usual exchangeability assumption is not satisfied. The presentation will focus on models based on completely random measures, or suitable transformations thereof, that have an additive, a hierarchical and a nested structure. Some of their distributional properties relevant for prediction will be discussed. These theoretical results are, then, used for devising Markov chain Monte Carlo algorithms that will be implemented within some illustrative examples for analyzing data in the contexts of species sampling problems and survival analysis.

SPEAKER: Pietro Caputo (Università di Roma Tre)

TITLE: Random walk on sparse random directed graphs

ABSTRACT: random walk on a finite graph exhibits cutoff if its distance from stationarity remains close to the initial value for a certain number of iterations and then abruptly drops to near zero on a much shorter time scale. Originally discovered in the context $

SPEAKER: Carlo Orrieri, Dipartimento di Matematica, Sapienza Università di Roma

TITLE: Controlled Vlasov-type dynamics

ABSTRACT: The aim of the talk is to connect the optimal control of Vlasov-type PDEs with large systems of controlled interacting particles/agents. The main difference towards the classical mean-field theory, where the particle are freely interacting with each other, is the presence of a central planner influencing the dynamics. In this situation it is not obvious that the mean field limit should commute with the optimization. In order to develop a rigorous limit theory for this problem we employ the well-known concept of Gamma convergence. The talk is based on a work in project with M. Fornasier, S. Lisini and G. Savarè

SPEAKER: Marc Peigné, Université de Tours, France

TITLE: Random dynamical systems: contraction and recurrence properties

ABSTRAC Consider a proper metric space X and a sequence (Fn) of i.i.d. random continuous mappings from X to X. It induces the stochastic dynamical system (SDS) Xn = Fn ∘ ∘ ∘ F1(x) starting at x ∈ X. We study existence and uniqueness of invariant measures, under some assumptions of contraction on the Fn, as well as recurrence and ergodicity of this process. We will consider two main examples: the case where the Fn are affine maps of the real line and the case where Xn is the reflected random walk on the positive real line.

SPEAKER: Pham Thi Da Cam, Université de Tours, France

TITLE: The survival probability of a critical multi-type branching process in i.i.d. random environment

ABSTRACT: We consider Galton Watson branching processes of unique type and multi-type in fixed and in random environment. The main target is to observe the asymptotic behaviour of the survival probability of the population. In particular, we utilise the generating function method thanks to the recursive structure of the process in deterministic case and condition to the generating function of offspring distribution in random case.

SPEAKER: C. Lefèvre, Université Libre de Bruxelles, Département de Mathématique

TITLE: Epidemic Risk and Insurance Coverage

ABSTRACT: This paper aims to apply simple actuarial methods to build an insurance plan protecting against an epidemic risk in a population. The studied model is an extended SIR epidemic in which the removal and infection rates may depend on the number of registered removals. The costs due to the epidemic are measured through the expected epidemic size and infectivity time. The premiums received during the epidemic outbreak are measured through the expected susceptibility time. Using martingale arguments, a method by recursion is developed to calculate the cost components and the corresponding premium levels in this extended epidemic model. Some numerical examples illustrate the effect of removals and the premium calculation in an insurance plan. This is a joint work with P. Picard (ISFA, Lyon) and M. Simon (ULB).

SPEAKER: Abram Kagan (Dept. of Mathematics, Univ. of Maryland, College Park and Dept. of Probability and Statistics, Charles University, Prague)

TITLE: On quantifying dependence between random elements

ABSTRACT: In the introduction some relatively recent results on the maximum correlation coefficient will be presented in two setups. The first deals with partial sums of independent identically distributed random variables, and the second with the components of a bivariate Gaussian stationary process. The main part of the talk is devoted to a discussion of the properties required from a reasonable numerical measure of dependence. My own contribution is in the construction of a calibrated scale of models starting at independence and going away from it in a specified direction.

SPEAKER: G. Letac (Institut Mathematique de Toulouse, Université Paul Sabatier, Toulouse)

TITLE: The Wishart distribution which never was

ABSTRACT: Replacing the integer parameter in the family of chi square distributions by a continuous parameter leads to the family of the gamma distributions. A similar phenomena occurs with the non central chi square. What happens in higher dimensions, where the natural extension of the chi square family is the Wishart family? The extension to a continuous parameter is clarified by the Gyndikin theorem (1975), with a beautiful short proof by Shanbhag (1989). The case of the non central Wishart is really difficult, and its explicit solution has been conjectured by E. Mayerhoffer (2010). We prove this conjecture by a detailed analysis of the zonal polynomials of symmetric matrices and of the convolution of measures concentrated on singular semi positive matrices (Joint work with H. Massam).

SPEAKER: Eleonora Perversi (Dipartimento di Matematica, Universita' di Pavia)

TITLE: Inequality and risk aversion in economies open to altruistic attitudes

ABSTRACT: In recent years there has been a great interest in the phenomenon of economic inequality, especially in relation to its connection with other important aspects of an advanced economy, for example attitude towards risk, growth, financial developments, and so on. In this talk, based on a joint work with Eugenio Regazzini, I will introduce a model for the surplus/deficit distribution, which points out a relationship between agents' risk aversion and inequality. More precisely, on the one hand, the long-time surplus/deficit distribution turns out to be a weak Pareto law whose exponent is given by an affine transformation of the agents relative risk aversion index, supposed to be the same for every agent. On the other hand, it is proved that concentration in a weak Pareto law can be measured through a function of its exponent. This way, a link is established between inequality and risk aversion. Finally, some feasible actions of economic policies suitable for the control of inequality are derived.

SPEAKER: Paul Chleboun (University of Warwick)

TITLE: Large deviations of the empirical current in zero-range processes on a ring.

ABSTRACT: We examine atypical current fluctuations in totally asymmetric zero-range processes in one dimension with periodic boundary conditions. The zero-range processes is a stochastic lattice gas in which each lattice site can be occupied by, a-priori, an unbounded number of particles. Particles move to their neighbour at a rate which only depends on the occupation of the departure site. For large systems, by calculating the Jensen-Varadhan action functional, we are able to find the time dependent optimal profiles which realise currents below the typical value. Under certain conditions on the jump rates, we demonstrate that these systems can exhibit a dynamical phase transition, in which above a critical non-typically current the optimal macroscopic density profile is given by a traveling wave with a shock and anti-shock pair. While rare events below the critical current are realised by a condensate, whereby a non-zero fraction of all the particles accumulate on a single site in the thermodynamic limit. This gives rise to a non-convex rate function for the empirical current, which in turn leads to a breakdown of the equivalence between the conditioned dynamics and the s-ensemble cloning methods typically used in simulations to sample these rare events.

SPEAKER: Alexandre Boritchev (Universite' de Lyon)

TITLE: 1D and multi-d Burgers Turbulence as a model case for the Kolmogorov Theory

Abstract: The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all models of turbulence. In particular, K41 and corrections to it provide estimates of small-scale quantities such as increments and energy spectrum for a 3D turbulent flow. However, because of the well-known difficulties involved in studying 3D turbulent flows, there are no rigorous results confirming or infirming those predictions. Here, we consider a well-known simplified model for 3D turbulence: Burgulence, or turbulence for the 1D or multi-dimensional potential Burgers equation. In the space-periodic case with a stochastic white in time and smooth in space forcing term, we give sharp estimates for small-scale quantities such as increments and energy spectrum.

SPEAKER: Radko Mesiar (STU Bratislava)

TITLE: On the role of ultramodularity (and Schur concavity) in the construction of binary copulas

ABSTRACT: We discuss and stress the role of ultramodularity in special types of constructions of binary copulas. After recalling of some known ultamodularity-based results, we focus on a the so-called D-product of a copula and its dual. We show that for each copula D which is ultramodular and Schur concave on the left upper triangle of the unit square, this D-product of an arbitrary copula and its dual is again a copula. Several examples and counterexamples are given. Finally, some of our results are generalized to the case of semicopulas and quasi-copulas. Work in collaboration with E. P. Klement, A. Kolesarova and S. Saminger-Platz

Speaker: A. Erdely (Universidad Nacional Autonoma de Mexico)

Title: Modeling complex dependence using gluing and vine copulas

Abstract: Bivariate dependence may be of such complexity that no single family of known parametric copulas is able to give an acceptable goodnes of fit. The gluing copula approach may be of good help in decomposing a complex non monotone dependence into easier to model piecewise monotone dependencies. This is good news for the vine copula approach since bivariate copulas are building blocks of such approach in higher dimensions. An application to real data in economics and geophysics will be discussed.

Speaker: Oriane Blondel (Lione)

Title: Random walk on the East model (and other environments with spectral gap)

Abstract: The East model is a one-dimensional interacting particle system with non attractive spin-flip dynamics. In the physics literature, it is a key example of a model with glassy features. Here we take this model as a random environment and investigate the b$ Joint work with Luca Avena and Alessandra Faggionato.

Speaker: Vittoria Silvestri (Cambridge)

Title: Title: Fluctuation results for Hastings-Levitov planar growth

Abstract: In 1998 the physicists Hastings and Levitov introduced a family of continuum models to describe a range of physical phenomena of planar aggregation/diffusion. These consist of growing random clusters on the complex plane, which are built by iterated composition of random conformal maps. It was shown by Norris and Turner (2012) that in the case of i.i.d. maps the limiting shape of these clusters is a disc: in this talk I will show that the fluctuations around this shape are given by a random holomorphic Gaussian field F on {|z| > 1}, of which I will provide an explicit construction. When the cluster is allowed to grow indefinitely, I will show that the boundary values of F converge to a distribution-valued Fractional Gaussian Field on the unit circle, which is log-correlated, and critical in a sense that I will explain.

Speaker: Pierre Mathieu (Marsiglia)

Title: Random walks on (hyperbolic) groups.

Abstract: the first part of the talk will be an introduction to the general theory of random walks on groups with some classical results on entropy, rate of escape ... . For hyperbolic groups, these probabilistic objects have geometric counterparts in terms of Gromov boundary, quasi-conformal measures ... I will then discuss fluctuation results, in particular a recent work with A. Sisto on deviation inequalities for random walks on acylindrically hyperbolic groups.

Speaker: Yan Shu (Parigi)

Title: Hamilton Jacobi equations on graphs and applications

Abstract: We introduce a notion of gradient and an infimal-convolution operator that extend properties of solutions of Hamilton Jacobi equations to more general spaces, in particular to graphs. As a main application, the hypercontractivity of this class of in infimal-convolution operators is connected to some discrete version of the log-Sobolev inequality and to a discrete version of Talagrand's transport inequality.

Speaker: Max Fathi (Parigi)

Title: A gradient flow approach to large deviations for diffusion processes

Abstract: In the 80s, De Giorgi introduced the notion of abstract gradient flows, which allowed to define a notion of solutions to ordinary differential equations of the form x' = −grad F(x) on metric spaces (rather than Riemannian manifolds for the usual definition). In 2005, Ambrosio, Gigli and Savare showed that when we consider the space of probability measures on R d endowed with the Wasserstein metric, this notion allows to give an alternate formulation for Fokker-Planck equations. These equations are the PDEs whose solutions are the flow of marginals of solutions of stochastic differential equations of the form dX = −grad H(X)dt + dB . In this talk, I will explain how we can use this notion to study large deviations for sequences of SDEs. The main result is that proving a large deviation principle is equivalent to studying the limit of a sequence of functionals that appear in the abstract gradient flow formulation for Fokker-Planck equations. As an application, I will show how to obtain large deviations from the hydrodynamic scaling limit for a system of interacting continuous spins in a random environment.

Speaker: Andrea Puglisi (Sapienza Università di Roma)

Title: Dynamics of an intruder in a granular fluid: from dilute to dense experiment.

Abstract: A single experimental setup is the occasion for a tour into many issues in non-equilibrium statistical mechanics. In the experiment, a rotating intruder performs a Brownian-like dynamics under the influence of collisions with a shaken granular media. Dissipations in the form of inelastic collisions and dry tangential friction make the system inherently out of equilibrium. When the granular medium is diluted, the results agree with a Boltzmann-Lorentz description which, in the large mass limit, is well approximated by a Fokker-Planck equation for a Ornstein-Uhlenbeck process with Coulomb friction. When the intruder is shaped so as to break the symmetry under rotation-inversion, an average drift ('ratchet' or 'motor' effect) is observed, with properties depending on the dominant dissipation: friction or inelastic collisions. When the density of the surrounding medium increases, non-Markovian effects appear. The first consequence is a violation of the Einstein relation which -near equilibrium- describes the linear response to a small force provided by an external motor. The analysis of a generalized fluctuation-dissipation relation explains the nature of the violation: a joint effect of dissipation and coupling with the dense fluid. When the density is increased further and the jamming transition is approached, anomalous diffusion appears in the form of transient cage effects (subdiffusion) and -more surprisingly- superfdiffusion at large times.

Speaker: E. Kosygina (Baruch College and the CUNY Graduate Center)

Title: On the connection between homogenization of some stochastic Hamilton-Jacobi-Bellman equations and large deviations of diffusions with random drift in a random potential

Abstract: The connection mentioned in the title is well-known (see, for example, A.-S. Sznitman, Brownian motion in a Poissonian potential, 1993, PTRF). The goal of the talk is to review some necessary facts from large deviations theory and explain that this connection is actually an equivalence.

Speaker: E. Kosygina (Baruch College and the CUNY Graduate Center)

Title: Excited random walks in Markovian cookie environments on Z

Abstract: We consider a nearest-neighbor random walk on Z whose probability ω(x, n) to jump to the right from site x depends not only on x but also on the number of prior visits n to x. The collection (ω(x, n)) is sometimes called the “cookie environment” due to the following informal interpretation. Upon each visit to a site the walker eats a cookie from the cookie stack at that site and chooses the probability to jump to the right according to the “flavour” of the cookie eaten. Assume that the cookie stacks are i.i.d. and that only the first M cookies in each stack may have “flavour”. All other cookies are assumed to be “plain”, i.e. after their consumption the walker makes unbiased steps to one of its neighbours. The “flavours” of the first M cookies within the stack can be different and dependent. We discuss recurrence/transience, ballisticity, and limit theorems for such walks. The talk is based on joint works with Dolgopyat (University of Maryland), Mountford, Zerner.

Speaker: Stefano Olla (CEREMADE, Paris-Dauphine)

Title: Diffusione e superdiffusione dell'energia in catene di oscillatori

Abstract: In catene unidimensonali tipo FPU la conduttività termica e' infinita e ci si aspetta una superdiffusione dell'energia. In una catena di oscillatori armonici con collisioni stocastiche conservano l'energia e il momento, dimostriamo che il profilo di temperatura evolve macroscopicamente seguendo una equazione di diffusione frazionaria.

Speaker: Stella Brassesco (Instituto Venezolano de Investigaciones Científicas)

Title: The winding number of planar Brownian motion

Abstract: From an explicit formula for the joint density of the radial part and the winding number of a planar Brownian motion, we obtain asymptotic expansions (as t tends to infinity) for the density of the winding number. In particular, this expansion yields corrections of any order (in inverse powers of log(t)) to Spitzer's law (Trans. Am. Math. Soc, 1958) and to a local limit theorem proved by Delbaen, Kowalski and Nikehbali (Int. Math. Research Notices, 2014).

Speaker: Larry Goldstein (University of Southern California)

Title: Normal approximation for recovery of structured unknowns in high dimension: Steining the Steiner formula

Abstract: Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications. In particular, the discrete probability distribution L(VC) given by the sequence v{0},...,v{d} of conic intrinsic volumes of a closed convex cone C in Rd summarizes key information about the success of convex programs used to solve for sparse vectors, and other structured unknowns such as low rank matrices, in high dimensional regularized inverse problems. The concentration of VC implies the existence of phase transitions for the probability of recovery of the unknown in the number of observations. Additional information about the probability of recovery success is provided by a normal approximation for VC. Such central limit theorems can be shown by first considering the squared length GC of the projection of a Gaussian vector on the cone C. Applying a second order Poincar´e inequality, proved using Stein’s method, then produces a non-asymptotic total variation bound to the normal for L(GC). A conic version of the classical Steiner formula in convex geometry translates finite sample bounds and a normal limit for GC to that for VC. Joint with Ivan Nourdin and Giovanni Peccati.

Speaker: Kirone Mallick (CEA, Saclay)

Title: Fluctuations and large deviations in non equilibrium systems

Abstract: Studies of non-equilibrium fluctuations have held a center stage during the past few decades of the development of non-equilibrium statistical mechanics. Indeed, macroscopic fluctuations are supposed to display some universal behaviour, regardless of the precise microscopic dynamics of the process under study. In this talk, we shall consider a few examples of non-equilibrium fluctuations: the particle current in an interacting particle system, the diffusion of a tracer in a one-dimensional gas of particles with excluded mutual passage and the melting of an Ising quadrant. These different problems will be analyzed quantitatively with the help of the macroscopic fluctuation theory.

Speaker: MARCO RIBEZZI CRIVELLARI ((Universita' di Barcellona)

Title: Experimental measurements of entropy production at the nano-scale: an application for the 'fluctuation theorems'

Abstract: The study of non-equilibrium systems has led to several mathematically rigorous and general results on the statistics of entropy production in non-equilibrium systems. These results are generally known under the name of 'fluctuation theorems' and include the Gallavotti-Cohen theorem or the Jarzynski equality. My research focuses on performing experiments in which such results can be tested in real physical systems. I am interested in assessing their range of validity but even more interested in exploiting them to develop new measurement techniques. I will present some recent experiments, in which focused laser beams (optical tweezers) are used to perform thermodynamic transformations on single DNA molecules. As a first example I will show how to use fluctuation theorems to measure the free energy change across the transformation. As a second example I will discuss how, in specific cases, fluctuation theorems can be used to measure the full entropy production in a nano-scale system starting from a partial measurement through what we call an 'inference' procedure. This last example provides a new and general application of fluctuation theorems which we are only beginning to explore.

Speaker: ALESSANDRA BIANCHI

Title: A random walk in a Levy random environment

Speaker: FABRIZIO LEISEN

Title: Dependent Vectors of Random Probability Measures

Abstract: The definition of vectors of dependent random probability measures is a topic of interest in applications to Bayesian statistics. Indeed, they can be used for identifying the de Finetti mixing measure in the representation of the law of a partially exchangeable array of random elements taking values in a separable and complete metric space. In this talk we describe the construction of vectors of random probability measures based on the normalization of an exchangeable vector of completely random measures that are jointly infinitely divisible. The dependence can be achieved in many ways and in this talk will be shown some recent constructions based on Levy Copulas or the stick breaking representation. Finally, the dependence structure of the vectors is studied through some quantities of interest. This talk is a review of papers in collaboration with Antonio Lijoi, Federico Bassetti, Roberto Casarin, Weixuan Zhu and Dario Spano

Speaker: JULIEN SOHIER (Eindhoven - The Netherlands)

Title: A comparison between different cycle decompositions for Metropolis dynamics

Abstract: In the last decades the problem of metastability has been attacked on rigorous grounds via many different approaches and techniques which are briefly reviewed in this paper. It is then useful to understand connections between different point of views. In view of this we consider irreducible, aperiodic and reversible Markov chains with exponentially small transition probabilities in the framework of Metropolis dynamics. We compare two different cycle decompositions and prove their equivalence.

Speaker: CLAUDIO LANDIM (IMPA Rio de Janeiro, CNRS Rouen)

Title: Zero-temperature limit of the Kawasaki dynamics for the Ising lattice gas in a large two-dimensional torus

Abstract:We consider the Kawasaki dynamics at inverse temperature beta for the Ising lattice gas on a two-dimensional square of length 2L+1 with periodic boundary conditions. We assume that initially the particles form a square of length n, which may increase, as well as L, with beta. We show that in a proper time scale L^2, theta_beta particles form almost always a square and that this square evolves as a Brownian motion when the temperature vanishes.

Speaker: PAUL CHLEBOUN (Warwick University)

Title: Large deviations and metastability in a size-dependent zero-range process.

Abstract: We discuss a general approach to understand phase separation and metastability in stochastic particle systems that exhibit a condensation transition. Condensation occurs when, above some critical density, a finite fraction of all the particles in the system accumulate on a single lattice site. We present a detailed analysis of a particular size-dependent zero-range process which was introduced as a toy model for clustering in granular media. This model also captures all the relevant details of more generic condensing zero-range processes close to the critical point. Results on the equivalence of ensembles and metastability are based on large deviation principles for the maximum of triangular arrays of independent random variables conditioned on their sum. We derive the saddle point structure of the associated free energy landscape, which implies different mechanisms for the dynamics of the condensate depending on the system parameters. These results lead us to an interesting conjecture on the stationary dynamics of the condensate in the thermodynamic limit.

Speaker: LUCA AVENA, WIAS, Berlin (Germany)

Title: A local CLT for some convolution equations with applications to self-avoiding walks

Abstract: We discuss a fixed point method to obtain a local central limit theorem for distributions defined by certain renewal type equations. The main motivation for investigating these equations stems from applications to lace expansions, in particular to high dimensional weakly self-avoiding walks. As an application we introduce and treat such self-avoiding walks in continuous space. The error bounds obtained are sharper that the ones obtained by other methods. Joint work with E. Bolthausen and C. Ritzmann.

Speaker: GUSTAVO POSTA (Politecnico di Milano, Università La Sapienza)

Title: Decadimento dell'entropia per sistemi markoviani interagenti

Abstract: Verrà illustrata una tecnica per ottenere disuguaglianze funzionali che implicano il decadimento dell'entropia per alcune catene di Markov a tempo continuo. La tecnica è ispirata alle idee di Bakry ed Emery in questo contesto. Si otterranno stime del tempo di mixing, uniformi nel volume, per alcuni sistemi di particelle interagenti.

Speaker: LORENZO CARVELLI (Università La Sapienza)

Titile: Restart di algoritmi meta-euristici per problemi di ottimizzazione combinatoria

Abstract: La soluzione di problemi di ottimizzazione combinatoria di grande dimensione, come ad esempio il problema del commesso viaggiatore con migliaia di città, richiede spesso l'uso di algoritmi meta-euristici. Questi algoritmi sono di tipo generale e non dipendono dal particolare problema. Tra essi ci sono il Simulated Annealing, gli algoritmi genetici e quelli Ant Colony. Per diminuire le possibilità che alla fine dell'esecuzione l'algoritmo fornisca una soluzione subottimale, viene spesso utilizzata la tecnica del "restart" che consiste nell'inizializzare periodicamente l'algoritmo in modo random. Nonostante il restart sia molto utilizzato in pratica, ci sono pochi studi teorici al riguardo. La scelta del tempo di restart viene quindi effettuata sulla base di criteri empirici. In questo seminario, dapprima illustrerò alcuni risultati teorici del mio lavoro di tesi sulle condizioni in cui il restart migliora le possibilità di successo dell'algoritmo meta-euristico e sulla scelta del tempo di restart. I risultati teorici non sono purtroppo utili nella pratica in quanto basati sulla conoscenza della funzione di sopravvivenza del tempo necessario a trovare la soluzione tramite l'algoritmo meta-euristico. Successivamente descriverò una nuova procedura iterativa per l'ottimizzazione in tempo reale del restart dimostrandone la convergenza. La procedura da me proposta non è basata sulla conoscenza della soluzione del problema. Questi due fatti permettono di applicarla con successo. Applicherò infine la procedura ad un algoritmo Ant Colony per trovare la soluzione di alcune istanze del problema del commesso viaggiatore con centinaia o migliaia di città.

Speaker: FABIO CAMILLI (SBAI, Università La Sapienza)

Title: Equazioni di Hamilton-Jacobi su networks

Abstract: Si discuteranno alcuni recenti risultati riguardanti le equazioni di Hamilton-Jacobi definite su un network. Per equazioni ellittiche su networks si evidenzierà il ruolo chiave delle condizioni di transizione di tipo Kirchhoff sui nodi interni, ma si mostrerà che tale approccio non si adatta alle equazioni di Hamilton-Jacobi. Si introdurrà invece un'appropriata definizione di soluzione viscosità per caratterizzare la soluzione significativa del problema.

Speaker: PIETRO CAPUTO (Università Roma Tre)

Title: Large deviations in sparse random graphs: a local weak convergence approach

Abstract: Consider the Erdös-Renyi random graph on n vertices where each edge is present independently with probability p=c/n, with c>0 fixed. For large n, a typical realization locally behaves like the Galton-Watson tree with Poisson offspring distribution with mean c. We discuss large deviations from this typical behavior, within the framework of the local weak convergence introduced by Benjamini-Schramm and Aldous-Steele. The associated rate function is expressed in terms of an entropy functional on unimodular measures and takes finite values only at measures supported by trees. Along the way, we present a new configuration model which allows one to sample uniform random graphs with a given finite neighborhood distribution, provided the latter is supported by trees. We also present a new class of unimodular random trees, which generalizes the Galton-Watson tree with given degree distribution to the case of neighborhoods of arbitrary finite depth. This is joint work with Charles Bordenave.

Speaker: ALESSANDRA FAGGIONATO (Università La Sapienza)

Title: Fluctuations of random walks on quasi 1D lattices and applications to biophysical systems

Abstract: A broad class of kinetic models for molecular motors is given by random walks on quasi 1D lattices with random holding times, not necessarly exponential. We derive information on the asymptotic velocity (law of large numbers), gaussian fluctuations (invariance principle) and large fluctuations (large deviation principle). As applications, we consider some special models and give explicit formulas. We also discuss some theoretical results on Gallavotti-Cohen type symmetries for molecular motors. Joint work with D. Di Pietro, V. Silvestri.

Speaker: DIMITRI KOROLIOUK (Accademia delle Scienze Ucraina)

Title: Binary statistical experiments with persistent regression

Abstract: We deal with a mathematical model of binary statistical experiments, based on statistical data, for the validation of the elementary hypothesis about the presence or absence of a predefined attribute A in the set of elements that make up a complex system. It is assumed that: 1) All the elements that make up the system can gain or lose the given attribute A over time, that is the frequency of the attribute A is a dynamic variable; 2) The basic object of the study are the statistical experiments, which are characterized by relative frequencies of the presence or absence of the attribute A in the sample of fixed volume at each time instant. 3) It is assumed the dependency of (average) results of the next experiment on the average result of the present experiment. This relationship is called the property of persistent regression and used as fundamental condition for the subsequent analysis of the model.

Speaker: MASSIMILIANO GUBINELLI (Università Dauphine, Parigi)

Title: Stochastic quantization, paraproducts and all that

Abstract: The stochastic quantization equation is a simple model for the kind of problems linked to locality and non-triviality of quantum field theories. In this talk we review recent advances in undestanding of the functional analytic structure of solutions to non-linear SPDEs and their application to the study of the stochastic quantization of a scalar field in 3 dimensions. These advances have been possible thanks to a generalization of the theory of (controlled) rough paths which allows a pathwise formulation to stochastic differential equations driven by irregular signal. In particular we discuss the role of multiscale decomposition of distributions and of the notion of paraproduct in the analysis of this problem.

Speaker: GIACOMO DI GESU' (Università La Sapienza)

Title: Spectral asymptotics for discrete metastable diffusions via Witten complex techniques

Abstract: I will consider a metastable diffusion moving in a multiwell potential on the rescaled n-dimensional integer lattice. From a purely spectral point of view metastability effects correspond to the presence of nearly degenerate small eigenvalues of the generator, each one linked to a well of the potential. I will present a result providing complete asymptotic expansions of these small eigenvalues. The proof, inspired by previous work of B. Helffer, M. Klein and F. Nier in continuous setting, is based on tools of semiclassical analysis (Harmonic approximation, WKB expansions) and on a supersymmetric extension à la Witten of the generator on the level of discrete 1-forms.