# Venue

Monday: Talks and reception will be held in room 1003 in Meyer building (electrical engineering department).

Tuesday to Friday: Talks will be held in room room 112 in Bloomfield building (industrial engineering department).

# Schedule

# Long talk abstracts

** Rami Atar: The Skorohod map and priority. **

We introduce a Skorohod map acting in the space of paths with values in the space of finite measures over the real line, motivated by queueing models in which tasks are scheduled according to a continuous parameter priority (earliest deadline first, for example). We use it to obtain new results regarding uniqueness of fluid models and LLN-scale convergence. Joint work with Anup Biswas, Haya Kaspi and Kavita Ramanan

** Martin Barlow: Scaling limits of the uniform spanning tree. **

The uniform spanning tree (UST) has played a major role in recent developments in probability. In particular the study of its scaling limit led to the discovery of SLE by Oded Schramm. In this talk I will discuss the geometry of the UST in 2 dimensions, and what we can say about its scaling limit. Joint work with David Croydon (University of Warwick), Takashi Kumagai (RIMS, Kyoto).

** Yuliy Baryshnikov: Ranges of quantum random walks. **

Quantum random walks (on lattices) is a popular model exhibiting interesting interactions of the geometry of the (ballistically scaling) range of the system, and properties of the underlying coin, a finite-dimensional unitary operator. In tis talk we will explore some to these connections.

** Omer Bobrowski: Maximal cycles in random geometric complexes. **

Random geometric complexes are simplicial complexes generated by random point process (e.g. Poisson). In this talk we will review recent advances in the study of the homology (cycles in various dimensions) of these complexes. In particular, we will focus on recent work describing the size of the "largest" cycles that can be formed in random geometric complexes, and discuss the contribution of this study to applications in topological data analysis (TDA).

** Krzysztof Burdzy: On obliquely reflected Brownian motion. **

I will present some results on obliquely reflected Brownian motion in fractal domains. Time permitting, I will also discuss discrete approximations of reflected Brownian motion in fractal domains. Joint work with Zhenqing Chen, Donald Marshall and Kavita Ramanan.

** Don Dawson: Random walk, percolation and interacting diffusions in an ultrametric setting. **

Spatial stochastic models and their behaviour in different space and time scales have been intensively studied in Euclidean spaces for many years. In this lecture we review some developments in the study of the analogous questions in a class of ultrametric spaces and the relations between these two settings. In particular we consider random walks, percolation and interacting R^k_+ valued diffusions for k = 1, 2 and the classification of their behaviours in different space and time scales in this setting. This is based on joint research projects with Luis Gorostiza and Andreas Greven.

** Nathalie Eisenbaum: Permanental vectors with non-symmetric kernels. **

A permanental vector with a symmetric kernel and index 2 is a squared Gaussian vector. The definition of permanental vectors is a natural extension of the definition of squared gaussian vectors to non-symmetric kernels and to positive indexes. The only known permanental vectors either have a positive definite kernel or are infinitely divisible. Are they some others ? We will answer this question.

** Oren Louidor: The full extremal process of the discrete Gaussian free field in 2D. **

We show the existence of the limit of the full extremal process of the discrete Gaussian free field in 2D with zero boundary conditions. The limit is a clustered Poisson point process with a random intensity measure, which is conjecturally related to the critical Liouiville quantum gravity measure w.r.t. the continuous Gaussian free field. Several corollaries follow directly, e.g. a natural construction for the super-critical Gaussian multiplicative chaos and Poisson-Dirichlet statistics for the limiting Gibbs measure - both w.r.t. the CGFF. The proof is based on a novel concentric decomposition of the DGFF which effectively reduces the problem to that of finding asymptotics for the probability of a decorated non-homogenous random-walk required to stay positive. Entropic repulsion plays a key role in the analysis. Joint work with M. Biskup (UCLA).

** Edwin Perkins: The boundary of the support of super-Brownian Motion. **

We will study the edge of the support of 1-dimensional super-brownian motion. The local behaviour of the density, Hausdorff dimension of the boundary, rates of convergence of certain solutions to singular semilinear heat equations studied in the pde literature are all expressed in terms of a particular eigenvalue of a killed Ornstein-Uhlenbeck operator. Time permitting, we will discuss some possible connections with pathwise uniqueness questions for some related stochastic pde's. This is joint work with Carl Mueller and Leonid Mytnik.

** David Perry: A useful Pollaczek-Khinchine type formula for certain closed stochastic storage systems. **

We introduce a new and useful PollaczekKhinchine type formula for certain closed stochastic storage systems. Special cases of the formula are applied to several versions of the (S-1, S) inventory models, the Erlang loss model of the M/G/S/S queue and S machines interference models with random breakdowns and repairs. Joint work with Onno Boxma and Wolfgang Stadje.

** Kavita Ramanan: Pathwise differentiability of reflected diffusions in simple polyhedra. **

We establish pathwise differentiability with respect to the initial condition and drift coefficient of a large class of semimartingale reflected diffusions in simple polyhedra that arise as strong solutions to stochastic differential equations with reflection. We characterize the derivatives in terms of time-dependent Skorokhod-type problems. The proof relies on certain path properties of these reflected diffusions at non-smooth parts of the boundary, which might be of independent interest. We apply these results to characterize derivatives of stochastic flows of semimartingale reflected diffusions and obtain a Bismut-Elworthy type formula. This is joint work with David Lipshutz.

** Jay Roesn: Conditions for permanental processes to be unbounded. **

Permanental processes are generalizations of Gaussian processes. They replace Gaussian processes in Isomorphism Theorems for non-symmetric Markov Processes. To apply these Isomorphism Theorems it is important to know sample path properties of permanental processes. We present a Sudakov type inequality which gives lower bounds on permanental processes.

** Gennady Samorodnitsky: Climbing down Gaussian peaks. **

How likely is the high level of a continuous Gaussian random field on an Euclidean space to have a ``hole'' of a certain dimension and depth? Questions of this type are difficult, but in this paper we make progress on questions shedding new light in existence of holes. How likely is the field to be above a high level on one compact set (e.g. a sphere) and to be below a fraction of that level on some other compact set, e.g. at the center of the corresponding ball? How likely is the field to be below that fraction of the level anywhere inside the ball? We work on the level of large deviations.

** Jonathan Taylor: TBA **

TBA

** Balint Toth: Diffusivity of random walks in random environments. **

I will survey recent results on diffusive bounds and central limit theorem for some models of RWRE. Full CLT is established for random walks in divergence-free random drift field, under the $H_(-1)$ condition. Diffusive bounds are established for RWRE under the assumption of existence of absolutely continuous invariant distribution of the environment seen from the random walker.

** Sreekar Vadlamani: Kinematic Fundamental Formulae - past, present and... future. **

We shall try to trace the history of kinematic fundamental formulae, and while presenting some of the major milestones, we shall present some recent developments related to GKF. (joint with D. Marinucci)

** Shmuel Weinberger: Quasicrystals to Statistical Topology. **

The inhabitant of a regular tiling experiences a life no different from one of a torus. However, those who live on an aperiodic tiling have a richer, almost higher dimensional, life. I will discuss some ideas that are used to express and study this, and discuss its relevance to understanding asymptotic aspects of compact manifold theory, and, in particular their "statistical sampling theory".

** Ofer Zeitouni: Complexity of random Gaussian functions and second moments. **

The study of critical points of Gaussian fields in high dimension often proceeds by variants of the Kac-Rice lemma. In the context of spherical p-spin models, I will describe recent results that allow one to give a precise description of the maxima, including a description of extremal processes. An important ingredient is the second moment method, and some techniques that were originally developed in the context of low dimensional GFF.

# Short talk abstracts

** Paul Balanca: Uniform multifractal structure of stable super-Brownian motion. **

We describe in this talk the uniform multifractal structure of stable super-Brownian motion in high dimension $d\geq\tfrac{2}{\gamma-1}$. The resulting spectrum is composed of a main component which exists uniformly at every time $t>0$ and encompasses Hölder exponents in the interval $[\tfrac{2}{\gamma},\tfrac{2}{\gamma-1}]$. In addition to this dominant behaviour, points with larger masses, and thus index smaller than $\tfrac{2}{\gamma}$, appear at exceptional times, leading to the definition and characterisation of a second type multifractal spectrum on the time axis.

** Iddo Ben-Ari: Efficient Coupling for Diffusion with Redistribution**

Consider Brownian motion on a bounded interval which is redistributed back into the interval whenever hitting the boundary. The redistribution from each boundary point is according to some fixed probability distribution associated with it, and, conditioned on the boundary point ``hit", the redistribution is independent of the past. It is not hard to show that this is an exponentially ergodic Markov process. The convergence rate has an analytic characterization as the spectral gap for an operator that can be viewed as the infinitesimal generator of the process. In some distinguished cases the gap has an explicit numerical expression. The main thrust of the work presented is to provide a probabilistic and intuitive interpretation for the convergence rate through construction of an efficient coupling, that is a coupling in which the exponential tail of the coupling time coincides with the convergence rate. The coupling is elementary but obtaining efficiency is not straightforward due to the fact that the redistribution ``reshuffles" the system. This is in sharp contrast with reflected BM on an interval for which it is well known that essentially any coupling is efficient. Time permitting, I will also discuss probabilistic treatment of spectral problems for related models.

** Oleg Butkovsky: Path-by-path uniqueness of solutions of stochastic heat equation with a drift. **

It is well known from the results of A. Zvonkin, A. Veretennikov, N. Krylov, A. Davie, J. Mattingly and other probabilists that ordinary differential equations (ODEs) regularize in the presence of noise. Even if an ODE is "very bad" and has no solutions (or has many solutions) the addition of a random noise leads (almost surely) to a "nice" ODE with a unique solution. We investigate the same phenomenon for a heat equation with a drift. We proved that for almost all trajectories of random white noise the perturbed heat equation has a unique solution. Joint work with Leonid Mytnik.

** Yogeshwaran Dhandapani: Normal Convergence for geometric statistics of clustering point processes. **

I shall describe a central limit theorem for geometric functionals on point processes satisfying a "clustering condition". Examples of geometric functionals of interest are Morse critical points, intrinsic volumes of a Boolean model, edge length in the k-nearest neighbour graph. Non-trivial examples of point process that satisfy our requirements are zeros of Gaussian analytic functions, permanental and determinantal point processes. In some particular cases, I shall describe variance asymptotics which are also necessary for the central limit theorem. This is a joint work with Joseph Yukich and Bartek Blaszczyszyn.

** Fima Klebaner: SPDE limits for the age distribution in populations with high carrying capacity. **

We prove fluid and central limit approximations for measure valued ages under smooth demographic assumptions. Joint work with Fan, Hamza (Monash) and Jagers (Chalmers).

** Sunder Ram Krishnan: The Reach of Randomly Embedded Manifolds. **

Roughly speaking, the reach, or critical radius, of a manifold is a mea- sure of its departure from convexity that incorporates both local curvature and global topology. It plays a major role in many aspects of Differential Geometry, and more recently has turned out to be a crucial parameter in assessing the efficiency of algorithms for manifold learning. We study a sequence of random manifolds, generated by embedding a fixed, compact manifold into Euclidean spheres of increasing dimension via a sequence of Gaussian mappings, and show a.s. convergence of the corresponding critical radii to a constant. Somewhat unexpectedly, the constant turns out to be the same one that arises in studying the exceedence probabilities of Gaussian processes over stratified manifolds. Joint with Robert Adler, Jonathan Taylor, Shmuel Weinberger

** Ely Merzbach: Fractional Poisson processes and martingales. **

We present different definitions and properties of the Fractional Poisson Process. Using inverse subordinators and Mittag-Leffler functions, we give a new definition of a fractional Poisson process parametrized by points of the Euclidean plane. Some properties are studied and, in particular, we prove a long-range dependence property. We extend the Watanabe characterization for such fractional Poisson fields. Joint work with G. Aletti and N. Leonenko.

** Leonid Mytnik: Regularity of superprocesses with stable branching mechanism.**

We study regularity properties of the densities of the super-Brownian motion. In dimension one, the spectrum of singularities of the continuous densities is established.

** Eyal Neuman: Discrete SIR Epidermic Processes and their Relation to Extreme Values of Branching Random Walk. **

We study the behaviour of spatial SIR epidemic models in dimensions two and three. In these models, populations of size N are located at sites of the d-dimensional lattice, and infections occur between individuals at the same or at neighbouring sites with infection probability pN. Susceptible individuals, once infected, remain contagious for one unit of time and then recover, after which they are immune to further infection. We answer the question which was raised in Lalley, Perkins and Zheng (2014) and prove that there exist critical values p_c(N) > 0 such that for N large enough, if p_N > p_c(N), then the epidermic survives forever with positive probability. When p_N < p_c(N) we prove that the epidemic dies out in finite time with probability 1. We show that the behaviour of extreme values of branching random walk is a key ingredient in the proof of phase transition for the SIR epidermic processes. In this context, we prove that the support of the local time of supercritical branching random walk near criticality, grows in a linear speed. Finally the asymptotic law for the right most position reached by subcritical branching random walk is derived. This is joint work with Xinghua Zheng.

** Gideon Weiss: FCFS Routing to Skilled Servers. **

Consider two independent i.i.d. sequences of types C={c1,…,cI) and S={s1,…,sJ) with a bipartite compatibility graph between C and S, and FCFS (first come first served) matching of the sequences according to the compatibility graph. We reveal the extreme simplicity of this model, and its implications to some complex queueing models.

** Richard Wilson: Slepian models and optimal predictions for catastrophes. **

We construct Slepian random field models to provide predictions of extreme events (catastrophes) for spatio-temporal processes. These are based on an alarm region for a random field for fixed locations in space at a fixed time for predicting extremes for a related random field (possibly the same one) at given point locations, along the boundary of a region or within a region. An optimal alarm region is chosen using likelihood based principles. The results are evaluated by considering probabilities for false alarms and the risk of an extreme. Joint work with Anastassia Baxevani.