Abstracts are added as they become available.

See the Schedule page for the schedule.

**Asymptotic analysis of multiclass queues with random order of service**

*Reza Aghajani (University of California, San Diego)*

The random order of service (ROS) is a natural scheduling policy for systems where no ordering of customers can or should be established. Queueing models under ROS have been used to study molecular interactions of intracellular components in biology. However, these models often assume exponential distributions for processing and patience times, which is not realistic especially when operations such as binding, folding, transcription and translation are involved. We study a multi-class queueing model operating under ROS with reneging and generally distributed processing and patience times. We use measure-valued processes to describe the dynamic evolution of the network, and establish a fluid approximation for this representation. Obtaining a fluid limit for this network requires a multi-scale analysis of its fast and slow components, and to establish an averaging principle in the context of measure-valued process. In addition, under slightly more restrictive assumptions on the patience time distribution, we introduce a reduced, function-valued fluid model that is described by a system of non-linear Partial Differential Equations (PDEs). These PDEs, however, are non-standard and the analysis of their existence, uniqueness and stability properties requires new techniques.

**Averaging results for non-autonomous slow-fast systems of SPDEs**

*Sandra Cerrai (University of Maryland)*

We study the validity of an averaging principle for a slow-fast system of stochastic reaction-diffusion equations. We assume here that the coefficients of the fast equation depend on time so that the classical formulation of the averaging principle in terms of the invariant measure of the fast equation is no longer available. As an alternative, we introduce the time-dependent evolution family of measures associated with the fast equation. Under the assumption that the coefficients in the fast equation are almost periodic, the evolution family of measures is almost periodic. This allows us to identify the appropriate averaged equation and prove the validity of the averaging limit.

**Detection of anomalous path in a noisy network**

*Shirshendu Chatterjee (City University of New York)*

Consider a two dimensional finite graph having side length $O(n)$. Each vertex of the graph is associated with a random variable, and these are assumed to be independent. In this setting, we will consider the following hypothesis testing problem. Under the null hypothesis, all the random variables have common distribution $N(0, 1)$, while under the alternative (signal) hypothesis, there is an unknown path (with unknown initial vertex) having $O(n)$ edges (e.g. a "left to right crossing") along which the associated random variables have distribution $N(\mu_n, 1)$ for some $\mu_n > 0$, and the random variables away from the path have distribution $N(0, 1)$. We will describe the values of the mean shift $\mu_n$ for which one can reliably detect (in the minimax sense) the presence of the anomalous path, and for which it is impossible to detect. This is based on a joint work with Ofer Zeitouni.

**Resolvent kernel functions arising from some stochastic partial differential equations**

*Le Chen (University of Nevada, Las Vegas)*

For a stochastic partial differential equation, the second moment of its solution is one of the very first important quantities to study. In many cases, this second moment satisfies a linear Volterra equation of the second kind. I will show several examples when one can solve this integral equation explicitly. This is a purely analytic problem and the main tools that we use consist the Laplace and Fourier transforms and their inversions.

**Mirror model on the square lattice and triangular lattice**

*Yan Dai (University of Arizona)*

We consider a random walk model on the two-dimensional square lattice, starts at the origin and only turns left and right at each step with equal probability. Going straight and revisiting a bound that has been visited before are not allowed. In this model, turning left or right at each step can be viewed as a walk deflecting by a left or right mirror on each vertex. Therefore, we refer this random walk model as a mirror model. Here, we study the nature of the mirror model process on the square and triangular lattice. Localization, reversibility and self-touching properties have been investigated. We believe that the scaling limit of the mirror model on the lattice $\mathbb{Z}^2$ in a bounded domain between two boundary points is the chordal Schramm-Loewner evolution with $\kappa$ = 6 (SLE$_6$). We test this conjecture on both lattices and find a good agreement with predictions of chordal SLE$_6$.

**Robust tests for change-points in time series**

*Herold Dehling (Ruhr-University Bochum)*

We will present some recent results on robust tests for change-points in time series. The test statistics are based on two-sample U-processes and U-quantiles. We will analyze the large sample behavior of these processes when the underlying data are functionals of an absolutely regular process. In our work, we make heavy use of techniques that have been developed by Borovkova, Burton and Dehling (2001) in their study of empirical processes of U-statistics of dependent data. [This talk is based on joint work with Roland Fried (Dortmund) and Martin Wendler (Greifswald)]

**Fluctuation of Lévy processes : From Wiener-Hopf factorizations to inverse scattering problem**

*Sonia Fourati (INSA de Rouen)*

In the sequel, I shall call "unilateral problems" the computation of the distribution of the coordinates of extremal points of a real Lévy Process (eventually killed at an independant exponential time). This problem is equivalent to the computation of the first time that the Lévy process exits from a half line. The "bilateral problem" is the computation of the distribution of the joint distribution of the process of minimal and maximal values of times or equivalently, the distribution of the first time that the process exits from a bounded interval. In this talk, an overview of the Wiener-Hopf factorization as the solution of the "unilateral problem" for real Lévy processes will be given. This follows and honors the seminal work of Priscilla Greenwood. We will show that the "bilateral problem" is some what explicited in the same way as the unilateral one, but scalar functions are replaced by matrices 2-2, and the factorization of Wiener-Hopf type is the one that appears in the Inverse Scattering theory. Some solutions are given by closed explicit formulas.

*This work has been supported by the X-term Project from FEDER founds*

**Dirichlet and Poisson-Dirichlet approximation for genetic drift models**

*Han Liang Gan (Northwestern University)*

Genetic drift models are used to study how gene types and their frequencies vary over time. For many seemingly simple models, exact distributions are often intractable. In this talk we will discuss how when working with a finite type space, the distribution of gene frequencies can often be well approximated by the Dirichlet distribution. However in some scenarios a countably infinite type space is preferable, and the infinite-dimensional analogue to Dirichlet, namely Poisson-Dirichlet is required. Using Stein's method we will calculate explicit error bounds and rates of convergence for Dirichlet and Poisson-Dirichlet approximation.

**Stochastic neuron models**

*Priscilla Greenwood (University of British Columbia)*

I will describe some stochastic neuron models, some results about them, and some open problems which may interest probabilists.

**Analysis of a Kraichnan-type fluid model**

*Jingyu Huang (University of Utah)*

We study the turbulent transport of a passive scalar quantity in a stratified, 2-D random velocity field. It is described by the stochastic partial differential equation \begin{equation*} \partial_t \theta(t, x, y)= \nu \Delta \theta (t, x, y) + V(t, x) \partial _y \theta (t, x, y) \ \ \text{for}\ t \geq 0 \ \text{and}\ x, y \in \mathbb{R}\,, \end{equation*} where $V$ is some Gaussian noise. We show via a priori bounds that, typically, the solution decays with time. More interesting still, the decay is shown to be “macroscopically multifractal” in special settings. The detailed analysis is based on a probabilistic representation of the solution, which is likely to have other applications as well. This is based on joint work with Davar Khoshnevisan.

**Spectrum of Random Band Matrices**

*Indrajit Jana (Temple University)*

Random matrix theoey can be used as a mathematical tool in several scientific research including Nuclear Physics, Signal Processing, Numerical linear algebra etc. We consider the limiting spectral distribution of matrices of the form $\frac{1}{2b_{n}+1} (R + X)(R + X)^{*}$, where $X$ is an $n\times n$ random band matrix of bandwidth $b_{n}$ and $R$ is a non random band matrix of bandwidth $b_{n}$. We show that the Stieltjes transform of ESD of such matrices converges to the Stieltjes transform of a non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For $R=0$, the integral equation yields the Stieltjes transform of the Marchenko-Pastur law.

**Particle representations for a class of non-linear stochastic partial differential equations with boundary conditions**

*Christopher Janjigian (University of Utah)*

This talk will present a weighted particle representation for a class of non-linear stochastic partial differential equations with Dirichlet boundary conditions, focusing on a concrete example.

In this model, an infinite family of independent particles carry weights which evolve according to a system of stochastic differential equations driven by a common cylindrical noise and which interact through through the associated weighted empirical measure. To account for the boundary condition, when the particles hit the boundary their corresponding weights are assigned a pre-specified value.

Our results include the existence and uniqueness of a solution of the infinite dimensional system of stochastic differential equations modeling the location and the weights of the particles. We also prove that the associated weighted empirical measure V is the unique solution of the desired stochastic partial differential equation driven by W and satisfying the boundary condition, subject to moment conditions and compatibility with the noise.

Based on joint work with Dan Crisan and Tom Kurtz.

**Towards classification of noncommutative Bernoulli schemes**

*Michael Keane (Wesleyan University)*

In this lecture a joint work still in progress with Toshihiro Hamachi will be discussed, in which we attempt to establish when two noncommutative Bernoulli schemes are isomorphic, and when one is a homomorphic image of the other. Our underlying ideas are largely based on the finitary isomorphism theory developed with Meir Smorodinsky at Tel-Aviv University in the 1970's. These ideas can best be understood by recalling the important first example of Meshalkin (1964), in which he showed that the commutative Bernoulli schemes based on the probability vectors (1/4,1/4,1/4,1/4) and (1/2,1/8,1/8,1/8,1/8) are isomorphic. In this case, the corresponding commutative schemes are finitarily isomorphic, but for the associated noncommutative schemes, there is an injective *-isomorphism which is not onto from the von Neumann algebra of the second scheme into that of the first scheme.

**Stochastic particle systems related to grain boundary coarsening**

*Joe Klobusicky (Rensselaer Polytechnic Institute)*

Transport of organelles and other vital material in biological cells is conducted largely by specialized molecular motor proteins moving along actin or microtubule filaments. Their dynamics are inherently stochastic due to both the importance of thermal fluctuations on their length scale and the dependence of their motion on discrete binding events to ATP molecules. This presentation will begin by reviewing some of the prevalent methods for representing the dynamics of molecular motors on a single microtubule, and describe ongoing and emerging research questions concerning experimental observations for how molecular motors move through networks of microtubules. As a complement to detailed computational models for addressing these questions, we formulate some relatively simple modeling scenarios for which asymptotic stochastic procedures can relate, in mathematical terms, how the parameters describing motor function relate to their effective transport on larger scales. Our key technical concern is attempting to more faithfully couple the stochastic spatial dynamics and the attachment/detachment stochastic switching dynamics.

**Spatial stochastic models for molecular motors attaching and detaching from microtubules**

*Peter Kramer (Rensselaer Polytechnic Institute)*

Transport of organelles and other vital material in biological cells is conducted largely by specialized molecular motor proteins moving along actin or microtubule filaments. Their dynamics are inherently stochastic due to both the importance of thermal fluctuations on their length scale and the dependence of their motion on discrete binding events to ATP molecules. This presentation will begin by reviewing some of the prevalent methods for representing the dynamics of molecular motors on a single microtubule, and describe ongoing and emerging research questions concerning experimental observations for how molecular motors move through networks of microtubules. As a complement to detailed computational models for addressing these questions, we formulate some relatively simple modeling scenarios for which asymptotic stochastic procedures can relate, in mathematical terms, how the parameters describing motor function relate to their effective transport on larger scales. Our key technical concern is attempting to more faithfully couple the stochastic spatial dynamics and the attachment/detachment stochastic switching dynamics.

**Mixing time for unbiased dyadic tilings**

*David Levin (University of Oregon)*

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. A {\it dyadic tiling} of size $n$ is a tiling of the unit square by $n$ non-overlapping dyadic rectangles, each of area $1/n$, where a {\it dyadic rectangle} is any rectangle that can be written in the form $[a2^{-s}, (a+1)2^{-s}] \times [b2^{-t}, (b+1)2^{-t}]$ for $a,b,s,t \in \mathbb{Z}_{\geq 0}$. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most $O(n^{4.09})$, which implies that the mixing time is at most $O(n^{5.09})$. We complement this by showing that the relaxation time is at least $\Omega(n^{1.38})$, improving upon the previously best lower bound of $\Omega(n\log n)$ coming from the diameter of the~chain. Joint with A. Stauffer and S. Cannon.

**The tightness of the Kesten–Stigum reconstruction bound of symmetric model with multiple mutations**

*Wenjian Liu (City University of New York)*

It is well known that reconstruction problems, as the interdisciplinary subject, have been studied in numerous contexts including statistical physics, information theory and computational biology, to name a few. We consider a $2q$-state symmetric model, with two categories of $q$ states in each category, and 3 transition probabilities: the probability to remain in the same state, the probability to change states but remain in the same category, and the probability to change categories. We construct a nonlinear second-order dynamical system based on this model and show that the Kesten-Stigum reconstruction bound is not tight when $q \geq 4$.

**A quantitative theory of the hydrodynamic limit**

*Georg Menz (University of California, Los Angeles)*

About a joint work with Deniz Dizdar, Felix Otto, and Tianqi Wu. The hydrodynamic limit is a dynamic manifestation of the law of large numbers. A microscopic random process converges macroscopically to a deterministic process. In this talk, we discuss how to derive quantitative error bounds for the hydrodynamic limit of the Kawasaki dynamics via the two-scale approach. This seems to be the first quantitative statement of this kind.

**Birthday paradox, monochromatic subgraphs, and the second moment phenomenon**

*Somabha Mukherjee (University of Pennsylvania)*

What is the chance that among a group of $n$ friends, there are $s$ friends all of whom have the same birthday? This is the celebrated birthday problem which can be formulated as the existence of a monochromatic $s$-clique $K_s$ ($s$-matching birthdays) in the complete graph $K_n$, where every vertex of $K_n$ is uniformly colored with $365$ colors (corresponding to birthdays). More generally, for a general connected graph $H$, let $T(H, G_n)$ be the number of monochromatic copies of $H$ in a uniformly random coloring of the vertices of the graph $G_n$ with $c_n$ colors. In this paper we show that $T(H, G_n)$ converges to ${\mathrm{Pois}}(\lambda)$ whenever ${\mathrm E}\,T(H, G_n) \rightarrow \lambda$ and $\mathrm{Var}\,T(H, G_n) \rightarrow \lambda$, that is, the asymptotic Poisson distribution of $T(H, G_n)$ is determined just by the convergence of its mean and variance. Moreover, this condition is necessary if and only if $H$ is a star-graph. In fact, the second-moment phenomenon is a consequence of a more general theorem about the convergence of $T(H,G_n)$ to a finite linear combination of independent Poisson random variables. As an application, we derive the limiting distribution of $T(H, G_n)$, when $G_n\sim G(n, p)$ is the Erd\H os-Rényi random graph. Multiple phase-transitions emerge as $p$ varies from 0 to 1, depending on whether the graph $H$ is balanced or unbalanced.

**Anomalous diffusion and the Generalized Langevin Equation**

*Hung Nguyen (Tulane University)*

The Generalized Langevin Equation is commonly used to describe the velocity of microparticles in viscoelastic fluids. Formally, the Generalized Langevin Equation (GLE) is written \begin{align*} m \ddot{x}(t)&=-\gamma \dot{x}(t)-\Phi'(x(t))-\int_{-\infty}^t \!\!\!\! K(t-s)\dot{x}(s)ds +F(t)+\sqrt{2\gamma}\dot{W}(t) \end{align*} where $\Phi(x)$ is a non-linear potential well, $W(t)$ is a Brownian motion, and $F(t)$ is a stationary, mean zero and Gaussian process satisfying ${\mathrm{E}}{F(t)F(s)}=K(t-s)$. Describing the long-term behavior of sub-diffusive GLEs in non-linear potentials is a long-standing open problem. We will look at recent advances in establishing existence and uniqueness of a stationary distribution for an infinite-dimensional Markov representation of the GLE. If time permits, we will also discuss asymptotic behaviors of the GLE in different limits, namely, the small-mass limit and the white noise limit.

**Sofic and percolative entropies of Gibbs measures on regular infinite trees**

*Moumanti Podder (Georgia Institute of Technology)*

Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $\Phi$. Let $\{G_{n}\}$ be a sequence of finite graphs with vertex sets $V_n$ that locally converge to $T_{d}$. From $\Phi$ one can construct a sequence of corresponding models on the graphs $G_n$. Let $\{\mu_n\}$ be the resulting Gibbs measures. Here we assume that $\{\mu_{n}\}$ converges to some limiting Gibbs measure $\mu$ on $T_{d}$ in the local weak$^*$ sense, and study the consequences of this convergence for the specific entropies $|V_n|^{-1}H(\mu_n)$. We show that the limit supremum of $|V_n|^{-1}H(\mu_n)$ is bounded above by the \emph{percolative entropy} $H_{perc}(\mu)$, a function of $\mu$ itself, and that $|V_n|^{-1}H(\mu_n)$ actually converges to $H_{perc}(\mu)$ in case $\Phi$ exhibits strong spatial mixing on $T_d$. When it is known to exist, the limit of $|V_n|^{-1}H(\mu_n)$ is most commonly shown to be given by the Bethe ansatz. Percolative entropy gives a different formula, and we do not know how to connect it to the Bethe ansatz directly. We discuss a few examples of well-known models for which the latter result holds in the high temperature regime.

**Lyapunov exponents for operator cocycles**

*Anthony Quas (University of Victoria)*

We study random products of compact operators on Hilbert space. The Lyapunov exponents measure exponential rates of stretching in different directions. In general, Lyapunov exponents are known to be highly sensitive to perturbations of the system. In our context, we show that for noiselike perturbations, the Lyapunov exponents vary continuously.

**Concentration of measure for stochastic heat equation**

*Andrei Sarantsev (University of California, Santa Barbara)*

We show Talagrand concentration inequality, comparing Wasserstein distance with relative entropy, for a stochastic heat equation, continuing similar research by (Pal, 2012; Cattiaux, Guillin, 2013) on stochastic differential equations.

**An impossibility theorem of quantifying causal effect**

*Yue Wang (University of Washington)*

We consider how to quantify the causal effect from a random variable to a response variable. We propose several seemingly reasonable criteria for such causal quantities, and then prove that in certain cases they are incompatible. We also check how popular causal quantities fail to satisfy such criteria.

**Correlations and adaptation in enzymatic networks**

*Ruth Williams (University of California, San Diego)*

The contrast between stochasticity of biochemical networks and regularity of cellular behavior suggests that biological networks generate robust behavior from noisy constituents. Identifying the mechanisms that confer this ability on biological networks is essential to understanding cells. Here we use stochastic queueing models to investigate one potential mechanism. In living cells, enzymes perform the critical function of acting as catalysts to ensure that important reactions occur at rates fast enough to sustain life. We show how competition among different molecular species for the attention of a limited pool of adaptive shared enzymes can produce strong correlations between the different species.

**On hydrodynamic limits of young diagrams**

*Jianfei Xue (University of Arizona)*

We study different evolutional models of two-dimensional Young diagrams parametrized by an energy parameter. We show that under proper scalings the limit shape function of the diagrams solves certain nonlinear or linear PDE depending on the choice of energy.

**Estimate exponential memory decay in Hidden Markov Model and its applications**

*Felix Xiaofeng Ye (University of Washington)*

Inference in hidden Markov model has been challenging in terms of scalability due to dependencies in the observation data. In this paper, we utilize the inherent memory decay in hidden Markov models, such that the forward and backward probabilities can be carried out with subsequences, enabling efficient inference over long sequences of observations. We formulate this forward filtering process in the setting of the random dynamical system and there exist Lyapunov exponents in the i.i.d random matrices production. And the rate of the memory decay is known as $\lambda_2-\lambda_1$, the gap of the top two Lyapunov exponents almost surely. An efficient and accurate algorithm is proposed to numerically estimate the gap after the soft-max parametrization. The length of subsequences $B$ given the controlled error $\epsilon$ is $B=\log(\epsilon)/(\lambda_2-\lambda_1)$. We theoretically prove the validity of the algorithm and demonstrate the effectiveness with numerical examples. The method developed here can be applied to widely used algorithms, such as mini-batch stochastic gradient method. Moreover, the continuity of Lyapunov spectrum ensures the estimated B could be reused for the nearby parameter during the inference.

**Random self-similar trees and their applications**

*Ilya Zaliapin (University of Nevada, Reno)*

The talk focuses on self-similarity for tree graphs (trees). The significance of scale- invariance of trees is well-documented in areas ranging from structure of river networks to the Boltzmann kinetic theory of cluster dynamics. However, a probabilistic treatment of the problem has been restricted to particular Markov classes, such as binary Galton- Watson trees. At the same time, solid empirical evidence motivates a search for a flexible class of self-similar models that would extend beyond the Markov constraint to fit a variety of observed combinatorial and metric structures.

We suggest a general definition of self-similarity via the operation of generalized tree pruning. The latter encompasses a number of previously studied discrete and continuous pruning operations, including the tree erasure studied by Jacques Neveu and Horton pruning studied by Ed Waymire. The proposed framework unifies the self-similarity results of Waymire and Neveu for the critical binary Galton-Watson trees; it also suggests multiple other types of pruning that preserve the critical binary Galton-Watson distribution. Moreover, we introduce a hierarchical branching process that generates a rich class of self-similar non-Markov trees. We emphasize a natural appearance of a one- parametric family of critical Tokunaga trees that preserve many of the symmetries of the critical binary Galton-Watson trees, and reproduce the latter at a particular parameter value. Interestingly, Tokunaga trees have been discussed in the physics literature since the 1970s in connection with approximating a variety of dendritic structures such as river networks, dynamically limited aggregation (DLA), and percolation clusters.

Finally, we discuss a mapping between trees and continuous functions. This allows us to apply the tree self-similarity concepts in time domain and use our results to probe into selected problems of 1-D inviscid Burgers dynamics and self-similar stochastic processes.

(Joint work with Yevgeniy Kovchegov, Oregon State University, and Maxim Arnold, University of Texas at Dallas.)

**Drift parameter estimation for nonlinear stochastic differential equations**

*Hongjuan Zhou (University of Kansas)*

We study a parameter estimation problem for the following stochastic differential equation (SDE) driven by fractional Brownian motion (fBm) $$dX_t = -f(X_t) \theta dt + \sigma dB_t \,, \quad t\ge 0 \,,$$ where $X_0=x_0 \in \mathbb{R}^m$ is a given initial condition. For the diffusion part, $B=(B^1, \dots, B^d)$ is a $d$-dimensional fractional Brownian motion (fBm) of Hurst parameter $H \in (0,1)$, and the diffusion coefficient $\sigma=(\sigma_1, \dots, \sigma_d)$ is an $m \times d$ matrix, with $\sigma_j$, $j=1, \dots, d$ being given vectors in $\mathbb{R}^m$. For the drift part, the function $f: \mathbb{R}^m \rightarrow \mathbb{R}^{m \times l}$ satisfies some regularity and growth conditions. We assume that $\theta= (\theta_1, \dots, \theta_l) \in \mathbb{R}^l$ is an unknown constant parameter vector. We propose the least squares estimator for $\theta$ and prove the strong consistency.