Schedule for: 19w2282 - Retreat for Young Researchers in Probability and areas of Application

Note: the Lecture rooms are available after 16:00.

Beverages and a small assortment of snacks are available in the lounge on a cash honour system.

A buffet breakfast is served daily between 7:00am and 9:00am in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Transient dynamics, often observed in multi-scale systems, are roughly defined to be the interesting dynamical behaviors that display over finite time periods. For a class of randomly perturbed dynamical systems that arise in chemical reactions and population dynamics, and that exhibit persistence dynamics over finite time periods and extinction dynamics in the long run, we use quasi-stationary distributions (QSDs) to rigorously capture the transient states governing the transient dynamics. We study the noise-vanishing concentration of the QSDs to gain information about the transient states. 

The classical Perron-Frobenius theorem can be applied to Markov chains with a single primitive transition matrix to show that there is a unique stationary distribution for the chain, and that distributions relax exponentially quickly to that stationary distribution, where the rate is determined by the second-largest eigenvalue. We can generalize Markov chains: first to chains with randomly chosen transition matrices, then to cocycles of transfer operators, which describe how densities move around according to random underlying dynamics on a state space. I will describe a generalization of the classical Perron-Frobenius theorem that can be applied in this setting to give an analog of a stationary distribution and a relaxation rate, along with some of the definitions and background required to understand the theorem statement.

Starting from a sequence of positive real numbers (w_n), which we call weights, we construct a tree in a recursive manner: at time 1, the tree has only one vertex. Then at any step n+1, we add a new vertex to the tree and we choose its parent at random among the already existing vertices, in such a way that the k-th vertex (in order of creation) is chosen with probability proportional to w_k. This model generalises the well-known uniform recursive tree (URT) in the case of a constant sequence (w_n). In fact, it can also be shown that the trees constructed using affine preferential attachment can be described with this construction, using a random sequence of weights (w_n). We prove almost-sure scaling limits for the height, profile and degrees in the tree as the number of vertices tends to infinity. These results are related to proving scaling limits in the Gromov-Hausdorff-Prokhorov topology for a family of random growth models on graphs that generalises Rémy's algorithm.

We consider an interacting particle system on trees known as the frog model: initially, a single active particle begins at the root and i.i.d. Poisson(u) many inactive particles are placed at each non-root vertex.  Active particles perform discrete time simple random walk and activate the inactive particles they encounter.  It has been shown by Hoffman, Johnson, and Junge that on regular trees, there is a critical value uc separating recurrent and transient regimes.  Little is known, however, about the behavior of the frog model on random structures, and other graphs that do not posses a high degree of self-similarity.  In this talk, I'll discuss our recent results showing that for Galton-Watson trees with certain types of offspring distributions there does exist a critical value uc separating recurrent and transient regimes for almost surely every tree, thereby partially answering a question of Hoffman-Johnson-Junge.  I'll also discuss a related proof showing that for every non-amenable tree with bounded degree there exists a phase transition from transience to recurrence (with a non-trivial intermediate phase sometimes sandwiched in between) as u varies.  This is based on joint work with Marcus Michelen.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A buffet lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

Extreme precipitation, driven by complex spatiotemporal processes, is characterized by limited predictability. Modeling precipitation extremes and understanding their spatial and temporal variations are important for long-term planning. The talk focuses on describing stochastic methods to model extremes at global and regional scale. The global picture of how precipitation properties vary across time and space, specifically in regions where ground-based observations are scarce, is discussed. Uncertainties, quantified using Bayesian techniques, in modelling extremes are enumerated. Importance of studying the uncertainties and propagating them to future for accurate assessment of projected precipitation is emphasized.

We study the optimal liquidation problems in target zone models with dynamic programming methods. Such control problems allow for stochastic differential equations with reflections and random coefficients. The value function is characterized by a Neumann problem of backward stochastic partial differential equations (BSPDEs) with singular terminal conditions. The existence and the uniqueness of strong solution to such BSPDEs are addressed, which in turn yields the optimal feedback control.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

A physical network is naturally subject to noise influences from both external (extrinsic) and internal (intrinsic) sources. The extrinsic noises are usually environmentally related, while the intrinsic ones are typically due to internal uncertainties. A discrete-time, discrete-state (dtds) network with only extrinsic noises is commonly modeled by a discrete random dynamical system (RDS), but the one with only intrinsic noises is often modeled by a Markov chain. In this talk, we will consider a dtds network with both extrinsic and intrinsic noises under the framework of the so-called Markov random network (MRN). In particular, we will discuss the phenomenon and mechanism of desynchronizations for MRNs which arise as Markov-perturbations of a synchronized discrete RDS. Characterization of desynchronizations will be given from the view points of both probability distributions and dynamical systems. This is an ongoing joint work with Profs. Arno Berger, Hong Qian, and Yingfei Yi.

In this talk, we will study time-periodic Fokker-Planck equation (FPE) which arise from stochastic differential equations with time-periodic coefficients, including the existence and uniqueness of periodic probability solutions to FPE, and convergence of global probability solutions of the associated Cauchy problem of FPE. As an application, the long-time behavior of a stochastic damping Hamiltonian system is investigated.

2-day workshop participants are welcome to use BIRS facilities (Corbett Hall Lounge, TCPL, Reading Room) until 15:00 on Sunday, although participants are still required to checkout of the guest rooms by 12 noon. There is no coffee break service on Sunday afternoon, but self-serve coffee and tea are always available in the 2nd floor lounge, Corbett Hall.

We study random gluings of polygons where the genus is not fixed a priori. We will study the law of the degrees of thee largest vertices and show that they are described by a Poisson-Dirichlet process. We use probabilistic arguments, which contrasts sharply with the algebraic tools usually used to prove similar results. Based on joint work with Nicolas Curien and Bram Petri.

Let $(X,d)$ be a metric space and let V be a dense countable subset of X. Construct a random graph G on V by placing an edge between any two points in V with probability q if the distance between them is less than one (and do so independently for different pairs of points). We are interested in the almost-sure properties of G, or more specifically, of the isomorphism class of G. Such properties may be very sensitive to the metric space (though usually less to V and q). For example, Bonata and Janssen, who initiated the study of these graphs, showed that in the case of the $(R^d,L_\infty)$, two independent samples of G are almost surely isomorphic, whereas in the case of $(R^d,L_p)$ with $1$<$p<\infty$ (including the Euclidean case $p=2$), two such samples are almost surely non-isomorphic. We consider the case of a circle R/LZ of length L with its intrinsic metric, and show a surprising dependence of behavior on L. Joint with Omer Angel.