ISU Probability Seminar (Fall '16) Note: The seminar meets on Wednesdays from 2:10PM-3PM in Carver 204 unless specified otherwise. (1) August 30th, 2016. Chao Zhu (University of Wisconsin-Milwaukee) Title: On Feller and Strong Feller Properties of Regime-Switching Jump Diffusion Processes. Abstract: This work considers the martingale problem for a class of weakly coupled Levy type operators. It is shown that under some mild conditions, the martingale problem is well-posed and uniquely determines a strong Markov process $(X, \Lambda)$. The process $(X, \Lambda)$, called a regime-switching jump diffusion with Levy type jumps, is further shown to posses Feller and strong Feller properties via the coupling method. This is a joint work with Fubao Xi (Beijing Institute of Technology, China). (2) September 6th, 2016. Mark Huber (Claremont McKenna College) Title: The Fundamental Theorem of Perfect Simulation. Abstract: Some distributions are easy to simulate from, but many are not. Perfect simulation algorithms give a method for drawing random variates exactly from a wide class of high dimensional distributions. In the past this was accomplished approximately using Markov chains, but now several protocols for getting exact samples exist. In fact, a common principle underlies several of these methods, probabilistic recursion. In this talk I will show how a local correctness property of the recursion can automatically be extended to a global correctness property through what I call the Fundamental Theorem of Perfect Simulation. This idea can then be used as a tool for solving more complex problems. . (3) September 14th, 2016. Arturo Kohatsu-Higa (Ritsumeikan University, Japan) Title: Probabilistic Interpretation of the Parametrix Method. Abstract: The parametrix method has been used for many purposes and many different variations of it have been introduced in the past. In this talk, I will present some of the probabilistic interpretations and some of its applications that I have found interesting for me in the recent past. (4) September 21st, 2016. Christopher Hoffman (University of Washington) Title: Frogs on Trees Abstract: I will discuss a branching process known as the frog model on a regular tree. We will show that if the degree of the tree is three then the frog model is recurrent a.s. while in high dimensions it is transient a.s. I will also discuss other variants of this problem. This is joint work with Tobias Johnson and Matt Junge. (5) October 6th, 2016 2:10-3 in Math 401*. R.H. Stockbridge (University of Wisconsin--Milwaukee) Title: Long-term Average Control of Continuous Inventory Models Abstract: This talk examines a control problem when, in the absence of impulses, the process has continuous sample paths. Examples of such processes include the amount of water behind a dam, the amount of natural gas in a storage facility or (approximately) the value of a business cash account. The talk will be framed in terms of management of inventory. The inventory process is modelled by a one-dimensional stochastic differential equation on some interval in which the left boundary is attracting, so as to capture the effect that demand tends to decrease the inventory level, and the right boundary is non-attracting so that, on their own, returns of inventory do not increase the level to its upper limit. Orders instantaneously increase the inventory level and incur both positive fixed and level-dependent costs. The manager's influence on the inventory is limited solely to ordering policies that increase the current level; he is not allowed to take action so as to reduce the inventory level}. Thus this talk examines an impulse control problem in which the impulses only affect the process by jumps in a single direction. Minimal conditions are provided on the model which imply that an optimal ordering policy exists in the class of $(s,S)$ policies. Examination of the steady state behavior of $(s,S)$ policies leads to a two-dimensional nonlinear optimization problem for which an optimizing pair establishes the levels for an optimal $(s,S)$ policy within a large class of ordering policies. The results will be illustrated using the classical model of a drifted Brownian motion for the underlying diffusion process as well as this process with reflection at 0. In addition, a class of non-Markovian policies will be examined and an intuitive condition on the parameters yields the somewhat surprising result that each $(s,S)$ policy incurs a larger cost than a corresponding non-Markovian policy. Thus no $(s,S)$ policy is optimal. Abstract: I will discuss a branching process known as the frog model on a regular tree. We will show that if the degree of the tree is three then the frog model is recurrent a.s. while in high dimensions it is transient a.s. I will also discuss other variants of this problem. This is joint work with Tobias Johnson and Matt Junge. (5) October 19th, 2016 2:10-3. Philip Matchett-Wood (University of Wisconsin--Madison) Title: Low-degree factors of random polynomials Abstract: We study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers. It is known that certain models are very likely to produce random polynomials that are irreducible, and our project can be viewed as part of a general program of testing whether this is a universal behavior exhibited by many random polynomial models. Interestingly, though the question comes from algebra and number theory, we primarily use tools from combinatorics, including additive combinatorics, and probability theory. We prove for a variety of models that it is very unlikely for a random polynomial with integer coefficients to have a low-degree factor—suggesting that this is, in fact, a universal behavior. For example, we show that the characteristic polynomial of random matrix with independent +1 or −1 entries is very unlikely to have a factor of degree up to $n^{1/2-\epsilon}$. Joint work with Sean O’Rourke, and also joint work with Melanie Matchett Wood and UW-Madison undergraduates Christian Borst, Evan Boyd, Claire Brekken, and Samantha Solberg. (5) October 25th, 2016 2:10-3 in Carver 0128*. Michael Damron (Georgia Tech) Title: The travel time to infinity in first-passage percolation Abstract: On the two-dimensional square lattice, assign i.i.d. nonnegative weights to the edges with common distribution F. For which distributions F is there an infinite self-avoiding path with finite total weight? This question arises in first-passage percolation, the study of the random metric space Z^2 with the induced random graph metric coming from the above edge-weights. It has long been known that there is no such infinite path when F(0) < 1/2 (there are only finite clusters of zero-weight edges), and there is one when F(0) > 1/2 (there is an infinite cluster of zero-weight edges). The critical case, F(0) = 1/2, is considerably more difficult due to the presence of finite clusters of zero-weight edges on all scales. I will discuss work with W.-K. Lam and X. Wang in which we give necessary and sufficient conditions on F for the existence of an infinite finite-weight path. The methods involve comparing the model to another one, invasion percolation, and showing that geodesics in first-passage percolation have the same first order travel time as optimal paths in an embedded invasion cluster. (5) November 2st, 2016 2:10-3. Ananda Weerasinghe (Iowa State University) Title: Maximal Inequalities for the Ornstein-Uhlenbeck Process Abstract: In this work, Graverson and Peskir obtain Burkholder-Gundy type maximal inequalities for the one-dimensional O-U process. These inequalities are often quite useful in estimations related to stochastic differential equations. (5) November 9th, 2016 2:10-3. Ananda Weerasinghe (Iowa State University) Title: Maximal Inequalities for the Ornstein-Uhlenbeck Process Abstract: In this work, Graverson and Peskir obtain Burkholder-Gundy type maximal inequalities for the one-dimensional O-U process. These inequalities are often quite useful in estimations related to stochastic differential equations. (5) November 16th, 2016 2:10-3. Steven Noren (Iowa State University) Title: Favorite Sites of a Persistent Random Walk Abstract: A persistent random walk on the integers is a non-Markovian discrete-time process in which the walker is biased towards the direction it traveled most recently. In this joint work with Dr. Arka Ghosh and Dr. Alex Roitershtein, I will present the key ideas of the proof that the number of the most visited sites of the persistent random walk, for any magnitude of bias, exceeds 3 only finitely often with probability one, much like the case for the simple random walk. (5) November 30th, 2016 2:10-3. Krishna Athreya (ISU) Title:Title: N and S version of Scheffes Theorem, quantiles, L1minimzation etc Abstract: This talk will be pedagogical. It will be at the graduate student level. We will prove a n and s version of Scheffe's thm on probability densities and give some applications. We shall define quantiles, medians of arbitrary probability distributions on R and show that the median minimizes the mean deviation of a real valued random variable. We will also outline some open problems. (5) December 7th, 2016 2:10-3. Jinsu Kim (University of Wisconsin--Madison) Title: Ergodicity and Mixing Times of Stochastic Reaction Networks Abstract: Reaction networks are graphical configurations that can be used to describe many biological interaction networks. If the abundances of the constituent species of the system are low, we can model the system as a continuous time Markov jump process. In this talk, we will mainly focus on which conditions of the graph imply existence of a stationary distribution for the associated Markov process. I will also present results related to their mixing times, which gives the time required for the distribution of the Markov process to get close to the stationary distribution. (5) December 9th, 2016 3:10-4* (Talk in Carver 390). Iddo Ben-Ari (University of Connecticut) Title: The Bak-Sneppen Model of Biological Evolution and Related Models Abstract: The Bak-Sneppen model is a Markovian model for biological evolution that was introduced as an example for Self-Organized Criticality. In this model, a population of size N evolves according to the following rule. The population is arranged on a circle, or more generally a connected graph. Each individual is assigned a random fitness, uniform on [0,1], independent of the other fitness of the other individuals. At each unit of time, the least fit individual and its neighbors are removed from the population, and are replaced by new individuals. Despite being extremely simple, the model is known to be very challenging, and the evidence for Self-Organized Criticality provided by Bak and Sneppen was obtained through numerical simulations. I will review the main rigorous results on this model, mostly due to R. Meester and his coauthors, present some new results and open problems. I will then turn to a recent and more tractable variants of the model, in which on the one hand the spatial structure is relaxed, while on the other hand the population size is random. I will focus on the functional central limit for model, which has a somewhat unusual form.