STOCHASTIC ANALYSIS

Meets Wednesdays, 2 PM in Parker 236 (PLEASE NOTE NEW ROOM)

 

Regular speakers and their current areas of interest:


Olav Kallenberg: stochastic analysis and dynamical systems, with applications to statistical mechanics and biological evolution

Ming Liao: stochastic processes in Lie groups

Erkan Nane: fractional diffusions and iterated processes

Jurek Szulga


2014 - 2015

 

 

December 3

Speaker: Dr. Ming Liao

Title:  Population genetics and coalescent process, part 3

Abstract: In the previous discussion, we have mainly considered a haploid population.  Roughly speaking, an individual in a haploid population has only one copy of a gene, inherited from its single parent.  It is then relatively easy to trace the ancestors of a sample of genes taken at the present time, and it has been shown that, as the population size tends to infinity, the ancestral process converges in distribution to a continuous time Markov chain, called Kingman's coalescent process.  For a diploid population, each individual has two copies of a gene, one from each of its two parents, so the ancestral process of a gene sample has a quite complicated structure, but by a simple convergence result of Markov chains, it can be shown that the ancestral process still converges in distribution to Kingman's coalescent.

November 19

Speaker: Olav Kallenberg

Title: Inner and outer conditioning in point processes and particle systems

Abstract: A point process $\xi$ on a suitable space $S$ may be regarded as the model of a particle system in space. A basic problem is to study the conditional distribution of $\xi$ in a bounded set $B\subset S$, given the configuration of $\xi$ outside of $B$. In statistical mechanics, such studies go back to the fundamental work of J.W.\ Gibbs (1902), leading up to the theory of Gibbs measures. The merger of the physical and mathematical theories was initiated in the 1970's and 80's. In this talk I shall outline the elements of a general theory of conditioning, involving the notions of Palm and Gibbs kernels, obtained by dual disintegrations of a compound Campbell measure.

November 12

Speaker: Professor Jerzy Szulga

Title: On the semicentennial of Bell’s Inequality

Abstract: In November 1964 John Stewart Bell published a short but influential article “On the Einstein-Podolsky-Rosen Paradox.” It contained a simple probabilistic inequality, showing that the classical probability approach to quantum mechanics fails. I will talk about this and related issues in the context of quantum – or noncommutative probability, into which the classical probability can be embedded in a suitable sense.

 

November 5

Speaker: Dr. Erkan Nane

Title: A Strong Law of Large Numbers with Applications to Self-Similar Stable Processes:

Abstract: In this talk I will prove a strong law of large numbers (SLLN) under general moment condition for sequence of random variables. We  extend the maximal inequalities of Chobanyan, Levental and Salehi (2005), and obtain a general version of SLLN. This SLLN then can be applied conveniently to a wide class of self-similar processes with stationary increments including stable processes. This is a recent  joint work with A. Zeleke and Y. Xiao , Michigan State University

 

October 29

Speaker: Dr. Ming Liao

Title: Population genetics and the coalescent process, part 2

Abstract: After a review of Wright-Fisher model and Kingman's coalescent process, two extensions will be considered.  The first is the inclusion of neutral mutations, and the second is a structured coalescent process taking care of migration between subpopulations.


October 22

Speaker: Olav Kallenberg

Title: Random fractals in stochastic evolution processes

Abstract: We show how random fractals arise naturally in the study of stochastic evolution processes. Informally, a fractal is characterized by the properties of fractional dimension and scaling invariance (even known as self-similarity). A biological population consists of individuals that move around in space, multiply, and eventually die, all inherently random phenomena. In the diffusion limit one gets a measure-valued process, whose associated support process exhibits some basic fractal properties. Though the process itself may be thought of as a randomly evolving diffuse cloud, its associated ancestral process forms a discrete skeleton of coalescing branches.

October 15

Speaker: Dr. Erkan Nane

Title: Correlation structure of time-changed Pearson diffusions

Abstract: The stochastic solution to  diffusion equations with polynomial coefficients is called a Pearson diffusion. If the  time derivative is replaced by a distributed fractional derivative, the stochastic solution is called a fractional Pearson diffusion. We  develop a formula  for the covariance function of a fractional Pearson diffusion in steady state, in terms of generalized Mittag-Leffler functions. That formula shows that fractional Pearson diffusions are long-range dependent, with a correlation that falls off like a power law, whose exponent equals the smallest order of the distributed fractional derivative. This is a recent Joint work with Jebessa Mijena.

 

October 8

Speaker: Dr. Ming Liao

Title:  Markov chains, population genetic models and coalescent process

Abstract:  I will start with a brief review of Markov chains, discrete and continuous times, as covered in our Applied Stochastic Processes course (math7820), and then will mention the Wright-Fisher model both in forward and backward times. The backward model traces the ancestry of a sample in present, and in the limit leads to Kingman's coalescent process. 

October 1

Speaker: Dr. Olav Kallenberg

Title: Some highlights of random measure theory

Abstract: I am currently writing a book on Random Measures, which is due to appear within the next couple of years. In this talk and its possible sequels, I am planning to give an informal introduction to some of the main ideas and results. Though this is one of the fundamental theories underlying the modern theory of stochastic processes, much of this material is relatively unknown, and many results are new. 

 

September 24

Speaker: Dr. Jerzy Szulga

Title: “Quantum probability and Lorentz transformation from an elementary point of view. Part II”

Abstract: Previously, a background was presented (quantum probability, Maxwell equations). Three topics will be addressed (almost no higher mathematics, i.e., beyond undergraduate algebra and calculus):

1.Discontinuity of the eigensystem of a Maxwell matrix.

2. Surjectivity of the exponential map, Maxwell -> Lorentz.

3. Pauli coding, sliders and jaw operators.

 

September 17

Speaker: Dr. Jerzy Szulga

Title: Quantum probability and Lorentz transformation from an elementary point of view

Abstract: I will refresh the framework of quantum probability, which essentially adapts the theory of subspaces of the space of linear continuous operators on a Hilbert space.  Then I will discuss some aspects of the Lorentz group and show how to derive several of its properties. These properties are well known but usually they  are presented within a quite advanced setup of differential geometry and tensor theory. In fact, at least some of these results can be obtained just by using undergraduate  linear algebra and calculus, and not even tediously.

 

 

September 10 

Speaker: Dr. Erkan Nane

Topic:  Intermittence and  time fractional stochastic partial differential equations.

Abstract: In this talk, I consider time fractional stochastic heat type equations. The time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. In this talk I discuss: (i) existence and uniqueness of solutions and existence of a continuous version of the solution; (ii) absolute moments of the solutions of this equation grow exponentially; and (iii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time.  These results extend the results of Mohammud Foondun and Davar Khoshnevisan, (Intermittence and nonlinear parabolic stochastic partial differential equations, Electron. J. Probab. 14 (2009), no. 21, 548--568) and Daniel Conus and Khoshnevisan (On the existence and position of the farthest peaks of a family of stochastic heat and wave equations, Probab. Theory Related Fields 152 (2012), no. 3-4, 681--701)  on the parabolic stochastic heat equations. --- This is a recent joint work with Jebessa B Mijena.


September 3 

Speaker: Dr. Ming Liao

Title:  Independent increments and semimartingale property on Lie groups

Abstract:  On a Euclidean space, it is well known that a process with independent increments is a semimartingale if and only if its (deterministic) drift has a finite variation.  The same holds on a Lie group.  Various representations of a process with independent increments in a Euclidean space are not affected by whether it is a semimartingale, but the proofs of these formulae may be easier in the semimartingale case.  However, such a process in a Lie group has a simpler and more direct martingale representation when it is a semimartingale.


August 27

Speaker: Olav Kallenberg

Topic: Tangential existence and comparison

Abstract: By a fundamental theorem of Jacod, a semimartingale X has independent increments iff its local characteristics are a.s. nonrandom. This seems to suggest that we may reduce the study of X to the independence case by a simple conditioning. Though this doesn't work in general, we can always construct an approximating tangential process with conditionallly independent increments. In this talk, I shall try to explain and elaborate on these ideas.

 

Last updated: 12/01/2014