STOCHASTIC ANALYSIS

Meets Wednesdays, 4 PM in Parker 224

 

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



Speaker: Dr. Erkan Nane

Title: Time fractional  stochastic partial differential equations: Intermittency

Abstract: Stochastic partial differential equations  are interesting to mathematicians because they give rise to an enormous number of challenging open problems. They  have  applications in various disciplines that range from applied mathematics, statistical mechanics, and theoretical physics, to theoretical neuroscience, theory of complex chemical reactions, including polymer science, fluid dynamics, and mathematical finance.

I will  talk about the right time-fractional SPDE, talk about its connections to a higher order parabolic type SPDEs. I will also discuss physical motivation, and mention some  recent results on time-fractional spdes including intermittency.

 

Speaker: Dr. Olav Kallenberg

Title: Boundedness and convergence of single and multiple stochastic integrals

Abstract: The most basic questions regarding stochastic integrals, both single and multiple, are whether they converge or diverge, and whether a sequence of such integrals tends in probability to zero. Using some general existence and comparison theorems for tangential processes, we can now answer such questions in surprising generality.



Speaker: Dr. Ming Liao

Title: Feller processes in Polish spaces

Abstract: Feller processes are a powerful tool in stochastic analysis. They are defined by a simple set of analytical conditions, and as consequences, possess useful properties such as right continuous paths with left limits, strong Markov properties and quasi-left continuity. In the classical literature, these processes are defined on locally compact and second countable spaces, but the theory can be developed on the more general Polish spaces (i.e., complete and separable metric spaces). I will describe this theory based on van Casteren's recent book Markov Processes, Feller Semigroups and Evolution Equations.





March 19,  2014

Speaker: Dr. Jerzy Szulga

Title: Paley-Marcinkiewicz-Zygmund property and its consequences

Abstract: In 1930’s Paley and Zygmund became interested in random Fourier series , where was a sequence of i.i.d. random variables, most notably Rademacher (random signs), or Steinhaus (uniform on the unit circle), or Gaussian. Such series can be viewed as random series with coefficients from a Banach space (of continuous functions, , Orlicz, etc.). The striking 1920’s Khinchin’s inequality for real Rademacher series has been extended to Banach space valued coefficients and the best constant has been found. In 1930s Marcinkiewicz and Zygmund analyzed a similar “reversal” of Jensen’s inequality for sums of independent random variables, and in 1960s Rudin investigated a similar property for portions of Walsh series (Walsh functions are characters of the dyadic group and can be seen as products of Rademacher functions). The seminar talk will describe the nexus of such “inverse” phenomena, e.g., classical inequalities and relations valid in inverse, contraction without convexity, hypercontraction (to be explained), etc., with potential application to statistics, harmonic analysis, random chaos, and all kinds of things stochastic.



Combined Colloquium and Stochastics Seminar

Wednesday, March 26, 2014 PLEASE NOTE DAY (Check e-mail for possible room change)

Speaker: Dr. Dongsheng Wu, University of Alabama Huntsville

Title: Regularity of Local Times of Gaussian Random Fields

Abstract available here

Faculty host: Erkan Nane

 

March 5, 2014


Speaker: Dr. Xiaoying (Maggie) Han

Title:  Stochastic Filtering and its Applications

Abstract:  Stochastic filtering plays an important role in signal processing, communications, finance, etc. Commonly used methods include Kalman filter methods and particle filter methods.  I will first provide the framework of stochastic filtering problems, including linear and nonlinear filtering.  Then I will introduce a numerical particle filtering scheme and an implicit filtering scheme that I've been working on lately.



NEW DAY THIS WEEK ONLY

THURSDAY, FEBRUARY 27, Parker Hall 224, 4:00

Speaker: Dr. Erkan Nane

Title:  Stochastic partial differential equations: Intermittency

Abstract: Stochastic partial differential equations  are interesting to mathematicians because they give rise to an enormous number of challenging open problems. They  have  applications in various disciplines that range from applied mathematics, statistical mechanics, and theoretical physics, to theoretical neuroscience, theory of complex chemical reactions, including polymer science, fluid dynamics, and mathematical finance.

I will  talk about intermittency, and intermittency fronts.

 

February 19, 2014

Speaker: Olav Kallenberg

Title: Some invariance properties involving stochastic integrals with respect to L'evy and related processes 

Abstract: L'evy processes are defined as processes with stationary, independent increments. In contrast to the special case of Brownian motion they may have jump discontinuities, which make the definition of stochastic integrals more subtle. Even more general (and more subtle) is the case of exchangeable processes, which require stochastic integration with respect to general semi-martingales. In this talk I will discuss some invariance properties of such integrals obtained at various points throughout my career.

February 12, 2014

Speaker: Dr. Ming Liao
 
Title:  Stochastic flows generated by stochastic differential equations

Abstract:  Just like an ordinary differential equation which generates a flow of diffeomorphisms, a stochastic differential equation generates a stochastic flow of diffeomorphisms on a compact manifold, which forms a random dynamical system.  I will first review some basic theory, including Lyapunov exponents, which are the nonrandom exponential rates at which the tangent vectors are stretched or compressed by the stochastic flow.  I will then mention some related results, including my own work in this field.

 

 

February 5, 2014

Speaker: Jerzy Szulga

Title: A probabiistics approach of Torence Tao to the famous Szemerédi’s Lemma from graph theory (based on T. Tao’s 2006 paper in Contribution to Discrete Mathematics, Vol. 1, No 1, 8-28)

Abstract: Endre  Szemerédi’s proved in 1975 a certain graph-theoretical result to show that every set of integers of positive density (*) contains arbitrarily long arithmetic progressions. His lemma (SL)  found numerous applications in number theory, computer sciences, combinatorics, etc. In particular, Ben Green and Torence Tao (Ann. Math. 2008) used  the SL to prove (*) for the set of primes. In the paper Tao gives the SL an analytic appearance and employs the language and properties of conditional expectations. This will be the topic of the talk.



January 29, 2014  Canceled snow.jpg



Speaker: Dr. Xiaoying (Maggie) Han
 
Title:  Stochastic Filtering and its Applications

Abstract:  Stochastic filtering plays an important role in signal processing, communications, finance, etc. Commonly used methods include Kalman filter methods and particle filter methods.  I will first provide the framework of stochastic filtering problems, including linear and nonlinear filtering.  Then I will introduce a numerical particle filtering scheme and an implicit filtering scheme that I've been working on lately.

 

January 22, 2014

Speaker: Olav Kallenberg

Topic: Stochastic differential equations: weak, strong, and functional solutions

Abstract: The theory of SDEs is a core topic of modern probability theory, taught routinely in graduate courses in the area. It has also been, along with other aspects of stochastic calculus, one of my basic research areas throughout my career. In this talk, my aims are 1) to outline the basic theory of existence and uniqueness, 2) to comment on the history of the subject, 3) to explain some basic connections to PDEs, and 4) to describe my own result on functional solutions, extending the celebrated work of Yamada and Watanabe.


This talk is also part of my efforts to reach out to other groups in the department interested in SDEs. 
 

 
 
January 15, 2014 at 4 pm PLEASE NOTE TIME CHANGE

Speaker: Dr. Erkan Nane

Title:  Stochastic partial differential equations: Intermittency

Abstract: Stochastic partial differential equations are interesting to mathematicians because they give rise to an enormous number of challenging open problems. They have  applications in various disciplines that range from applied mathematics, statistical mechanics, and theoretical physics to theoretical neuroscience, theory of complex chemical reactions, including polymer science, fluid dynamics, and mathematical finance.

I will give an outline of the existence of solutions to nonlinear SPDEs and talk about intermittency.



December 4, 2013 Last Seminar of Semester


Speaker: Olav Kallenberg

Title: Some weak and strong ergodic theorems for random particle systems

Abstract: Ergodic theorems play a fundamental role in the study of stochastic processes and their applications, especially to statistical mechanics. After reviewing some basic classical results, I will consider smoothing limits of random measures, which in turn can be used to derive Poisson limit theorems for infinite particle systems. Much of this material is previously known, but some of the central results I obtained myself during the past Thanksgiving holidays. (The breaks and holidays give the best opportunity for concentrated efforts on hard problems.) This will all be included as part of my new book on Random Measures, currently under preparation. 

 

November 20, 2013

Speaker: Dr. Jerzy Szulga

Topic: I will present the key points of the paper Lévy Flights in Evolutionary Ecology, by W.A. Woyczyński et al.

In October 22 ceremony, held in Paris, my mentor and PhD supervisor, also a coauthor of our several joint papers, Wojbor Woyczyński of Case Western Reserve University, was awarded, together with a team of French mathematicians and biologists, the 2013 Prix La Recherche in the Field of Mathematics., for the aforementioned paper, published  in  Journal of Mathematical Biology (2012) 65: 677-707

(The jury, chaired this year by  Albert Fert, the 2007 Nobel Laureate in Physics,  awards  one prize  annually in each of the 12 areas of science and technology.)

The considered  evolution process  νt meanders in the metric space of measures on a closed Euclidean domain X.  Each vector x in X represents a phenotypic “trait”.  Individual are subject to a stochastic birth and death process but upon birth a random mutation x → x+Z may occur. This mechanism turns νt into a time homogeneous Markov process.

While at the beginning νt is just a counting measure, i.e., the straight sum of Dirac’s deltas, in time it becomes convoluted. For the long time behavior it requires a renormalization with deterministic or random parameters.

There are three main issues to present.

First is the rigorous construction of the evolution process to place it in the fruitful mathematical framework.

The second is the description of its characteristics, mainly with the help of SDE’s (stochastic differential equations). The SDE’s involve Lévy processes (a.k.a. “flights”) but of pure jump type, in contrast to the well known SDE’s based on the diffusion  Lévy process such as the Brownian Motion.  So, the classical Laplacian, driving these SDE’s, gives room to fractional Laplacians.

The third issue is the limit behavior, in spirit similar to approximation (hence simulation) of Brownian Motion by Random Walk.

Although the contents of the paper (and of supporting materials) is highly technical, even cumbersome, I will try to make it accessible to general mathematical public.

 

November 13, 2013

Speaker: Dr. Ming Liao

Title:  Levy processes and Fourier analysis on compact Lie groups

Abstract:  A process with independent and stationary increments is called a Levy process.  A Brownian motion is a continuous Levy process, but a general Levy process has jumps, thus provides a more general model in applications.  Levy processes may be defined in groups because increments may be defined in terms of the group structure.  For Levy processes in compact Lie groups, the Fourier analysis based on Peter-Weyl Theorem provides a convenient tool for study.  When the Levy process has an L^2 distribution density, it may be expanded into a Fourier series, and this allows  us to determine how fast the Levy process converges to the uniform distribution (that is, the normalized Haar measure).  We may also obtain useful conditions under which the Levy process has an L^2, or even smooth, density.

 

October 30, 2013

Speaker: Dr. Erkan Nane

Title: Stochastic partial differential equations

Abstract: Stochastic partial differential equations have  applications in various disciplines that range from applied mathematics, statistical mechanics, and theoretical physics, to theoretical neuroscience, theory of complex chemical reactions, including polymer science, fluid dynamics, and mathematical finance.

Additionally, I will give a short introduction to the study of SPDEs and state some recent surprising results for time fractional SPDEs.

 

October 23, 2013

Speaker: Dr. Olav Kallenberg

Title: Dobrushin's theorem and beyond

Abstract: Around 1956 the applied mathematician Dobrushin, famous for his work in statistical mechanics, proved that if you take a stationary set of points (particles) in space and scramble their positions at random (e.g. by letting them perform independent Brownian motions), you end up in the limit with a Poisson process, the most random of all point configurations. (Actually his proof was flawed, and it took more than a decade before a rigorous proof was given.) I will give a modern version of the theorem (that I proved last night), indicate how it leads to some interesting problems in real analysis, and discuss some extensions and connections to certain challenging problems in stochastic geometry.

 

October 2, 2013

Speaker: Dr. Olav Kallenberg

Title: History of Brownian motion + proof of 3rd arcsine law
 
Abstract: We often hear or read comments about the history of Brownian motion. However, the speakers or writers usually get things totally wrong. My main purpose in this talk is to set the record straight. The errors usually arise from a confusion between the physical phenomenon and the mathematical object of BM. For the former, the main contributors are van Leevenhoek, Brown, Einstein, Langevin, and Ornstein-Uhlenbeck. For the latter, the main contributors are Bachelier, Wiener, L'evy, Doeblin, It^o, Kakutani, Donsker, Skorohod, Strassen, ... Historical comments were made about all of those. --- A second purpose of my talk was to outline an elementary proof of L'evy's third arcsine law for BM, usually regarded as a deep result. 


August 28, 2013
 
Speaker: Dr. Olav Kallenberg
Title: Stationary and invariant densities and disintegration kernels

Abstract: Let $\xi$ and $\eta$ be jointly stationary random measures on a common space $S$, subject to the measurable action of a group $G$. If $\xi\ll\eta$ a.s., we would expect the existence of a stationary, product-measurable density process (RN-derivative) $X$ on $S$, so that $\xi=X\cdot\eta$ a.s. This holds when $S$ is Borel and $G$ is a locally compact, second countable topological group. The proof of this fact is surprisingly hard (took me the whole summer) and requires ideas from real analysis, differential geometry, topological groups, and probability theory. Key ingredients include the facts that 1) the centered balls in a Riemannian manifold $M$ form a differentiation basis for any locally finite measure, and 2) any topological group $G$ as stated contains an open subgroup $G_0$ (hence with discrete coset space $G/G_0$), which is isomorphic to the projective limit of a sequence of Lie groups $G_n$.

Last updated: 04/14/2014