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
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
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 $G_n$.