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Probability seminar
- Title:
PROPERTIES OF SUPERPROCESSES
- Speaker:
Julia Ruscher (Microsoft Research)
- Time: 2:30 p.m., Monday, November 23, 2009
- Room: MEB 235
- Abstract:
A large class of random spatial processes involves populations that undergo reproduction (birth and death happen randomly according to a certain
mechanism) in addition to random spatial motion (e.g. Brownian motion). Superprocesses are exotic Markov processes in that sense that they are the scaling limits of many such processes. We will introduce classes of Superprocesses and look at their relationship and properties, in particular the Hausdorff dimension of the support of Superprocesses.
- Title:
DIRICHLET FORMS AND REFLECTING BROWNIAN MOTION
- Speaker:
Mauricio Duarte (University of Washington)
- Time: 2:30 p.m., Monday, November 16, 2009
- Room: MEB 235
- Abstract:
The theory of Dirichlet forms is yet another example of the rich interplay between Analysis and Probability. The probabilistic part, initiated by the fundamental work of Fukushima in the early 70's, connects symmetric Markov processes and Dirichlet forms in a very powerful way. More recently, the connection has been extended to non-symmetric forms. This extension makes possible the characterization of all forms that are related to `nice' Markov processes, under the notion of quasi-regularity.
In this talk, we will show how to construct a Markov process out of a Dirichlet form, and as an application, we will show a construction of Brownian motion with oblique reflection in a smooth domain.
- Title:
STABLE-LIKE PROCESSES
- Speaker:
Richard Bass (University of Connecticut)
- Time: 2:30 p.m., Monday, November 9, 2009
- Room: MEB 235
- Abstract:
The class of stable-like processes is a subset of
the class of multidimensional jump processes. They stand in
the same relationship to stable processes as multidimensional
diffusions do to Brownian motion. I'll describe several models
of stable-like processes, and then talk about relatively recent
results, such as uniqueness of martingale problems and
Harnack inequalities. Then I'll talk about very recent results on
the regularity of potentials of stable-like processes.
- Title:
CLASSICAL REPRESENTATIONS FOR THE QUANTUM ISING MODEL
- Speaker:
Geoffrey Grimmett (Cambridge University and Microsoft)
- Time: 2:30 p.m., Monday, November 2, 2009
- Room: MEB 235
- Abstract:
The quantum Ising model in $d$ dimensions may be mapped to a $(d+1)$-dimensional `continuous' Ising model of classical type. This may be solved using an approach developed by Aizenman and others under the name `random-current representation'. We prove the sharpness of the phase transition, and establish two inequalities for critical exponents. The value of the ground-state critical point may be calculated rigorously in one dimension, and the corresponding transition is continuous. (Joint work with Jakob Bj\"ornberg.)
- Title:
INTERSECTION EXPONENTS
FOR BIASED RANDOM WALKS ON CYLINDERS
- Speaker:
Brigitta Vermesi (IPAM)
- Time: 2:30 p.m., Monday, October 19, 2009
- Room: MEB 235
- Abstract:
In the past decade, there have been significant advances in the study of 2-dimensional critical systems in statistical physics, in particular due to the introduction of Schramm Loewner Evolution (SLE). For example, some critical exponents for planar Brownian motion have been computed exactly using SLE. But what can we say about the same exponents in the case of 3-dimensional Brownian motion? We approach the question by looking at a simpler model: biased random walks on d-dimensional cylinders. In this talk, I will describe the random walk problem and explain how it relates to the 3-dimensional Brownian motion case. This leads to a conjecture about exponents for Brownian motion.
- Title:
MIXING IN TIME AND SPACE OF GIBBS MEASURES
- Speaker:
Allan Sly (Microsoft Research)
- Time: 2:30 p.m., Monday, October 12, 2009
- Room: MEB 235
- Abstract:
The mixing time of the Glauber dynamics is often closely related to the
spatial mixing properties of the measure such as uniqueness and the
reconstruction problem. Such questions are of interest in probability,
statistical physics and theoretical computer science. I will discuss some
recent progress in understanding the mixing time of the Glauber dynamics
for the Ising model and for random colorings of random graphs.
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