Probability Seminar List
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The CUNY Probability Seminar is typically held on Tuesdays at 4pm in the
CUNY Graduate Math Department. The exact dates, times and locations are mentioned below.
Seminar List for Spring 2003
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| Speaker | Haya Kaspi, Technion |
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| Title | Splitting/Coalescence Phenomena in Skew Brownian Motion |
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| Abstract | Skew Brownian motion is a process that satifies the stochastic differential
equation
where Bt is a given Brownian motion, b Î [-1,1] is a fixed constant
and Lxt is the symmetric local time of X at 0, i.e.
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Lxt = |
lim
e® 0
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\frac12e |
ó
õ
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t
0
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1(-e,e)(Xxs)ds . |
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The existence and uniqueness of strong solutions to the above equation,
starting at each real x, was proved by Harrison and Shepp. In the special
case
b = 1 the solution to the above equation is the reflected Brownian
motion. For any b Î (-1,1) the absolute value of X is a reflected
Brownian motion. The set of measure 0 on which the solution is not unique
depends on the starting point x, and it turns out that there is no
strong uniqueness for all starting points simultaneously.
In this talk we shall consider a stochastic flow in which individual particles
follow skew Brownian motions, with each one of these processes driven by
the same Brownian motion and starting at a different points in space and time.
Due to the lack of the simultaneous strong uniqueness for
the whole system of stochastic differential equations, the flow contains
lenses, i.e., pairs of skew Brownian motions which start at the same time
at the same point, bifurcate, and then coalesce in a finite time.
We shall discuss both qualitative and quantitative (distributional) results
on the geometry of the flow and lenses.
The talk is based on a joint work with Krzysztof Burdzy.
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| Speaker | Jay Rosen, CUNY |
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| Title | The fractal nature of late points for two dimensional random walk |
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| Abstract | [1]
() [2]
Let Tn(x) denote the time of first visit of a point x on the
lattice torus Zn2 = Z2/nZ2 by the simple random walk.
The size of the set of a,n-late points
Ln(a) = {x Î Zn2:Tn(x) ³ a\frac4p(nlogn)2}
is approximately n2(1-a), for a Î (0,1)
( Ln(a) is empty if a > 1 and n is large enough).
These sets have interesting clustering and fractal properties:
we show that for b Î (0,1) a disc of radius nb centered at
non-random x typically contains about n2 b(1-a/b2)
points from Ln(a) (and is empty if b < Ö{a}),
whereas choosing the center x of the disc uniformly
in Ln(a) boosts the typical number a,n-late points in it
to n2b(1-a). We also show that
the number of pairs of a,n-late points within distance nb
of each other is about nr(a,b) and provide an explicit
expression for the function r. In doing this, we correct the
predictions of Brummelhuis and Hilhorst (1991),
showing that the number of pairs
of late points does not correspond to the number of late points
in a typical disc centered at a late point. |
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| Speaker | Vladimir Dobric, Lehigh Uninversity |
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| Title | Gaussian-Markov processes and their natural wavelets representations |
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| Abstract | Every Gaussian-Markov process is characterized by its
canonically associated martingale process. A reproducing kernel
Hilbert space of the Gaussian-Markov process can be naturally
decomposed into an orthogonal sum of subspaces. The decomposition
takes place in the Hilbert space of square-integrable functions with
respect to the measure generated by the covariance function of its
associated martingale. The subspaces are reproducing kernel Hilbert
spaces corresponding to a decomposition of the process into a sum of
independent processes. The covariance functions of those processes
give rise to a wavelet bases for the reproducing Hilbert space of the
process. |
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| Speaker | Leif Jensen, Columbia University |
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| Title | Large Deviations for the Asymmetric Exclusion Process |
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| Abstract | In the simple exclusion process, particles perform random walks on a
lattice subject to a rule that only one particle may occupy a single site.
The law of large numbers for the process with asymmetric dynamics has
been known for some time. In the appropriate scaling limit the particle
density is the entropic solution to a partial differential equation in
conservation form. We establish a large deviations upper bound for this
process with a candidate rate function given by the positive part of the
Kruzkov entropy/entropy-flux functional for the proper macroscopic entropy
function. There are partial results for the lower bound that would be
complete if a certain PDE approximation result could be established.
Some partial results are also known for similar processes. The result
is proved by estimates on the relative entropy cost of adjusting the
dynamics of the process to make it exhibit deviant behavior. |
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| Speaker | Edgar Feldman, CUNY |
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| Title | Quantum Random Walks |
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| Speaker | William B. Johnson, Texas A&M University |
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| Title | Stochastic approximation properties in Banach spaces |
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| Abstract | In this talk I will report on some recent joint work with V. P. Fonf,
G. Pisier, and D. Preiss which will appear in Studia Math.
We show that a Banach space X has the stochastic
approximation property iff it has the stochasic basis
property, and these properties are equivalent to the
approximation property if X has non trivial type. This answers a
question Rosinski asked in 1980.
If for every Radon probability on X,
there is an operator from an
Lp space into X whose range has probability one, then X
is a quotient of an Lp space. This extends a 1979 theorem of
Sato's which dealt with the case p = 2.
In any infinite
dimensional Banach space
X there is a compact set
K so that for any Radon probability on X there is an
operator range of probability one that does not contain K. |
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| Speaker | Alexie Miasnokov, City College and CUNY Grad. Center |
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| Title | Probabilistic algorithms in group theory |
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| Abstract | I am going to discuss two recent developments in combinatorial group
theory:
generic properties of groups and probabilistic algorithms. |
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