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| Speaker |
Emmanuel Shertzer, Columbia University |
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| Title |
The Voter Model and the Potts Model in one dimension. |
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| Abstract |
The voter model can be seen as a simple model for describing the
propagation of opinions in a population where neighbors influence each
other. More precisely, every integer is assigned with an original
opinion at time t=0 and then updates its opinion by taking on the
opinion of one of its neighbors chosen uniformly at random with rate 1.
In the first part of the talk, I will show that such a model can easily
be described in terms of a system of coalescing random walks.
In the second part of the talk, I will introduce a variation of the
preceding model where the voters do not only change their mind under
the influence of their environment, but where they are also able to
come up with an opinion differing from their neighbors. This model is
closely related to a classical model in statistical physics called the
one dimensional stochastic Potts model. I will show that under the
appropriate scaling, this model converges to a continuum object which
can be constructed by a marking procedure of a family of coalescing
Brownian motions.
This is joint work with C. Newman and K. Ravishankar.
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| Speaker |
Anja Sturm, University of Delaware |
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| Title |
Coexistence and survival in some cancellative spin systems |
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| Abstract |
We consider variations of the usual voter model, which favor the type
that is locally less common.
These voter models are dual to systems of branching annihilating random
walks that preserve the parity
of the number of particles where we interpret sites occupied by a 1 as
a particle. Both sets of models into the category of cancellative spin
systems. We consider coexistence of types in the voter models
which is related to the survival of particles in the branching
annihilating random walk. We find conditions for the uniqueness of a
homogeneous coexisting invariant law as well as for convergence to this
law from homogeneous and coexisting initial laws. For a particular one
dimensional model we also show a complete convergence result for any
initial condition. This is based on comparison with oriented
percolation of the associated branching annihilating random walk.
This is joint work with Jan Swart (UTIA Prague). |
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|
| Speaker |
Dmitry Dolgopyat, University of Maryland |
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| Title |
Central Limit Theorem for Random walk in Markovian environment. |
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| Abstract |
We prove the central limit theorem for a random walk at Z^d
where the transition probabilities at different sites are governed
by independent finite state mixing Markov chains. This is a joint work
with Carlangelo Liverani. |
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|
| Speaker |
Janos Englander , UC Santa Barbara |
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| Title |
Strong Law of Large Numbers for Branching Diffusions |
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| Abstract | Let
X be the branching particle diffusion corresponding to a second order
semilinear elliptic operator on a Euclidean domain. Under appropriate
spectral theoretical assumptions on the operator, we prove that the
exponentially discounted random measures X(t) converge almost surely in
the vague topology as t tends to infinity. The exponential rate is the
generalized principal eigenvalue of the linear part of the operator,
which is assumed to be finite and positive.
This result was motivated by a cluster of articles due to Asmussen
and Hering dating from the mid-seventies as well as
the more recent work concerning analogous results for superdiffusions
by Englander, Turaev and Winter. We extend significantly the
results obtained by Asmussen and Hering in the seventies, and by Chen
and Shiozawa very recently. We also include some key examples of the
branching process literature. As far as the proofs are concerned, we
appeal to modern techniques concerning martingales and `spine'
decompositions or `immortal particle pictures'.
This is joint work with Andreas Kyprianou and Simon Harris (Bath, UK). |
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| Speaker |
Olympia Hadjiliadis, Brooklyn College, CUNY |
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| Title |
Formulas for Stopped Diffusion Processes with Stopping Times based on Drawdowns and Drawups |
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| Abstract | This
paper studies drawdown and drawup processes in a general diffusion
model. The main result is a formula for the joint distribution of the
running minimum and the running maximum of the process stopped at the
time of the first drop of size a. As a consequence, we obtain the
probabilities that a drawdown of size a precedes a drawup of size b and
vice versa. The results are applied to several examples of diffusion
processes, such as drifted Brownian motion, Ornstein-Uhlenbeck process,
and Cox-Ingersoll-Ross process. We also discuss applications of the
results to the problem of mis-identification of a two-sided change in
the drift of a diffusion process.
This is joint work with Libor Pospisil and Jan Vecer. |
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| Speaker |
Jose Blanchet , Columbia University |
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| Title |
Algorithms and Large Deviations |
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| Abstract |
This talk concentrates on the interplay between large
deviations theory and the design of efficient stochastic simulation
algorithms that are aimed at both estimating rare-event probabilities
and sampling stochastic processes conditional on a rare event. The
point is designing simulation estimators that can be easily
implemented and whose coefficient of variation remains uniformly
bounded as the event becomes rarer and rarer. Typically, a large
deviations result is helpful to guide the construction of the
estimator, but, as we shall see, the complexity analysis of the
algorithm often demands a refinement of the underlying large
deviations argument behind the rare-event probability of interest. In
this talk we illustrate the techniques both in light and heavy-tailed
stochastic processes. Applications to stochastic networks will be
given as motivation.
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|
| Speaker |
Victor de la Pena, Columbia University |
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| Title |
New Exponential Inequalities for Self-Normalized Martingales |
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| Abstract |
In this talk I will introduce a new class of exponential inequalities for self-normalized martingales.
I will show their usefulness by an application to hypothesis testing of the variance.
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|
| Speaker |
Souvik Ghosh, Columbia University |
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| Title |
LARGE DEVIATION PRINCIPLE FOR A CLASS OF LONG RANGE DEPENDENT INFINITELY DIVISIBLE PROCESS |
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| Abstract | We
make an attempt at understanding the effect of long range dependence on
the large deviation principle for the partial sums of an infinitely
divisible process. It has been observed in certain short memory
processes that the large deviation principle is very similar to that of
an i.i.d sequence. Whereas, if the process is long range dependent the
large deviations change dramatically. We want to see if such a
phenomenon holds for infinitely divisible processes.
We consider a stationary, mean zero infinitely divisible process (X n,n Î \mathbbZ) without a Gaussian component but with exponentially light tails. The process is characterized by its Lévy measure P on \mathbbR \mathbbZ
which is shift invariant. With the aim of modeling long range
dependence for such processes, we consider the situation where the Lévy
measure is the law of the paths of an irreducible null recurrent Markov
Chain with the marginals being the invariant measure p of the chain, i.e., for any n ³ 1 and A 0,¼,A n Î B(\mathbbR),
|
P |
æ
è
|
z Î \mathbbR \mathbbZ:(z 0,¼,z n) Î A 0×¼×A n |
ö
ø
|
= |
ó
õ
|
A 0
|
¼ |
ó
õ
|
A n
|
p(dz 0)P(z 0,dz 1)¼P(z n-1,dz n-1), |
|
where P(·,·) is the transition kernel of the
Markov chain. We study how the structure of this Markov chain affects
the large deviation principle for the partial sums of the process (X n).
References:
Alparslan, U.T., Samorodnitsky, G. (2007) Ruin probability with certain stationary stable claims generated by conservative flows ,
Advances in Applied Probability. 39, 360-384.
Ghosh, S. (2008)The effect of memory on large deviations of moving average processes and infinitely divisible processes , Thesis, Cornell University.
Mikosch, T., Samorodnitsky, G. (2000) The supremum of a negative drift random walk with dependent heavy-tailed steps ,
The Annals of Probability. 10, 1025-1064.
Rosinski, J., Samorodnitsky, G. (1996) Classes of mixing stable processes , Bernoulli. 2, 365-377.
|
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|
| Speaker |
Michael Kozdron, University of Regina |
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| Title |
TBA |
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|
| Speaker |
David Nualart, University of Kansas |
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| Title |
Central limit theorems for functionals of Gaussian processes. |
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| Abstract |
In this talk we first establish a central limit theorem for a
normalized sequence of random variables which are functionals of an
underlying Gaussian stochastic process, and they belong to a fixed
Wiener chaos. The convergence in law to the normal distribution is
equivalent to the convergence of the moments of order 4, and also to
the convergence of the square norm of the derivatives in the sense of
Malliavin calculus. In a second part of the talk we introduce the
fractional Brownian motion with Hurst parameter H, and discuss some of
its basic properties. We also present several asymptotic properties of
functionals of the fractional Brownian related to power variations and
to the Levy area.
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