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Series: Stochastics Seminar

The usual approach to KPZ is to study the scaling limit of particle
systems. In this work, we show that the partition function of directed
polymers (with a suitable boundary condition) converges, in a certain
regime, to the Cole-Hopf solution of the KPZ equation in equilibrium.
Coupled with some bounds on the fluctuations of directed polymers,
this approach allows us to recover the cube root fluctuation bounds for
KPZ in equilibrium. We also discuss some partial results for more
general initial conditions.

Series: Stochastics Seminar

We wish to understand ends of minimal surfaces contained in
certain subsets of R^3. In particular, after explaining how the
parabolicity and area growth of such minimal ends have been previously
studied using universal superharmonic functions, we describe an
alternative approach, yielding stronger results, based on studying
Brownian motion on the surface. It turns out that the basic results
also apply to a larger class of martingales than Brownian motion on a
minimal surface, which both sheds light on the underlying geometry and
potentially allows applications to other problems.

Series: Stochastics Seminar

We consider random walks on Z^d among nearest-neighbor random
conductances which are i.i.d., positive, bounded uniformly from above
but which can be arbitrarily close to zero. Our focus is on the
detailed properties of the paths of the random walk conditioned to
return back to the starting point after time 2n. We show that in the
situations when the heat kernel exhibits subdiffusive behavior ---
which is known to be possible in dimensions d \geq 4-- the walk gets
trapped for time of order n in a small spatial region. This proves that
the strategy used to infer subdiffusive lower bounds on the heat kernel
in earlier studies of this problem is in fact dominant. In addition, we
settle a conjecture on the maximal possible subdiffusive decay in four
dimensions and prove that anomalous decay is a tail and thus zero-one
event. Joint work with Marek Biskup, Alexander Vandenberg and Alexander
Rozinov.

Series: Stochastics Seminar

Let $\M$ be a smooth connected manifold endowed with a smooth measure
$\mu$ and a smooth locally subelliptic diffusion operator $L$ which is
symmetric with respect to $\mu$. We assume that $L$ satisfies a
generalized curvature dimension inequality as introduced by
Baudoin-Garofalo \cite{BG1}. Our goal is to discuss functional
inequalities for $\mu$ like the Poincar\'e inequality, the log-Sobolev
inequality or the Gaussian logarithmic isoperimetric inequality.

Series: Stochastics Seminar

L-moments are expectations of certain linear combinations of order
statistics. They form the basis of a general theory which covers the
summarization and description of theoretical probability distributions,
the summarization and description of observed data samples, estimation
of parameters and quantiles of probability distributions, and hypothesis
tests for probability distributions. L-moments are in analogous to the
conventional moments, but are more robust to outliers in the data and
enable more secure inferences to be made from small samples about an
underlying probability distribution. They can be used for estimation
of parametric distributions, and can sometimes yield more efficient
parameter estimates than the maximum-likelihood estimates. This talk
gives a general summary of L-moment theory and methods, describes some
applications ranging from environmental data analysis to financial risk
management, and indicates some recent developments on nonparametric
quantile estimation, "trimmed" L-moments for very heavy-tailed
distributions, and L-moments for multivariate distributions.

Series: Stochastics Seminar

In this talk we will describe the different behaviors of solutions of the random Schrödinger with long-range correlations. While in the case of arandom potential with rapidly decaying correlations nontrivial phenomenaappear on the same scale, different phenomena appear on different scalesfor a random potential with slowly decaying correlations nontrivial .

Series: Stochastics Seminar

We review various classifications of probability
distributions based on their tail heaviness. Using
a characterization of medium-tailed distributions
we propose a test for testing the null hypothesis
of medium-tail vs long- or short-tailed distributions.
Some operating characteristics of the proposed
test are discussed.

Series: Stochastics Seminar

The marginal distribution of identically distributed random variables
having a constant sum is called a completely mixable distribution. In
this talk, the concept, history and present research of the complete
mixability will be introduced. I will discuss its relevance to existing
problems in the Frechet class, i.e. problems with known marginal
distributions but unknown joint distribution and its applications in
quantitative risk management.

Series: Stochastics Seminar

Excursion sets of stationary random fields have attracted much attention in recent years.They have been applied to modeling complex geometrical structures in tomography, astro-physics and hydrodynamics. Given a random field and a specified level, it is natural to studygeometrical functionals of excursion sets considered in some bounded observation window.Main examples of such functionals are the volume, the surface area and the Euler charac-teristics. Starting from the classical Rice formula (1945), many results concerning calculationof moments of these geometrical functionals have been proven. There are much less resultsconcerning the asymptotic behavior (as the window size grows to infinity), as random variablesconsidered here depend non-smoothly on the realizations of the random field. In the talk wediscuss several recent achievements in this domain, concentrating on asymptotic normality andfunctional central limit theorems.

Series: Stochastics Seminar

We consider the problem of estimating the covariance matrix. Factormodels and random effect models have been shown to provide goodapproximations in modeling multivariate observations in many settings. These models motivate us to consider a general framework of covariancestructures, which contains sparse and low rank components. We propose aconvex optimization criterion, and the resulting estimator is shown torecover exactly the rank and support of the low rank and sparsecomponents respectively. The convergence rates are also presented. Tosolve the optimization problem, we propose an iterative algorithm basedon Nesterov's method, and it converges to the optimal with order 1/t2for any finite t iterations. Numerical performance is demonstratedusing simulated data and stock portfolio selection on S&P 100.(This is joint work with T. Tony Cai.)