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Series: Stochastics Seminar

Recall that the notion of generalized function is introduced for the functions that are not defined point-wise, and is given as a linearfunctional over test functions. The same idea applies to random fields.In this talk, we study the long term asymptotics for the quenchedexponential moment of V(B(s)) where B(s) is d-dimensional Brownian motion,V(.) is a generalized Gaussian field. We will discuss the solution to anopen problem posed by Carmona and Molchanov with an answer different fromwhat was conjectured; the quenched laws for Brownian motions inNewtonian-type potentials, and in the potentials driven by white noise orby fractional white noise.

Series: Stochastics Seminar

1-bit compressed sensing combines the dimension reduction
of compressed sensing with extreme quantization -- only the sign of
each linear measurement is retained. We discuss recent
convex-programming approaches with strong theoretical guarantees. We
also discuss connections to related statistical models such as sparse
logistic regression.
Behind these problems lies a geometric question about random
hyperplane tessellations. Picture a subset K of the unit sphere, as
in the continents on the planet earth. Now slice the sphere in half
with a hyperplane, and then slice it several times more, thus cutting
the set K into a number of sections. How many random hyperplanes are
needed to ensure that all sections have small diameter? How is the
geodesic distance between two points in K related to the number of
hyperplanes separating them? We show that a single geometric
parameter, the mean width of K, governs the answers to these
questions.

Series: Stochastics Seminar

We consider stochastic dynamical systems defined on networks that exhibit the phenomenon of collective metastability---by this we mean network dynamics where none of the individual nodes' dynamics are metastable, but the configuration is metastable in its collective behavior. We will concentrate on the case of SDE with small white noise for concreteness. We also present some specific results relating to stochastic perturbations of the Kuramoto system of coupled nonlinear oscillators. Along the way, we show that there is a non-standard spectral problem that appears in all of these calculations, and that the important features of this spectral problem is related to a certain homology group.

Series: Stochastics Seminar

I will talk about two model problem concerning a diffusion with a cellular drift (a.k.a array of opposing vortices). The first concerns the expected exit time from a domain as both the flow amplitude $A$ (or more precisely the Peclet number) goes to infinity, AND the cell size (or vortex seperation) $\epsilon$ approaches $0$ simultaneously. When one of the parameters is fixed, the problem has been extensively studied and the limiting behaviour is that of an effective "homogenized" or "averaged" problem. When both vary simultaneously one sees an interesting transition at $A \approx \eps^{-4}$. While the behaviour in the averaged regime ($A \gg \eps^{-4}$) is well understood, the behaviour in the homogenized regime ($A \ll \eps^{-4}$) is poorly understood, and the critical transition regime is not understood at all. The second problem concerns an anomalous diffusive behaviour observed in "intermediate" time scales. It is well known that a passive tracer diffusing in the presence of a strong cellular flows "homogenizes" and behaves like an effective Brownian motion on large time scales. On intermediate time scales, however, an anomalous diffusive behaviour was numerically observed recently. I will show a few preliminary rigorous results indicating that the stable "anomalous" behaviour at intermediate time scales is better modelled through Levy flights, and show how this can be used to recover the homogenized Brownian behaviour on long time scales.

Series: Stochastics Seminar

I will review recent progress concerning nonparametric estimation of log-concave densities and related families in $R^1$ and $R^d$. In the case of $R^1$, I will present limit theory for the estimators at fixed points at which the population density has a non-zero second derivative and for the resulting natural mode estimator under a corresponding hypothesis. In the case of $R^d$ with $d\ge 2$ will briefly discuss some recent progress and sketch a variety of open problems.

Series: Stochastics Seminar

In this talk I will describe a theory of matrix completion for the extreme
case of noisy 1-bit observations. In this setting, instead of observing a
subset of the real-valued entries of a matrix M, we obtain a small number
of binary (1-bit) measurements generated according to a probability
distribution determined by the real-valued entries of M. The central
question I will address is whether or not it is possible to obtain an
accurate estimate of M from this data. In general this would seem
impossible, but I will show that the maximum likelihood estimate under a
suitable constraint returns an accurate estimate of M when $\|M\|_{\infty}
\le \alpha$ and $\rank(M) \le r$. If the log-likelihood is a concave
function (e.g., the logistic or probit observation models), then we can
obtain this maximum likelihood estimate by optimizing a convex program. I
will also provide lower bounds showing that this estimate is near-optimal
and illustrate the potential of this method with some preliminary numerical
simulations.

Series: Stochastics Seminar

The periodic generalized Korteweg-de Vries equation (gKdV) is a canonical dispersive partial differential equation with numerous applications in physics and engineering. In this talk we present invariance of the Gibbs measure under the flow of the gauge transformed periodic quartic gKdV. The proof relies on probabilistic arguments which exhibit nonlinear smoothing when the initial data are randomized. As a corollary we obtain almost sure global well-posedness for the (ungauged) quartic gKdV at regularities where this PDE is deterministically ill-posed.

Series: Stochastics Seminar

In small dimension a random geometric graph behaves
very differently from a standard Erdös-Rényi random graph. On the other
hand when the dimension tends to infinity (with the number of vertices being
fixed) both models coincides. In this talk we study the behavior of the clique
number of random geometric graphs when the dimension grows with the
number of vertices.

Series: Stochastics Seminar

Gaussian fields with logarithmically decaying correlations, such as branching Brownian motion and the 2D Gaussian free field, are conjectured to form a new universality class of extreme value statistics (notably in the work of Carpentier & Ledoussal and Fyodorov & Bouchaud). This class is the borderline case between the class of IID random variables, and models where correlations start to affect the statistics. In this talk, I will report on the recent rigorous progress in describing the new features of this class. In particular, I will describe the emergence of Poisson-Dirichlet statistics. This is joint work with Olivier Zindy.

Series: Stochastics Seminar

There is a long history on the study of zeros of random polynomials whose coefficients are independent, identically distributed, non-degenerate random variables. We will first provide an overview on zeros of random functions and then show exact and/or asymptotic bounds on probabilities that all zeros of a random polynomial are real under various distributions. The talk is accessible to undergraduate and graduate students in any areas of mathematics.