Fall 2015

The transition semigroup $T_t$ of the Ornstein-Uhlenbeck process on $\mathbb{R}^n$ has a famous regularizing property called hypercontractivity: if $\mu$ is standard Gaussian measure, $f \in L^p(\mu)$, and $q > p$, then for large enough $t$, we have $T_t f \in L^q(\mu)$. This property is known to be closely related to the logarithmic Sobolev inequality. Strong hypercontractivity is a phenomenon that arises if we work in the complex world: if you replace $\mathbb{R}^n$ by $\mathbb{C}^n$ and consider $f$ which is holomorphic, then $T_t f \in L^q(\mu)$ for smaller $t$. In other words, in this setting the regularizing happens ``faster''. In this talk, I will explain the relationship between these concepts, and then discuss recent work in which we carry it over to a setting where $\mathbb{C}^n$ is replaced by a complex nilpotent Lie group, and Gaussian measure is replaced by a so-called hypoelliptic heat kernel. (You do not need to know anything about Lie groups for this talk.) This is joint work with Bruce Driver, Leonard Gross, and Laurent Saloff-Coste.

On a $d$-ary tree place some number (random or otherwise) of sleeping frogs at each site, as well as one awake frog at the root. Awake frogs perform simple random walk and wake any "sleepers" they encounter. A longstanding open problem: Does every frog wake up? It turns out this depends on $d$ and the amount of frogs. The proof uses two different recursions and two different versions of stochastic domination. Joint with Christopher Hoffman and Tobias Johnson.

We consider a one dimensional random walk in a random environment (RWRE) with a positive speed $\lim_{n\to\infty}\frac{X_n}{n}=v_\alpha>0$. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities $P_\omega(X_n < xn)$ with $x \in (0,v_\alpha)$ decay approximately like $\exp\{-n^{1-1/s}\}$ for a deterministic $s > 1$. More precisely, they showed that $n^{-\gamma} \log P_\omega( X_n < x n)$ converges to $0$ or $-\infty$ depending on whether $\gamma > 1-1/s$ or $\gamma < 1-1/s$. We improve on this by showing that $n^{-1+1/s} \log P_\omega( X_n < x n)$ oscillates between $0$ and $-\infty$, almost surely. This had previously been shown by Gantert only in a very special case of random environments.

In this talk we will discuss the small time behavior of transition densities of diffusion processes that are of weak Hörmander’s type. In particular, we work with nilpotent diffusion processes and develop a large deviation principle according to its graded geometric structure. This is a joint work with G. Ben Arous.

I will present an overview of the stochastic homogenization of parabolic equations in spatio-temporal media. The topic can be interpreted as a continuous version of a random walk in a time-dependent random environment. In particular, I will discuss quenched error estimates which decay at an algebraic rate. This talk is based on joint work with Charles Smart.

We focus in this talk on statistical problems for noisy differential equations driven by an additive fractional Brownian motion, either in a 1-d or general d-dimensional context. I will first review the standard procedures in order to estimate the Hurst parameter, the diffusion coefficient and finally the drift coefficient of the equation. Then I will give an account on two new results in this direction:

1. A new class of estimators in ergodic situations, when the dependence of the drift with respect to the parameter in not linear.
2. The LAN property for ergodic equations, which yields a lower bound for the convergence of estimators.

These results are based on some ongoing works with Eulalia Nualart (Barcelona) and Fabien Panloup (Toulouse).

I will start with a brief introduction on what the Sato Grassmanian is, and then give a probabilistic description using stochastic areas.

Probabilists and theoretical statisticians are generally experts in using the Fourier, Laplace, and Mellin transforms for exact distributional calculations, and for establishing weak convergence. The Stieltjes transform rose in popularity after its successful applications in RMT; a striking case is establishing the WSL (Wigner's semi-circle law) for various random matrix ensembles. These applications often involve dealing with the Stieltjes transform asymptotics only in the rapidly decaying case, e.g., the Gaussian.

The speaker will show the classic asymptotic expansions in the rapidly decaying case, and move relatively quickly on to the case of algebraic decay. He will take the Cauchy case as a testing example and derive the expansions. He will then show general asymptotic expansions for essentially the general algebraic decay case.

He will then return to the rapidly decaying case and complete the circle by sketching the classic Stieltjes transform proof of the WSL for one ensemble.

He will likely indicate the problem of generalized Stieltjes transforms(GST), for the interest quantum physicists have in the GST.

We will look at lattice paths coming from dual pivot quicksort with the motivation to analyze the optimal running time of this algorithms. The main property is the average number of zeros in such these paths. We will see an exact and an asymptotic result of this expectation along with a brief introduction to the methods and tools used.

The first aim of this to talk is to present an original methodology for developing the spectral representation of a class of non-self-adjoint (NSA) invariant semigroups. This class is defined in terms of self-similar semigroups on the positive real line and we name it the class of generalized Laguerre semigroups. Our approach is based on an in-depth analysis of an intertwining relationship that we establish between this class and the classical Laguerre semigroup which is self-adjoint. We proceed by discussing substantial difficulties that one may face when studying the spectral representation of NSA operators.

Finally, we also show that the spectral representation enables us to get precise information regarding the speed of convergence towards stationarity. In particular, we observe in some cases the hypocoercivity phenomena which, in our context, can be interpreted in terms of the spectral norms.

This talk is based on a joint work with M. Savov (Bulgarian Academy of Sciences).

Contraction method has been used in probability to show the convergence of a sequence(suitably normalized) of random variables satisfying a recursion. Examples are the number of comparisons needed by the Quicksort algorithm to sort a list of n numbers, the size of tries, etc. It can also be used to establish convergence of important characteristics of random networks. In particular, we studied the degree profile of random hierarchical lattice networks. At every step, each edge is either serialized (with probability p) or parallelized (with probability 1-p). We establish an asymptotic Gaussian law for the number of nodes of outdegree 1, and show how to extend the derivations to encompass asymptotic limit laws for higher outdegrees. The recursive equations which we get involves coefficients and toll terms depending on the recursion variable and thus are not in the standard form of the contraction method. Yet, an adaptation of the contraction method goes through, showing that the method has promise for a wider range of random structures and algorithms.

We constructed a white noise theory for the Canonical Lévy Process by Solé, Utzet, and Vives. The construction is based on the alternative construction of the chaos expansion of square integrable random variable and its white noise characterization. We showed a Clark-Ocone theorem in $L^2(P)$ and under the change of measure Canonical Lévy process in the generalized distribution spaces. We shall give an example using a Donsker Delta approach which is employed on a binary option to solve the mean-variance hedging problem. This research is a joint work with Dr. Frederi Viens.

We use a method of rotations to study the $L^p$ boundedness, $1 < p < \infty$, of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetric $\alpha$-stable processes, $0 < \alpha < 2$, and other closely related Fourier multipliers which have potential applications to the study of the $L^p$ boundedness of the Beurling-Ahlfors transform.

A random $n$-permutation may be generated by sequentially removing random cards $C_1,...,C_n$ from an $n$ card deck $D = \{1,...,n\}$. The permutation $\sigma$ is simply the sequence of cards in the order they are removed. This permutation is itself uniformly random, as long as each random card $C_t$ is drawn uniformly from the remaining set at time $t$. We consider a variant of this simple procedure in which one is given a choice between $k$ random cards from the remaining set at each step, and selects the lowest numbered of these for removal. This induces a bias towards selecting lower numbered of the remaining cards at each step, and therefore leads to a final permutation which is more ``ordered'' than in the uniform case (i.e. closer to the identity permutation $id = (1,2,3,...,n)$).

We quantify this effect in terms of two natural measures of order: The number of inversions $I$ and the length of the longest increasing subsequence $L$. For inversions, we establish a weak law of large numbers and central limit theorem, both for fixed and growing $k$. For the longest increasing subsequence, we establish the rate of scaling, in general, and existence of a weak law in the case of growing $k$. We also show that the minimum strategy, of selecting the minimum of the $k$ given choices at each step, is optimal for minimizing the number of inversions in the space of all online $k$-card selection rules.