Spring 2015

We study the properties of solutions of stochastic differential equations driven by processes generating loops in free nilpotent groups. We are in particular interested in existence and smoothness for the density.

In this talk we discuss random walks in a time-dependent zero-drift random environment in $\mathbb{Z}^d$. We prove a quenched invariance principle under an appropriate moment condition. The proof is based on the use of a maximum principle for parabolic difference operators.

We consider the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles define solutions to variational formulas that characterize limit shapes and yield new results for Busemann functions, geodesics and the competition interface. This is joint work with Nicos Georgiou and Timo Seppäläinen.

We discuss a technique, going back to work of Molchanov, for determining the small-time asymptotics of the heat kernel (equivalently, the large deviations of the corresponding diffusion) at the cut locus of a sub-Riemannian manifold (valid away from any abnormal geodesics). We relate the leading term of the expansion to the structure of the cut locus, especially to conjugacy, and explain how this can be used to find general bounds as well as to compute specific examples, in both the Riemannian and sub-Riemannian contexts. We also show how this approach leads to restrictions on the types of singularities of the exponential map that can, generically, occur along minimal geodesics. Further, we discuss the asymptotics for the gradient and Hessian of the logarithm of the heat kernel on a Riemannian manifold, giving a characterization of the cut locus in terms of the behavior of the log-Hessian. In particular, the leading term in the expansion of the log-Hessian comes from the variance "of the minimal geodesics" with respect to the small-time Brownian bridge. Much of this work is joint with Davide Barilari, Ugo Boscain, and Grégoire Charlot.

We study the Taylor expansion for the solution of a differential equation driven by $p$-rough paths with $p>2$. We will prove a general theorem concerning the convergence of the Taylor expansion on a nonempty interval provided that the vector fields are analytic on a ball centered at the initial point. We will also derive criteria that enable us to study the rate of convergence of the Taylor expansion. Our deterministic results can be applied to stochastic differential equations driven by fractional Brownian motions with Hurst parameter $1/4 < H < 1/2$ and continuous Gaussian processes with finite $2D$ $\rho$-variation. At last, we give a Castell expansion and a tail estimate with exponential decay for the remainder terms of the solutions of the differential equations driven by continuous centered Gaussian process with finite $2D$ $\rho$-variation and fractional Brownian motion with Hurst parameter $H>1/4$.

Let $(S, \mathcal{S}, \pi)$ be a measure space. Let $f : S \rightarrow\mathbb{R}$ be $\mathcal{S}$ measurable and integrable with respect to $\pi$. Let $\lambda \equiv \int_S fd\pi$. Our goal is to estimate $\lambda$. If $\pi$ is a probability measure (i.e. $\pi(S)=1$), there are currently two well-known statistical procedures for this. One is based on iid sampling from $\pi$, also called IID Monte Carlo (IIDMC); the other method is Markov chain Monte Carlo (MCMC). Both IIDMC and MCMC require that the target distribution $\pi$ is a probability distribution or at least a totally finite measure. In this talk, we discuss Monte Carlo methods to statistically estimate the integrals of a class of functions with respect to some distributions that may not be finite (i.e. $\pi(S) = \infty$) based on regenerative stochastic processes in continuous time such as Brownian motion.

The starting point of our analysis is a formula for a hypoelliptic heat kernel on nilpotent groups introduced by Agrachev, Boscain, Gauthier, and Rossi. This formula is explicit as long as we can find all unitary irreducible representations. In this talk, we consider $n$-step nilpotent Lie groups. First we apply Kirillov's orbit method to describe these representations for n-step nilpotent Lie groups. This allows us to write the corresponding hypoelliptic heat kernel using an integral formula over a Euclidean space

A celebrated result of H.J Brascamp and E.H. Lieb (with subsequent proofs by many) says that the ground state eigenfunction for the Laplacian in convex regions (and of Schrödinger operators with convex potentials in $\mathbb{R}^n$) is log-concave. There are many applications of this result. A proof can be given from a similar result for the finite dimensional distributions of Brownian motion. Some years ago the speaker raised similar questions (and made some conjectures) when the Brownian motion is replaced by other Lévy processes. In this talk we elaborate on this topic and outline a proof of a result of Moritz Kassmann and Luis Silvestre (2013) concerning superharmonicity of eigenfunctions for certain fractional powers of the Laplacian. Our proof is joint work with Dante DeBlassie of New Mexico State University.

We will introduce fractal properties of rough differential equations driven by frational Brownian motion with Hurst parameter $H>1/4$. We will first survey some known results on density and tail estimates of such processes. Then we will show the Hausdorff dimension of the sample paths is equal to $\min(d, 1/H)$, where $d$ is the dimension of the process. Also we will show that with positive probability, the level sets in the form of $\{t: X_t=x \}$ has Hausdorff dimension $1-dH$ when $dH < 1$, and are almost surely empty otherwise.

We obtain large deviation results for a two time-scale model of jump-diffusion processes. The process on the two time scales are fully inter-dependent, the slow process has small perturbative noise and the fast process is ergodic. We will discuss both probabilistic and PDE methods that can be used to obtain these results. Some applications of these results to finance will also be discussed.

This is joint work with Jean-Pierre Fouque, Jin Feng, Martin Forde, and Lea Popovic.

Consumers are engaged in more social networking and E-commerce activities these days and are increasingly storing their documents and media in the online storage. Businesses are relying on Big Data analytics for business intelligence and are migrating their traditional IT infrastructure to the cloud. These trends cause the online data storage demand to rise faster than Moore's Law. Erasure coding techniques are used widely for distributed data storage since they provide space-optimal data redundancy to protect against data loss.

Cost-effective, network-accessible storage is a strategic infrastructural capability that can serve many businesses. These customers, however, have very diverse requirements of latency, reliability, cost, security etc. In this talk, I will describe how to characterize latency, reliability, cost, and the trade-offs involved in these. In order to characterize latency, we give and analyze a novel scheduling algorithm. I will describe that limited bandwidth between data centers allow us to design new coding schemes that help improve mean time to data loss of the system by 10^20 for (51,30) erasure code as compared to a standard Reed-Solomon code. Finally, I will focus on joint optimization of customer requirements, present new approaches for content placement and content access, and validate the results using implementations on an open source distributed file system on a public test grid.

Excited random walks (also called cookie random walks) are a model of a self-interacting random motion where the transition probabilities depend on the past behavior of the walk through the local time at the present site. More specifically, given a collection $\{ \omega_x(j) \}_{x\in \mathbb{Z},\, j\geq 1} \in (0,1)^{\mathbb{Z} \times \mathbb{N}}$, upon the $j$-th visit to the site $x$ the random walk steps to the right (resp. left) with probability $\omega_x(j)$ (resp. $1-\omega_x(j)$). Most of the results known for excited random walks assume either

• non-negative cookies: $\omega_x(j)\geq 1/2$ for all $x,j$
• or boundedly many cookies per site: there is an $M<\infty$ such that $\omega_x(j) = 1/2$ for all $x\in\mathbb{Z}$ and $j>M$.

Until very recently, very little was known in the case when at each site $x$ there were infinitely many $j$ with $\omega_x(j)>1/2$ and $\omega_x(j) < 1/2$. In this direction, Kozma, Orenshtein, and Shinkar studied the case of periodic cookie sequences; that is, there is a fixed periodic sequence $\{p(j)\}_{j\geq 1}$ with $\omega_x(j) = p(j)$ for all $x,j$. Under this assumption they proved an explicit criterion for recurrence/transience of the excited random walk.

In this talk we will consider a different model where for each $x\in \mathbb{Z}$ the sequence $\{\omega_x(j)\}_{j\geq 1}$ comes from an independent copy of a finite state Markov chain. This model generalizes both the case of periodic cookie sequences and many instances of boundedly many cookies per site. We are able to extend many of the known results from the boundedly many cookies case to our setup, including a criterion for recurrence/transience, a law of large numbers with an explicit criterion for non-zero speed, and limiting distributions in the transient case.

We consider two kinds of inhomogeneous corner growth models with independent waiting times $\{W(i, j): i, j \geq 1 \}$:

1. $W(i, j)$ is distributed exponentially with parameter $a_i + b_j$ for each $i$, $j$.
2. $W(i, j)$ is distributed geometrically with fail parameter $a_i b_j$ for each $i$, $j$.

These generalize exactly-solvable i.i.d. models with exponential or geometric waiting times. The parameters $(a_n)$ and $(b_n)$ are random with a joint distribution that is stationary with respect to the nonnegative shifts and ergodic (separately) with respect to the positive shifts of the indices. Then the shape functions of models (1) and (2) satisfy variational formulas in terms of the marginal distributions of $(a_n)$ and $(b_n)$. For certain choices of these marginal distributions, we still get closed-form expressions for the shape function as in the i.i.d. models.