Spring 2013

It is known that $d$-dimensional fractional Brownian motion (fBm) with parameter $H$ hits points with positive probability if and only if $d<1/H$. Very few results are available for Gaussian proceses beyond fBm. We present tools from the Malliavin calculus and a general strategy from potential theory to help us find upper and lower bounds on hitting probabilities for points and other sets, for general Gaussian processes, and will explain what problems remain open, and what non-Gaussian extensions could look like.

This is joint work with Eulalia Nualart.

The Stein method allows to measure the distance between the laws of two random variables. Recently, this method combined with the Malliavin calculus, led to several interesting results. We will present the basic facts related to this theory and we will give some recent applications to limit theorems.

The rough path theory for Banach spaces has been developed for a while and relatively mature at its current stage. However, the theory does not properly identify the fully geometric framework of rough paths evolving on manifolds. For this purpose, recently Cass, Litterer and Lyons introduced a notion of rough path on manifolds, in which they consider rough paths on a manifold $M$ as maps sending Banach valued 1-forms on $M$ to Banach-valued rough paths in the classical sense.

In this talk, we propose another approach based on which we are able to make an invariant definition of 'fractional Brownian motions' on manifolds. After that, I will propose several questions regarding our framework. The talk is based on an ongoing project with Elton Hsu.

We will preview a singular integral operator: the Beurling-Ahlfors transform $B$. By associating martingales to the operator, we can use a theorem of Burkholder to estimate its $L^p$ norm. In recent papers, it has become evident that the martingales have important orthogonality structures in them, that lead to better estimates. In this talk I will present a general version of Burkholder's theorem, revealing the full extent of orthogonality accommodated by his proof. This will lead to further development for the theory of orthogonality in martingales.

We investigate the Polya process, which underlies an urn of colored balls growing in real time, under a possibly random ball addition scheme. A partial differential equation governs the evolution of the process. Some special cases are amenable to an asymptotic solution: The diagonal Polya process and the Ehrenfest process will serve as an illustration.

Applications of standard (discrete) urns and their analogue when embedded in real time include several classes of random trees that have applications in computer science, epidemiology and philology. Given time, we shall present some of these applications.

In this talk, we show the existence of an asymptotic expansion of the trace operator \[e^{-t(-\Lambda^{\alpha/2}+V)}-e^{-t(-\Lambda^{\alpha/2})}\] in $\mathbb{R}^d$, where $\Lambda^{\alpha/2}$ is the infinitesimal generator of an $\alpha$-stable process and $V$ is a potential in the class of rapidly decaying functions at infinity.

We consider large deviations of random walks in a random environment on the strip $\mathbb{Z} \times \{1,2,\ldots,d\}$.   Large deviations for random walks in random environments have been studied in a variety of different types of graphs, but only in the one-dimensional nearest-neighbor case is there a known variational formula relating the quenched and averaged rate functions. We will generalize the argument for the one-dimensional case to that of a strip of finite width and prove quenched and averaged large deviation principles with a variational formula relating the two rate functions. The main novelty in our approach will be to use an idea of Furstenburg and Kesten to obtain probabilistic formulas for the limits of certain products of random matrices.

We present sharp tail asymptotic estimates for the density and the distribution function of the sum of n correlated log-normal random variables. Despite the simplicity of the problem formulation and the importance of this result for applications, the full solution has been absent from the literature so far. Applications to systematic construction of scenarios for stress testing and to the computation of the Value at Risk in the multi-dimensional Black-Scholes model will be discussed.

Joint work with Archil Gulisashvili (Ohio).

This talk investigates several properties related to densities of solutions $(X_t)_{t\in[0,1]}$ to differential equations driven by a fractional Brownian motion with Hurst parameter $H>1/4$. We first determine conditions for strict positivity of the density of $X_t$. Then we obtain some exponential bounds for this density when the diffusion coefficient satisfies an elliptic type condition. Finally, still in the elliptic case, we derive some bounds on the hitting probabilities of sets by fractional differential systems in terms of Newtonian capacities.

Joint work with C. Ouyang, E. Nualart, S. Tindel.

We consider the following stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H>1/4$: $$ X_t = x + \sum_{i=1}^n \int_0^t V_i(X_s^x) dB_s^i, $$ where the $V_i$ are $C^\infty$-bounded vector fields. Under elliptic assumption on the vector fields, we provide some functional inequalities satisfied by the distribution of the solution $X_t^x$ by using integration by parts formula in Malliavin calculus.

Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. The rigorous connection has recently been made between the physics of the microscopic many-body dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrödinger equation (NLS). I'll discuss recent progress with Gerard Ben Arous and Benjamin Schlein on a central limit theorem for the quantum many-body dynamics, a step towards large deviations for Bose-Einstein condensation.

Many disciplines have used urn models to simulate more complex problems. Characteristics and properties of urn models are therefore of interest, as they have numerous applications. With a primary focus on tenable urns, seven different categories of replacement matrices emerge. Furthermore, the eigenvalue structure of a replacement matrix takes on certain qualities for tenable urns. Investigating 2x2 urns and larger strongly tenable balanced urns, it is shown that the principal eigenvalues are real-valued and non-negative.

We consider a random walk among i.i.d. obstacles on Z under the condition that the walk starts from the origin and reaches a remote location y. The obstacles are represented by a killing potential, which takes value M>0 with probability p and value 0 with probability 1-p, 0 < p < 1, independently at each site of the lattice. We consider the walk under both quenched and annealed measures. It is known that under either measure the crossing time from 0 to y of such walk, H(y), grows linearly in y. More precisely, the expectation of H(y)/y converges to a limit as y goes to infinity. The reciprocal of this limit is called the asymptotic speed of the conditioned walk. We study the behavior of the asymptotic speed in two regimes:

1. as p goes to 0 for M fixed (``sparse'')
2. as M goes to infinity for p fixed (``spiky'')

We observe and quantify a dramatic difference between the quenched and annealed settings.

Joint work with Thomas Mountford (EPFL, Lausanne)