Past Stochastic Analysis Seminar

We obtain a new identity giving a quintuple law of overshoot, time of

overshoot, undershoot, last maximum, and time of last maximum of a general Levy

process at ?rst passage. The identity is a simple product of the jump measure

and its ascending and descending bivariate renewal measures. With the help of

this identity, we consider applications for passage problems of stable

processes, recovering and extending results of V. Vigon on the bivariate jump

measure of the ascending ladder process of a general Levy process and present

some new results for asymptotic overshoot distributions for Levy processes with

regularly varying jump measures.

(Parts of this talk are based on joint work with Ron Doney and Claudia

Kluppelberg)

In recent years, the use of random planar maps as discretized random surfaces has received a considerable attention in the physicists community. It is believed that the large-scale properties, or the scaling limit of these objects should not depend on the local properties of these maps, a phenomenon called universality.

By using a bijection due to Bouttier-di Francesco-Guitter between certain classes of planar maps and certain decorated trees, we give instances of such universality

phenomenons when the random maps follow a Boltzmann distribution where each face with degree $2i$ receives a nonnegative weight $q(i)$. For example, we show that under

certain regularity hypothesis for the weight sequence, the radius of the random map conditioned to have $n$ faces scales as $n^{1/4}$, as predicted by physicists and shown in the case of quadrangulations by Chassaing and Schaeffer. Our main tool is a new invariance principle for multitype Galton-Watson trees and discrete snakes.

The aging of spin-glasses has been of much interest in the last decades. Since its explanation in the context of real spin-glass models is out of reach, several effective models were proposed in physics literature. In my talk I will present how aging can be rigorously proved in so called trap models and what is the mechanism leading to it. In particular I will concentrate on conditions leading to the fact that one of usual observables used in trap models converges to arc-sine law for Levy processes.

Random Walks in Dirichlet Environment play a special role among random walks in random environments since the annealed law corresponds to the law of an edge oriented reinforced random walks. We will give few results concerning the ballistic behaviour of these walks and some properties of the asymptotic velocity. We will also compare the behaviour of these walks with general random walks in random environments in the limit of small disorder

We show that the solutions of stochastic differential equations converge in

the rough path metric as the coefficients of these equations converge in a

suitable lipschitz norm. We then use this fact to obtain results about

differential equations driven by the Brownian rough path.

It is now known that the overall behaviour of a simple random walk (SRW) on

supercritical (p>p_c) percolation cluster in Z^d is similiar to that of the SRW

in Z^d. The critical case (p=p_c) is much harder, and one needs to define the

'incipient infinite cluster' (IIC). Alexander and Orbach conjectured in 1982

that the return probability for the SRW on the IIC after n steps decays like

n^{2/3} in any dimension. The easiest case is that of trees; this was studied by

Kesten in 1986, but we can now revisit this problem with new techniques.

The Yang-Mills energy is a non-negative functional on the space of connections on a principal bundle over a Riemannian manifold. At a heuristical level, this energy determines a Gibbs measure which is called the Yang-Mills measure. When the manifold is a surface, a stochastic process can be constructed - at least in two different ways - which is a sensible candidate for the random holonomy of a connection distributed according to the Yang-Mills measure. This process is constructed by using some specifications given by physicists of its distribution, namely some of its finite-dimensional marginals, which of course physicists have derived from the Yang-Mills energy, but by non-rigorous arguments. Without assuming any familiarity with this stochastic process, I will present a large deviations result which is the first rigorous link between the Yang-Mills energy and the Yang-Mills measure.

The estimation of heat kernels has been of much interest in various settings. Often, the spaces considered have some kind of uniformity in the volume growth. Recent results have shown that this is not the case for certain random fractal sets. I will present heat kernel bounds for spaces admitting a suitable resistance form, when the volume growth is not uniform, which are motivated by these examples.

We study diploid branching particle models and its behaviour when rapid

stirring, i.e. rapid exchange of particles between neighbouring spatial

sites, is added to the interaction. The particle models differ from the

``usual'' models in that they all involve two types of particles, male

and female, and branching can only occur when both types of particles

are present. We establish the existence of nontrivial stationary

distributions for various models when birth rates are sufficiently large.

We aim at presenting some aspects of stochastic calculus via regularization

in relation with integrator processes which are generally not semimartingales.

Significant examples of those processes are Dirichlet processes, Lyons-Zheng

processes and fractional (resp. bifractional) Brownian motion. A Dirichlet

process X is the sum of a local martingale M and a zero quadratic variation

process A. We will put the emphasis on a generalization of Dirichlet processes.

A weak Dirichlet process is the sum of local martingale M and a process A such

that [A,N] = 0 where N is any martingale with respect to an underlying

filtration. Obviously a Dirichlet process is a weak Dirichlet process. We will

illustrate partly the following application fields.

Analysis of stochastic integrals related to fluidodynamical models considered

for instance by A. Chorin, F. Flandoli and coauthors...

Stochastic differential equations with distributional drift and related

stochastic control theory.

The talk will partially cover joint works with M. Errami, F. Flandoli, F.

Gozzi, G. Trutnau.

I will show rough path estimates for smooth L^p functions whose derivatives are in L^q. The application part related to (linear or nonlinear) Fourier analysis will be also discussed.

This talk attempts to survey key aspects of the mathematics that has been developed in recent years towards an explicit understanding of the structure of exponential functionals of Brownian motion, starting with work of Yor's in the 1990s

It is known that a continuous path of bounded variation

can be reconstructed from a sequence of its iterated integrals (called the signature) in a similar way to a function on the circle being reconstructed from its Fourier coefficients. We study the radius of convergence of the corresponding logarithmic signature for paths in an arbitrary Banach space. This convergence has important consequences for control theory (in particular, it can be used for computing the logarithm of a flow)and the efficiency of numerical approximations to solutions of SDEs. We also discuss the nonlinear structure of the space of logarithmic signatures and the problem of reconstructing a path by its signature.

We introduce conditions in the spirit of $(T)$ and $(T')$ of the discrete setting, that imply, when $d \geq 2$, a law of large numbers with non-vanishing limiting velocity (which we refer to as 'ballistic behavior') and a central limit theorem with non-degenerate covariance matrix. As an application of our results, we consider the class of diffusions where the diffusion matrix is the identity, and give a concrete criterion on the drift term under which the diffusion in random environment exhibits ballistic behavior.

/notices/events/abstracts/abstracts/stochastic-analysis/ht05/burrage.shtml

It is well-known that the only space-time scaling limits of Galton-Watson processes are continuous-state branching processes. Their genealogical structure is most explicitly expressed by discrete trees and R-trees, respectively. Weak limit theorems have been recently established for some of these random trees. We study here a Markovian forest growth procedure that allows to construct the genealogical forest of any continuous-state branching process with immigration as an a.s. limit of Galton-Watson forests with edge lengths. Furthermore, we are naturally led to continuous forests with edge lengths. Another strength of our method is that it yields results in the general supercritical case that was excluded in most of the previous literature.

/notices/events/abstracts/stochastic-analysis/ht05/draief.shtml

It became apparent during the last decade that in several questions in classical complex analysis extremal configurations are fractal, making them very difficult to attack: it is not even clear how to construct or describe extremal objects. We will argue that the most promising approach is to consider conformally self-similar random configurations, which should be extremal almost surely.