DH 3rd floor SR

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 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.

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.

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.

A firm issues a convertible bond. At each subsequent time, the bondholder

must decide whether to continue to hold the bond, thereby collecting coupons, or

to convert it to stock. The bondholder wishes to choose a conversion strategy to

maximize the bond value. Subject to some restrictions, the bond can be called by

the issuing firm, which presumably acts to maximize the equity value of the firm

by minimizing the bond value. This creates a two-person game. We show that if

the coupon rate is below the interest rate times the call price, then conversion

should precede call. On the other hand, if the dividend rate times the call

price is below the coupon rate, call should precede conversion. In either case,

the game reduces to a problem of optimal stopping. This is joint work with Mihai

Sirbu.