DH 3rd floor SR

Weakly self-avoiding walk (WSAW) is obtained by giving a penalty for every

self-intersection to the simple random walk path. The Edwards model (EM) is

obtained by giving a penalty proportional to the square integral of the local

times to the Brownian motion path. Both measures significantly reduce the

amount of time the motion spends in self-intersections.

The above models serve as caricature models for polymers, and we will give

an introduction polymers and probabilistic polymer models. We study the WSAW

and EM in dimension one.

We prove that as the self-repellence penalty tends to zero, the large

deviation rate function of the weakly self-avoiding walk converges to the rate

function of the Edwards model. This shows that the speeds of one-dimensional

weakly self-avoiding walk (if it exists) converges to the speed of the Edwards

model. The results generalize results earlier proved only for nearest-neighbor

simple random walks via an entirely different, and significantly more

complicated, method. The proof only uses weak convergence together with

properties of the Edwards model, avoiding the rather heavy functional analysis

that was used previously.

The method of proof is quite flexible, and also applies to various related

settings, such as the strictly self-avoiding case with diverging variance.

This result proves a conjecture by Aldous from 1986. This is joint work with

Frank den Hollander and Wolfgang Koenig.

We give a construction of Brownian motion in a Weyl chamber, by a

multidimensional generalisation of Pitman's theorem relating one

dimensional Brownian motion with the three dimensional Bessel

process. There are connections representation theory, especially to

Littelmann path model.

We consider Brownian motion on a manifold conditioned not to leave

the tubular neighborhood of a closed riemannian submanifold up

to some fixed finite time. For small tube radii, it behaves like the

intrinsic Brownian motion on the submanifold coupled to some

effective potential that depends on geometrical properties of

the submanifold and of the embedding. This characterization

can be applied to compute the effect of constraining the motion of a

quantum particle on the ambient manifold to the submanifold.

We consider a queuing system with three queues (nodes) and one server.

The arrival and service rates at each node are such that the system overall

is overloaded, while no individual node is. The service discipline is the

following: once the server is at node j, it stays there until it serves all

customers in the queue.

After this, the server moves to the "more expensive" of the two

queues.

We will show that a.s. there will be a periodicity in the order of

services, as suggested by the behavior of the corresponding

dynamical systems; we also study the cases (of measure 0) when the

dynamical system is chaotic, and prove that then the stochastic one

cannot be periodic either.

We discuss estimating the growth exponents for positive solutions to the

random parabolic Anderson's model with small parameter k. We show that

behaviour for the case where the spatial variable is continuous differs

markedly from that for the discrete case.

The probabilistic approach to homogenization can be adapted to fully

degenerate situations, where irreducibility is insured from a Doeblin type

condition. Using recent results on weak sense Poisson equations in a

similar framework, obtained jointly with A. Veretennikov, together with a

regularization procedure, we prove the homogenization result. A similar

approach can also handle degenerate random homogenization.

A reveiw of results about spectral analysis of generators of

some stochastic lattice models (a stochastic planar rotators model, a

stochastic Blume-Capel model etc.) will be presented. Then I'll discuss new

results by R.A. Minlos, Yu.G. Kondratiev and E.A. Zhizhina concerning spectral

analysis of the generator of stochastic continuous particle system. The

construction of one-particle subspaces of the generators and the spectral

analysis of the generator restricted on these subspaces will be the focus of

the talk.