Past Stochastic Analysis Seminar

Under the nondegenerate condition as in the diffusion case, we show

that the linear stochastic jump diffusion process projected on the

unite sphere has an uni que invariant probabolity measure. The

Lyapunov exponentcan be represented as an integral over the

sphere. These results were extended to the degenerated and Levy jump

cases.

Anderson localisation is an important phenomenon describing a

transition between insulation and conductivity. The problem is to analyse

the spectrum of a Schroedinger operator with a random potential in the

Euclidean space or on a lattice. We say that the system exhibits

(exponential) localisation if with probability one the spectrum is pure

point and the corresponding eigen-functions decay exponentially fast.

So far in the literature one considered a single-particle model where the

potential at different sites is IID or has a controlled decay of

correlations. The present talk aims at $N$-particle systems (bosons or

fermions) where the potential sums over different sites, and the traditional

approach needs serious modifications. The main result is that if the

`randomness' is strong enough, the $N$-particle system exhibits

localisation.

The proof exploits the muli-scale analysis scheme going back to Froehlich,

Martinelli, Scoppola and Spencer and refined by von Drefus and Klein. No

preliminary knowledge of the related material will be assumed from the

audience, apart from basic facts.

This is a joint work with V Chulaevsky (University of Reims, France)

I will present a new formula for diffusion processes which involving

Ito integral for the transition probability functions. The nature of

the formula I discovered is very close to the Kac formula, but its

form is similar to the Cameron-Martin formula. In some sense it is the

Cameron-Martin formula for pinned diffusions.

I will consider a stochastic process ( \xi_u; u \in

\Gamma_\infty ) where \Gamma_\infty is the set of vertices of an

infinite binary tree which satisfies some recursion relation

\xi_u= \phi(\xi_{u0},\xi_{u1}, \epsilon_u) \text { for each } u \in \Gamma_\infty.

Here u0 and u1 denote the two immediate daughters of the vertex u.

The random variables ( \epsilon_u; u\in \Gamma_\infty), which

are to be thought of as innovations, are supposed independent and

identically distributed. This type of structure is ubiquitous in models

coming from applied proability. A recent paper of Aldous and Bandyopadhyay

has drawn attention to the issue of endogeny: that is whether the process

( \xi_u; u \in \Gamma_\infty) is measurable with respect to the

innovations process. I will explain how this question is related to the

existence of certain dynamics and use this idea to develop a necessary and

sufficient condition [ at least if S is finite!] for endogeny in terms of

the coupling rate for a Markov chain on S^2 for which the diagonal is

absorbing.

We shall review recent progress in the understanding of

isoperimetric inequalities for product probability measures (a very tight

description of the concentration of measure phenomeonon). Several extensions

of the classical result for the Gaussian measure were recently derived by

functional analytic methods.

The Hopfield model took his name and its popularity within the theory

of formal neural networks. It was introduced in 1982 to describe and

implement associative memories. In fact, the mathematical model was

already defined, and studied in a simple form by Pastur and Figotin in

an attempt to describe spin-glasses, which are magnetic materials with

singular behaviour at low temperature. This model indeed shows a very

complex structure if considered in a slightly different regime than

the one they studied. In the present talk we will focus on the

fluctuations of the free energy in the high-temperature phase. No

prior knowledge of Statistical mechanics is required to follow the

talk.

Joe Doob, who died recently aged 94, was the last survivor of the

founding fathers of probability. Doob was best known for his work on

martingales, and for his classic book, Stochastic Processes (1953).

The talk will combine an appreciation of Doob's work and legacy with

reminiscences of Doob the man. (I was fortunate to be a colleague of

Doob from 1975-6, and to get to know him well during that year.)

Following Doob's passing, the mantle of greatest living probabilist

descends on the shoulders of Kiyosi Ito (b. 1915), alas now a sick

man.

We are interested in a microscopic stochastic description of a

population of discrete individuals characterized by one adaptive

trait. The population is modeled as a stochastic point process whose

generator captures the probabilistic dynamics over continuous time of

birth, mutation and death, as influenced by each individual's trait

values, and interactions between individuals. An offspring usually

inherits the trait values of her progenitor, except when a mutation

causes the offspring to take an instantaneous mutation step at birth

to new trait values. Once this point process is in place, the quest

for tractable approximations can follow different mathematical paths,

which differ in the normalization they assume (taking limit on

population size , rescaling time) and in the nature of the

corresponding approximation models: integro or integro-differential

equations, superprocesses. In particular cases, we consider the long

time behaviour for the stochastic or deterministic models.

Complete stochastic volatility models provide prices and

hedges. There are a number of complete models which jointly model an

underlying and one or more vanilla options written on it (for example

see Lyons, Schonbucher, Babbar and Davis). However, any consistent

model describing the volatility of options requires a complex

dependence of the volatility of the option on its strike. To date we

do not have a clear approach to selecting a model for the volatility

of these options

A version of Lyons theory of rough path calculus which applies to a

subclass of rough paths for which more geometric interpretations are

valid will be presented. Application will be made to the Brownian and

to the (fractional) support theorem.

The convergence to stationarity of many finite ergodic Markov

chains presents a sharp cut-off: there is a time T such that before

time T the chain is far from its equilibrium and, after time T,

equilibrium is essentially reached. We will discuss precise

definitions of the cut-off phenomenon, examples, and some partial

results and conjectures.

The question whether the measure of a Levy process starting from x>0 and "conditioned to stay positive" converges to the corresponding obiect for x=0 when x tends to 0 is rather delicate. I will describe work with Loic Chaumont which settles this question, essentially in all cases of interest. As an application, I will show how to use this result and excursion theory to give simpler proofs of some recent results about the exit problem for reflected processe derived from spectrally one-sided Levy processes due to Avram. Kyprianou and Pistorius.

We formulate a problem of super-hedging under gamma constraint by

taking the portfolio process as a controlled state variable. This

leads to a non-standard stochastic control problem. An intuitive

guess of the associated Bellman equation leads to a non-parabolic

PDE! A careful analysis of this problem leads to the study of the

small time behaviour of double stochastic integrals. The main result

is a characterization of the value function of the super-replication

problem as the unique viscosity solution of the associated Bellman

equation, which turns out to be the parabolic envelope of the above

intuitive guess, i.e. its smallest parabolic majorant. When the

underlying stock price has constant volatility, we obtain an

explicit solution by face-lifting the pay-off of the option.

The Brownian snake (with lifetime given by a normalized

Brownian excursion) arises as a natural limit when studying random trees. This

may be used in both directions, i.e. to obtain asymptotic results for random

trees in terms of the Brownian snake, or, conversely, to deduce properties of

the Brownian snake from asymptotic properties of random trees. The arguments

are based on Aldous' theory of the continuum random tree.

I will discuss two such situations:

1. The Wiener index of random trees converges, after

suitable scaling, to the integral (=mean position) of the head of the Brownian

snake. This enables us to calculate the moments of this integral.

2. A branching random walk on a random tree converges, after

suitable scaling, to the Brownian snake, provided the distribution of the

increments does not have too large tails. For i.i.d increments Y with mean 0,

a necessary and sufficient condition is that the tails are o(y^{-4}); in

particular, a finite fourth moment is enough, but weaker moment conditions are

not.

The celebrated Levy-Khintchine formula provides us an explicit

structure of Levy processes on $R^d$. In this talk I shall present a

structure result for quasi-regular semi-Dirichlet forms, i.e., for

those semi-Dirichlet forms which are associated with right processes

on general state spaces. The result is regarded as an extension of

Levy-Khintchine formula in semi-Dirichlet forms setting. It can also

be regarded as an extension of Beurling-Deny formula which is up to

now available only for symmetric Dirichlet forms.

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.