Past Stochastic Analysis Seminar

Fragmentation processes model the evolution of a particle that split as time goes on. When small particles split fast enough, the fragmentation is intensive and the initial mass is reduced to dust in finite time. We encode such fragmentation into a continuum random tree (CRT) in the sense of Aldous. When the splitting times are dense near 0, the fragmentation CRT is in turn encoded into a continuous (height) function. Under some mild hypotheses, we calculate the Hausdorff dimension of the CRT, as well as the maximal H

Given a countable set of sites S an a transition matrix p(x,y) on that set, we consider a process of particles evolving on S according to the following rule: each particle waits an exponential time and then jumps following a Markov chain governed by p(x,y); the particle keeps jumping until it reaches an empty site where it remains for another exponential time. Unlike most interacting particle systems, this process fails to

have the Feller property. This causes several technical difficulties to study it. We present a method to prove that certain measures are invariant for the process and exploit the Kolmogorov zero or one law to study some of its unusual path properties.

According to the Stokes-Einstein law, microscopic particles subject to intense bombardment by (much smaller) gas molecules perform Brownian motion with a diffusivity inversely proportion to their radius. Smoluchowski, shortly after Einstein's account of Brownian motion, used this model to explain the behaviour of a cloud of such particles when, in addition their diffusive motion, they coagulate on collision. He wrote down a system of evolution equations for the densities of particles of each size, in particular identifying the collision rate as a function of particle size.

We give a rigorous derivation of (a spatially inhomogeneous generalization of) Smoluchowski's equations, as the limit of a sequence of Brownian particle systems with coagulation on collision. The equations are shown to have a unique, mass-preserving solution. A detailed limiting picture emerges describing the ancestral spatial tree of particles making up each particle in the current population. The limit is established at the level of these trees.

We describe a nice example of duality between coagulation and fragmentation associated with certain Dirichlet distributions. The fragmentation and coalescence chains we derive arise naturally in the context of the genealogy of Yule processes.

We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analytic results, and interacting particle system models, which don't, at present, incorporate all the competitive strategies that a population might adopt. The second is a simplification of the first in which competition is only supposed to act within lattice sites and the total population size within each lattice point is a constant. In a special case, this second model is dual to a branching-annihilating random walk. For each model, using a comparison with N-dependent oriented percolation, we show that for certain parameter values both populations will coexist for all time with positive probability.

As a corollary we deduce survival for all time of branching annihilating random walk for sufficiently large branching rates.

We also present conjectures relating to the role of space in the survival probabilities for the two populations.

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Ergodic Markov processes possess invariant measures. In the case if transition probabilities or SDE coefficients depend on a parameter, it is important to know whether these measures depend regularly on this parameter. Results of this kind will be discussed. Another close topic is whether approximations to Markov diffusions possess ergodic properties similar to those of the limiting processes. Some partial answer to this question will be presented.

I shall talk on recent results on behaviour of solutions of

2D Navier-Stokes Equation (and some other related equations), perturbed by a random force, proportional to the square root of the viscosity. I shall discuss some properties of the solutions, uniform in the viscosity, as well as the inviscid limit.

Hamiltonian Feynman path integrals, or Feynman (path) integrals over

trajectories in the phase space, are values, which some

pseudomeasures, usually called Feynman (pseudo)measures (they are

distributions, in the sense of the Sobolev-Schwartz theory), take on

functions defined on trajectories in the phase space; so such

functions are integrands in the Feynman path integrals. Hamiltonian

Feynman path integrals (and also Feynman path integrals over

trajectories in the configuration space) are used to get some

representations of solutions for Schroedinger type equations. In the

talk one plans to discuss the following problems.

We present a large deviation result for the behaviour of the

end-point of a diffusion under the influence of a strong drift. The rate

function can be explicitely determined for both attracting and repelling

drift. It transpires that this problem cannot be solved using

Freidlin-Wentzel theory alone. We present the main ideas of a proof which

is based on the Girsanov-Formula and Tauberian theorems of exponential type.

Let {X_n} be a sequence of bounded, real-valued random variables.

Assume that the partial-sums processes {S_n}, where S_n=X_1+...+X_n,

satisfies the large deviation principle with a convex rate-function, I().

Given an observation of the process {X_n}, how would you estimate I()? This

talk will introduce an estimator that was proposed to tackle a problem in

telecommunications and discuss it's properties. In particular, recent

results regarding the large deviations of estimating I() will be presented.

The significance of these results for the problem which originally motivated

the estimator, estimating the tails of queue-length distributions, will be

demonstrated. Open problems will be mentioned and a tenuous link to Oxford's

Mathematical Institute revealed.

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.