Past Stochastic Analysis Seminar

We examine the classical orthogonal polynomial ensembles using integration by parts for the underlying Markov operators, differential equations on Laplace transforms and moment equations. Equilibrium measures are described as limits of empirical spectral distributions. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. Applications to sharp deviation inequalities on largest eigenvalues are discussed.

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Although the numerical methods for stochastic differential equations (SDEs) have been well studied, there are few results on the numerical solutions for SDEwMSs. The main aim of this talk is to investigate the invariant measure of numerical solutions of SDEwMSs and discuss their convergence.

Levy trees are random continuous trees that are obtained as

scaling limits of Galton-Watson trees. Continuous tree means here real tree, that is a certain class of path-connected metric spaces without cycles. This class of random trees contains in particular the continuum random tree of Aldous that is the limit of the uniform random tree with N vertices and egde length one over the square root of N when N goes to infinity. In this talk I give a precise definition of the Levy trees and I explain some interesting fractal properties of these trees. This talk is based on joint works with J-F Le Gall and M. Winkel available on arxiv : math.PR/0501079 (published in

PTRF) math.PR/0509518 (preprint)

math.PR/0509690 (preprint).

I will present two-sided estimates for the heat kernel of the elliptic

Schr

Given a family of independent events in a probability space, the probability

that none of the events occurs is of course the product of the probabilities

that the individual events do not occur. If there is some dependence between the

events, however, then bounding the probability that none occurs is a much less

trivial matter. The Lov

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A self-interacting random walk is a random process evolving in an environment depending on its past behaviour.

The notion of Edge-Reinforced Random Walk (ERRW) was introduced in 1986 by Coppersmith and Diaconis [2] on a discrete graph, with the probability of a move along an edge being proportional to the number of visits to this edge. In the same spirit, Pemantle introduced in 1988 [5] the Vertex-Reinforced Random Walk (VRRW), the probability of move to an adjacent vertex being then proportional to the number of visits to this vertex (and not to the edge leading to the vertex). The Self-Interacting Diffusion (SID) is a continuous counterpart to these notions.

Although introduced by similar definitions, these processes show some significantly different behaviours, leading in their understanding to various methods. While the study of ERRW essentially requires some probabilistic tools, corresponding to some local properties, the comprehension of VRRW and SID needs a joint understanding of on one hand a dynamical system governing the general evolution, and on the other hand some probabilistic phenomena, acting as perturbations, and sometimes changing the nature of this dynamical system.

The purpose of our talk is to present our recent results on the subject [1,3,4,6].

Bibliography

[1] M. Bena

Numerous physical systems are justifiably modelled as Markov processes. However,

in practical applications the (usually implicit) assumptions concerning accurate

measurement of the system are often a fair departure from what is possible in

reality. In general, this lack of exact information is liable to render the

We review the analytic transformations allowing to construct standard bridges from a semistable Markov process, with indec 1/2, enjoying the time inversion property. These are generalized and some of there properties are studied. The new family maps the space of continuous real-valued functions into a family which is the topic of our focus. We establish a simple and explicit formula relating the distributions of the first hitting times of each of these by the considered semi-stable process

When two single server queues have the same arrivals process, this is said to be a `fork-join queue'. In the case where the arrivals and service processes are Brownian motions, the queue lengths process is a reflecting Brownian motion in the nonnegative orthant. Tan and Knessl [1996] have given a simple explicit formula for the stationary distribution for this queueing system in a symmetric case, which they obtain as a heavy traffic limit of the classical discrete model. With this as a starting point, we analyse the Brownian model directly in further detail, and consider some related exit problems.

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.

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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.

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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.

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.