Past Stochastic Analysis Seminar

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.

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.

Consider a spectrally one-sided Levy process X and reflect it at

its past infimum I. Call this process Y. We determine the law of the

first crossing time of Y of a positive level a in terms of its

'scale' functions. Next we study the exponential decay of the

transition probabilities of Y killed upon leaving [0,a]. Restricting

ourselves to the case where X has absolutely continuous transition

probabilities, we also find the quasi-stationary distribution of

this killed process. We construct then the process Y confined in

[0,a] and prove some properties of this process.

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.

http://www.maths.ox.ac.uk/notices/events/abstracts/stochastic-analysis/…

Consider a graph with n vertices placed randomly in the unit

square, each connected by an edge to its nearest neighbour in a

south-westerly direction. For many graphs of this type, the centred

total length is asymptotically normal for n large, but in the

present case the limit distribution is not normal, being defined in

terms of fixed-point distributions of a type seen more commonly in

the analysis of algorithms. We discuss related results. This is

joint work with Andrew Wade.

We consider a system of interacting Fisher-Wright diffusions

which arise in population genetics as the diffusion limit of a spatial

particle model in which frequencies of genetic types are changing due to

migration and reproduction.

For both models the historical processes are constructed,

which record the family structure and the paths of descent through space.

For any fixed time, particle representations for the

historical process of a collection of Moran models with increasing particle

intensity and of the limiting interacting Fisher-Wright diffusions are

provided on one and the same probability space by means of Donnelly and

Kurtz's look-down construction.

It will be discussed how this can be used to obtain new

results on the long term behaviour. In particular, we give representations for

the equilibrium historical processes. Based on the latter the behaviour of

large finite systems in comparison with the infinite system is described on

the level of the historical processes.

The talk is based on joint work with Andreas Greven and Vlada

Limic.

We consider the Cahn Hilliard Equation in the line, perturbed by

the space derivative of a space--time white noise. We study the

solution of the equation when the initial condition is the

interface, in the limit as the intensity of the noise goes to zero

and the time goes to infinity conveniently, and show that in a scale

that is still infinitesimal, the solution remains close to the

interface, and the fluctuations are described by a non Markovian

self similar Gaussian process whose covariance is computed.

After a brief introduction to the basics of Rough Paths I'll

explain recent work by Peter Friz, Dan Stroock and myself proving that a

Brownian path conditioned to be uniformly close to a given smooth path

converges in distribution to that path in the Rough Path metric. The Stroock

Varadhan support theorem is an immediate consequence.

The novel part of the argument is to

obtain the estimate in a way that is independent of the particular norm used

in the Euclidean space when one defines the uniform norm on path space.

In this talk, we consider a class of non-linear stochastic partial

differential equations. We represent its solutions as the weighted

empirical measures of interacting particle systems. As a consequence,

a simulation scheme for this class of SPDEs is proposed. There are two

sources of error in the scheme, one due to finite sampling of the

infinite collection of particles and the other due to the Euler scheme

used in the simulation of the individual particle motions. The error

bound, taking into account both sources of error, is derived. A

functional limit theorem is also derived. The results are applied to

nonlinear filtering problems.

This talk is based on joint research with Kurtz.

We construct and study different surface measures on the space of

paths in a compact Riemannian manifold embedded into the Euclidean

space. The idea of the constructions is to force a Brownian particle

in the ambient space to stay in a small neighbourhood of the manifold

and then to pass to the limit. Finally, we compare these surface

measures with the Wiener measure on the space of paths in the

manifold.