Past Stochastic Analysis Seminar

Relatively little is known about the ability of numerical methods for stochastic differential equations (SDE

Diffusion limited aggregation (DLA) is a random growth model which was

originally introduced in 1981 by Witten and Sander. This model is prevalent in

nature and has many applications in the physical sciences as well as industrial

processes. Unfortunately it is notoriously difficult to understand, and only one

rigorous result has been proved in the last 25 years. We consider a simplified

version of DLA known as the Eden model which can be used to describe the growth

of cancer cells, and show that under certain scaling conditions this model gives

rise to a limit object known as the Brownian web.

We study the parabolic Anderson problem, i.e., the heat equation on the d-dimentional

integer lattice with independent identically distributed random potential and

localised initial condition. Our interest is in the long-term behaviour of the

random total mass of the unique non-negative solution, and we prove the complete

localisation of mass for potentials with polynomial tails.

First I will introduce Poisson random measures and their connection to Levy processes.  Then SPDE

We will see how Dynkin's isomorphism emerges from the "loop soup" introduced by

Lawler and Werner.

Sequential Monte Carlo Samplers are a class of stochastic algorithms for

Monte Carlo integral estimation w.r.t. probability distributions, which combine

elements of Markov chain Monte Carlo methods and importance sampling/resampling

schemes. We develop a stability analysis by functional inequalities for a

nonlinear flow of probability measures describing the limit behaviour of the

methods as the number of particles tends to infinity. Stability results are

derived both under global and local assumptions on the generator of the

underlying Metropolis dynamics. This allows us to prove that the combined

methods sometimes have good asymptotic stability properties in multimodal setups

where traditional MCMC methods mix extremely slowly. For example, this holds for

the mean field Ising model at all temperatures.

We report on two joint works with Jeremy Quastel and Alejandro Ramirez, on an

interacting particle system which can be viewed as a combustion mechanism or a

chemical reaction.

We consider a model of the reaction $X+Y\to 2X$ on the integer lattice in

which $Y$ particles do not move while $X$ particles move as independent

continuous time, simple symmetric random walks. $Y$ particles are transformed

instantaneously to $X$ particles upon contact.

We start with a fixed number $a\ge 1$ of $Y$ particles at each site to the

right of the origin, and define a class of configurations of the $X$ particles

to the left of the origin having a finite $l^1$ norm with a specified

exponential weight. Starting from any configuration of $X$ particles to the left

of the origin within such a class, we prove a central limit theorem for the

position of the rightmost visited site of the $X$ particles.

We consider Burgers type nonlinear SPDEs with L

We follow Arnold's approach of Euler equation as a geodesic on the group of

diffeomorphisms. We construct a geometrical Brownian motion on this group in the

case of the two dimensional torus, and prove the global existence of a

stochastic perturbation of Euler equation (joint work with F. Flandoli and P.

Malliavin).

Other diffusions allow us to obtain the deterministic Navier-Stokes equation

as a solution of a variational problem (joint work with F. Cipriano).

In St Anne's College

In this talk, the convergence analysis of a class of weak approximations of

solutions of stochastic differential equations is presented. This class includes

recent approximations such as Kusuoka's moment similar families method and the

Lyons-Victoir cubature on Wiener Space approach. It will be shown that the rate

of convergence depends intrinsically on the smoothness of the chosen test

function. For smooth functions (the required degree of smoothness depends on the

order of the approximation), an equidistant partition of the time interval on

which the approximation is sought is optimal. For functions that are less smooth

(for example Lipschitz functions), the rate of convergence decays and the

optimal partition is no longer equidistant. An asymptotic rate of convergence

will also be presented for the Lyons-Victoir method. The analysis rests upon

Kusuoka-Stroock's results on the smoothness of the distribution of the solution

of a stochastic differential equation. Finally, the results will be applied to

the numerical solution of the filtering problem.

We consider multi-dimensional Gaussian processes and give a novel, simple and

sharp condition on its covariance (finiteness of its two dimensional rho-variation,

for some rho <2) for the existence of "natural" Levy areas and higher iterated

integrals, and subsequently the existence of Gaussian rough paths. We prove a

variety of (weak and strong) approximation results, large deviations, and

support description.

Rough path theory then gives a theory of differential equations driven by

Gaussian signals with a variety of novel continuity properties, large deviation

estimates and support descriptions generalizing classical results of

Freidlin-Wentzell and Stroock-Varadhan respectively.

(Joint work with Nicolas Victoir.)