Past Stochastic Analysis Seminar

Reflected Brownian motion (RBM) in a two-dimensional wedge is a well-known stochastic process. With an appropriate drift, it is positive recurrent and has a stationary distribution, and the invariant measure is absolutely continuous with respect to Lebesgue measure. I will give necessary and sufficient conditions for the stationary density to be written as a finite sum of exponentials with linear exponents. Such densities are a natural generalisation of the stationary density of one-dimensional RBM. Using geometric ideas reminiscent of the reflection principle, I will give an explicit formula for the density in such cases, which can be written as a determinant. Joint work with Ton Dieker.

We study randomized (i.e. Monte Carlo) algorithms to compute expectations of Lipschitz functionals w.r.t. measures on infinite-dimensional spaces, e.g., Gaussian measures or distribution of diffusion processes. We determine the order of minimal errors and corresponding almost optimal algorithms for three different sampling regimes: fixed-subspace-sampling, variable-subspace-sampling, and full-space sampling. It turns out that these minimal errors are closely related to quantization numbers and Kolmogorov widths for the underlying measure. For variable-subspace-sampling suitable multi-level Monte Carlo methods, which have recently been introduced by Giles, turn out to be almost optimal.

Joint work with Jakob Creutzig (Darmstadt), Steffen Dereich (Bath), Thomas Müller-Gronbach (Magdeburg)

In ordinary percolation, sites of a lattice are open with a given probability and one investigates the existence of infinite clusters (percolation). In dynamical percolation, the sites randomly flip between the states open and closed and one investigates the existence of "atypical" times at which the percolation structure is different from that of a fixed time.

1. I will quickly present some of the original results for dynamical percolation (joint work with Olle Haggstrom and Yuval Peres) including no exceptional times in critical percolation in high dimensions.

2. I will go into some details concerning a recent result that, for the 2 dimensional triangular lattice, there are exceptional times for critical percolation (joint work with Oded Schramm). This involves an interesting connection with the harmonic analysis of Boolean functions and randomized algorithms and relies on the recent computation of critical exponents by Lawler, Schramm, Smirnov, and Werner.

3. If there is time, I will mention some very recent results of Garban, Pete, and Schramm on the Fourier spectrum of critical percolation.

Joint work with Martin Dyer (Leeds) and Leslie Goldberg (Liverpool).

A spin system may be modelled as a graph, in which edges (bonds) indicate interactions between adjacent vertices (sites). A configuration of the system is an assignment of colours (spins) to the vertices of the graph. The interactions between adjacent spins define a certain distribution, the Boltzmann distribution, on configurations. To sample from this distribution it is usually necessary to simulate one of a number of Markov chains on the space of all configurations. Theoretical analyses of the mixing time of these Markov chains usually assume that spins are updated at single vertices chosen uniformly at random. Actual simulations, in contrast, may make (random) updates according to a deterministic, usually highly structured pattern. We'll explore the relationships between systematic scan and random single-site updates, and also between classical uniqueness conditions from statistical physics and more recent techniques in mixing time analysis.

In the past the theory of rough paths has proven to be an elegant tool for deriving support theorems and large deviation principles. In this talk I will explain how this approach can be used in the analysis of stochastic flows generated by Kunita SDE's. As driving processes I will consider general Banach space valued Wiener processes

The Skorokhod reflection problem, originally introduced as a means for constructing solutions to stochastic differential equations in bounded regions, has found applications in many areas of Probability, for example in queueing-like stochastic dynamical systems; its uses range from methods for proving limit theorems to representations of local times of diffusions and control. In this talk, I will present several applications, e.g. to Levy stochastic networks and to queueing-like systems driven by local times of Levy processes, and give an order-theoretic approach to the problem by extending the domain of functions involved from the real line to a fairly arbitrary partially ordered set. I will also discuss how Palm probabilities can be used in connection with the Skorokhod problem to obtain information about stationary solutions of certain systems.

I consider the Metropolis Markov Chain based on the nearest neighbor random walk on the positive half of the Mean Field Ising Model, i.e., on those vectors from $\{−1, 1\}^N$ which contain more $1$ than $−1$. Using randomly-chosen paths I prove a lower bound for the Spectral Gap of this chain which is of order $N^-2$ and which does not depend on the inverse temperature $\beta$. In conjunction with decomposition results such as those in Jerrum, Son, Tetali and Vigoda (2004) this result may be useful for bounding the spectral gaps of more complex Markov chains on the Mean Field Ising Model which may be decomposed into Metropolis chains. As an example, I apply the result to two Multilevel Markov Chain Monte Carlo algorithms, Swapping and Simulated Tempering. Improving a result by Madras and Zheng (2002), I show that the spectral gaps of both algorithms on the (full) Mean Field Ising Model are bounded from below by the reciprocal of a polynomial in the lattice size $N$ and in the inverse temperature $\beta$.

It is an interesting exercise to compute the iterated integrals of Brownian Motion and to calculate the expectations (of polynomial functions of these integrals).

Recent work on constructing discrete measures on path space, which give the same value as Wiener measure to certain of these expectations, has led to promising new numerical algorithms for solving 2nd order parabolic PDEs in moderate dimensions. Old work of Krylov associated finitely additive signed measures to certain constant coefficient PDEs of higher order. Recent work with Levin allows us to identify the relevant expectations of iterated integrals in this case, leaving many interesting open questions and possible numerical algorithms for solving high dimensional elliptic PDEs.

APOLOGIES - this seminar is cancelled.

Professor Terry Lyons will talk instead on signed probability measures and some old results of Krylov.

Log-Sobolev inequalities with weighted square field are derived from a class of super Poincaré inequalities. As applications, stronger versions of Talagrand's transportation-cost inequality are provided on Riemannian manifolds. Typical examples are constructed to illustrate these results.

We discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces. More precisely, we consider a random planar map M(n), which is uniformly distributed over the set of all planar maps with n faces in a certain class. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n to the power 1/4. We then discuss the convergence in distribution of the resulting random metric spaces as n tends to infinity, in the sense of the Gromov-Hausdorff distance between compact metric spaces. This problem was stated by Oded Schramm in his plenary address paper at the 2006 ICM, in the special case of triangulations.

In the case of bipartite planar maps, we first establish a compactness result showing that a limit exists along a suitable subsequence. Furthermore this limit can be written as a quotient space of the Continuum Random Tree (CRT) for an equivalence relation which has a simple definition in terms of Brownian labels attached to the vertices of the CRT. Finally we show that any possible limiting metric space is almost surely homomorphic to the 2-sphere. As a key tool, we use bijections between planar maps and various classes of labelled trees.

We study the dynamic of a very simple chain of three anharmonic oscillators with linear nearest-neighbour couplings. The first and the last oscillator furthermore interact with heat baths through friction and noise terms. If all oscillators in such a system are coupled to heat baths, it is well-known that under relatively weak coercivity assumptions, the system has a spectral gap (even compact resolvent) and returns to equilibrium exponentially fast. It turns out that while it is still possible to show the existence and uniqueness of an invariant measure for our system, it returns to equilibrium much slower than one would at first expect. In particular, it no longer has compact resolvent when the potential of the oscillators is quartic and the spectral gap is destroyed when it grows even faster.

Self-interacting diffusions are solutions to SDEs with a drift term depending

on the process and its normalized occupation measure $\mu_t$ (via an interaction

potential and a confinement potential): $$\mathrm{d}X_t = \mathrm{d}B_t -\left(

\nabla V(X_t)+ \nabla W*{\mu_t}(X_t) \right) \mathrm{d}t ; \mathrm{d}\mu_t = (\delta_{X_t}

- \mu_t)\frac{\mathrm{d}t}{r+t}; X_0 = x,\,\ \mu_0=\mu$$ where $(\mu_t)$ is the

process defined by $$\mu_t := \frac{r\mu + \int_0^t \delta_{X_s}\mathrm{d}s}{r+t}.$$

We establish a relation between the asymptotic behaviour of $\mu_t$ and the

asymptotic behaviour of a deterministic dynamical flow (defined on the space of

the Borel probability measures). We will also give some sufficient conditions

for the convergence of $\mu_t$. Finally, we will illustrate our study with an

example in the case $d=2$.

A wide variety of problems arising in applications require the sampling of a

probability measure on the space of functions. Examples from econometrics,

signal processing, molecular dynamics and data assimilation will be given.

In this situation it is of interest to understand the computational

complexity of MCMC methods for sampling the desired probability measure. We

overview recent results of this type, highlighting the importance of measures

which are absolutely continuous with respect to a Guassian measure.

I will describe some applications of the main techniques of rough paths

theory to problems not related to SDE

Gradient bounds are a very powerful tool to study heat kernel measures and

regularisation properties for the heat kernel. In the elliptic case, it is easy

to derive them from bounds on the Ricci tensor of the generator. In recent

years, many efforts have been made to extend these bounds to some simple

examples in the hypoelliptic situation. The simplest case is the Heisenberg

group. In this talk, we shall discuss some recent developments (due to H.Q. Li)

on this question, and give some elementary proofs of these bounds.

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.)

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