Past Stochastic Analysis Seminar

 

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

theory to problems not related to SDE

  Schramm Loewner Evolutions (SLE) are random planar curves (if κ ≤ 4) or growing compact sets generated by a curve (if κ > 4). We consider more general L

  A dimer configuration of a graph is a subset of the edges, such that every vertex is contained in exactly one edge of the subset. We consider dimer configurations of the honeycomb lattice on the cylinder, which are known to be equivalent to configurations of interlaced particles. Assigning a measure to the set of all such configurations, we show that the probability that particles are located in any subset of points on the cylinder can be written as a determinant, i.e. that the process is determinantal. We also examine Markov chains of interlaced particles on the circle, with dynamics equivalent to RSK.  

 

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.

 

  The "Cubature on Wiener space" algorithm can be regarded as a general approach to high order weak approximations. Based on this observation we will derive many well known weak discretisation schemes and optimise the computational effort required for a given accuracy of the approximation. We show that cubature can also help to overcome some stability difficulties. The cubature on Wiener space algorithm is frequently combined with partial sampling techniques and we outline an extension to these methods to reduce the variance of the samples. We apply the extended method to examples arising in mathematical finance. Joint work of G. Gyurko, C. Litterer and T. Lyons  

  The talk is based on my recent joint work with Zhechun Hu and Wei Sun. We consider a nonlinear filtering problem for general right continuous Markov processes associated with semi-Dirichlet forms. We show that in our general setting the filtering processes satisfy also DMZ (Duncan-Mortensen-Zakai) equation. The uniqueness of the solutions of the filtering equations are verified through their Wiener chaos expansions. Our results on the Wiener chaos expansions for nonlinear filters with possibly unbounded observation functions are novel and have their own interests. We investigate further the absolute continuity of the filtering processes with respect to the reference measures and derive the density equations for the filtering processes.

  Recently, comparison with branching processes has been used to determine the asymptotic behaviour of the diameter (largest graph distance between two points that are in the same component) of various sparse random graph models, giving results for $G(n,c/n)$ as special cases. In ongoing work with Nick Wormald, we have studied $G(n,c/n)$ directly, obtaining much stronger results for this simpler model.  

 

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

 

  The talk will be self-contained and does not require specific knowledge on the Ising model. I will present the basic results concerning the Wulff crystal of the Ising model and I will study its behaviour near the critical point. Finally I will show how to apply these results to the problem of image segmentation.  

  We present a link between polymer pinning by a columnar defect in a random medium and a particular model of interacting particles on the line, related to polynuclear growth. While the question of whether an arbitrarily small intensity for the defect always results in pinning is still open, in a 'randomized' version of the model, which is closely related to the zero-temperature Glauber dynamics of the Ising model, we are able to obtain explicit results and a complete understanding of the process. This is joint work with Vladas Sidoravicius and Maria Eulalia Vares.  

  We start from the stochastic Ising model(or Glauber Dynamics) and have a short review of some important topics in Particle Systems such as ergodicity, convergence rates and so on. Then an abstract nonlinear model will be introduced by an evolution differential equation. We will build the existence and uniqueness theorem, and give some nice properties such as convergence exponentially and monotonicity for the abstract systems. To apply our abstract theory, we will study a family of nonlinear interacting particle systems generalized from Glauber Type Dynamics(we call them nonlinear Glauber Type Dynamics) and prove that such generalization can be done in infinitely many ways. For nonlinear Glauber Type Dynamics, we have two interesting inequalities related to Gibbs measures. Finally, we will concentrate on one specific nonlinear dynamics, and provide the relation between nonlinear system and the linear one, and that between Gibbs measures and tangent functionals to a nonlinear transfer operator.

 

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