Past Stochastic Analysis Seminar

The classical maximum principle for optimal control of solutions of stochastic differential equations (developed by Pontryagin (deterministic case), Bismut, Bensoussan, Haussmann and others), assumes that the system is Markovian and that the controller has access to full, updated information about the system at all times. The classical solution method involves an adjoint process defined as the solution of a backward stochastic differential equation, which is often difficult to solve.

We apply Malliavin calculus for Lévy processes to obtain a generalized maximum principle valid for non-Markovian systems and with (possibly) only partial information available for the controller. The backward stochastic differential equation is replaced by expressions involving the Malliavin derivatives of the quantities of the system.

The results are illustrated by some applications to finance

This talk gives a survey on a series of work which I and co-authors have been doing for 10 years. I will start from the Feynman-Kac type formula for Dirichlet forms. Then a necessary and sufficient condition is given to characterize the killing transform of Markov processes. Lastly we shall discuss the regular subspaces of linear transform and answer some problems related to the Feynman-Kac formula

Counting paths, or walks, is an important ingredient in the classical representation theory of compact groups. Using Brownian paths gives a new flexible and intuitive approach, which allows to extend some of this theory to the non- cristallographic case. This is joint work with P. Biane and N. O'Connell

The Bakry Emery criterion asserts that a probability measure with a strictly positive generalized curvature satisfies a logarithmic Sobolev inequality, and by results of Bakry and Ledoux an isoperimetric inequality of Gaussian type. These results were complemented by a theorem of Wang: if the curvature is bounded from below by a negative number, then under an additional Gaussian integrability assumption, the log-Sobolev inequality is still valid.

The goal of this joint work with A. Kolesnikov is to provide an extension of Wang's theorem to other integrability assumptions. Our results also encompass a theorem of Bobkov on log-concave measures on normed spaces and allows us to deal with non-convex potentials when the convexity defect is balanced by integrability conditions. The arguments rely on optimal transportation and its connection to the entropy functional

Some recent joint results with N. V. Krylov on the convergence of solutions of finite difference schemes are presented.

The finite difference schemes, considered in the talk correspond to discretizations (in the space variable) of second order parabolic and of second order elliptic (possibly degenerate) equations.

Space derivatives of the solutions to the finite difference schemes are estimated, and these estimates are applied to show that the convergence of finite difference approximations for equations in the whole space can be accelerated to any given rate. This result can be applied to stochastic PDEs, in particular to the Zakai equation of nonlinear filtering, when the signal and observation noises are independent.

Some recent joint results with N. V. Krylov on the convergence of solutions of finite difference schemes are presented.

The finite difference schemes, considered in the talk correspond to discretizations (in the space variable) of second order parabolic and of second order elliptic (possibly degenerate) equations.

Space derivatives of the solutions to the finite difference schemes are estimated, and these estimates are applied to show that the convergence of finite difference approximations for equations in the whole space can be accelerated to any given rate. This result can be applied to stochastic PDEs, in particular to the Zakai equation of nonlinear filtering, when the signal and observation noises are independent.

This is a joint work with Bruno Schapira, and it is a work in progress.

We study recurrence and transience properties of some edge reinforced random walks on the integers: the probability to go from $x$ to $x+1$ at time $n$ is equal to $f(\alpha_n^x)$ where $\alpha_n^x=\frac{1+\sum_{k=1}^n 1_{(X_{k-1},X_k)=(x,x+1)}}{2+\sum_{k=1}^n 1_{X_{k-1}=x}}$. Depending on the shape of $f$, we give some sufficient criteria for recurrence or transience of these walks

Born in France around 1780, the optimal transport problem has known a scientific explosion in the past two decades, in relation with dynamical systems and partial differential equations. Recently it has found unexpected applications in Riemannian geometry, in particular the encoding of Ricci curvature bounds

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 $\beta$.

APOLOGIES - this seminar is cancelled.

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

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.

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.