Oxford-Man Institute

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.

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

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.

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.

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

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 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)

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.

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.