Foundations of Stochastic Analysis (11w5077)

Objectives

Stochastic analysis has a double nature. At one end, it has branched into a number of specific fields, often inspired by scientific applications. At the other end, it consists of a collection of methods underpinning all other uses and developments in the field.

Some of the best known and most active specific areas of stochastic analysis include the Schramm-L"owner Evolution (SLE), random matrices, stochastic partial differential equations (SPDE), rough path theory and Black-Scholes theory in mathematical finance.

Over the years, the foundations of stochastic analysis included various specific topics, such as the general theory of Markov processes, the general theory of stochastic integration, the theory of martingales, Malliavin calculus, the martingale-problem approach to Markov processes, and the Dirichlet form approach to Markov processes.

To create some focus for the very broad topic of the conference, we chose a few areas of concentration, including

$bullet$ Dirichlet forms

$bullet$ Analysis on fractals and percolation clusters

$bullet$ Jump type processes

Dirichlet form theory provides a powerful tool that connects the probabilistic potential theory and analytic potential theory. Recently Dirichlet forms found its use in effective study of fine
properties of Markov processes on spaces with minimal smoothness, such as reflecting Brownian motion on non-smooth domains, Brownian motion and jump type processes on Euclidean spaces and fractals, and Markov processes on trees and graphs. It has been shown that Dirichlet form theory is an important tool in study of various invariance principles, such as the invariance principle for reflected Brownian motion on domains with non necessarily smooth boundaries and the invariance principle for Metropolis algorithm. Dirichlet form theory can also be used to study a certain type of SPDEs.

Fractals are used as an approximation of disordered media. The analysis on fractals is motivated by the desire to understand properties of natural phenomena such as polymers, and growth of molds and crystals. By definition, fractals are mathematical objects that are very rough and lack smoothness, so one can not use the standard
analytic methods that were developed for Euclidean spaces and for manifolds. It turns out that Dirichlet form theory is well suited for
studying fractals---significant progress has been made in this area in the last fifteen years. Detailed study of heat kernel estimates and parabolic Harnack principle on fractals require techniques both from probability and analysis. Stability of such estimates under perturbations of operators and spaces can be proved by translating the problem into some analytic and geometric conditions. Such equivalent conditions are often obtained in the framework of graphs and general metric measure spaces, and Dirichlet forms are one of the key tools for the analysis. As an example, we mention that this approach turned out to be very useful in the analysis of random walks on random media such as percolation clusters. We believe that Dirichlet forms can play a more important role in studying scaling limits of nearest neighbor random walk and long range random walk on percolation clusters---this is an example of a concrete research project to be discussed at the conference.

In recent years there was an explosion of activity in the area of jump type processes. There are diverse reasons for this increased
interest in the area. One is that many physical and economic systems are best modeled by discontinuous Markov process. On the theoretical
side, jump type processes provide a hard but elegant challenge for mathematical methods, because the infinitesimal generators of jump
type Markov processes are non-local operators. Research on Markov processes generated many new results for non-local operators and for pseudo-differential operators as well, such as heat kernel estimates, parabolic Harnack principle and a priori H"older estimates of parabolic functions. These new directions in the development of the de Giorgi-Moser-Nash theory for non-local operators made it possible to give general criteria for convergence of Markov chains with jumps, and they also provided a new approach to long-range random walk in random media.

The topics described above will serve as catalysts for the discussions but they are not intended to limit the scope of the conference. We believe that the invited participants proposed below will bring their own perspective on the foundations of stochastic analysis and help trigger activity in various exciting areas.