Stochastic Analysis and its Applications (17w5119)

Arriving in Banff, Alberta Sunday, October 22 and departing Friday October 27, 2017


(University of Washington)

(University of British Columbia)

Jeremy Quastel (University of Toronto)


Stochastic analysis, at one end, 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 stochastic partial differential equations (SPDE), Gaussian free fields and Schramm-L\"owner Evolution (SLE), random matrices, rough path theory and Black-Scholes theory in mathematical finance.

Over the years, the stochastic analysis and its applications included various specific topics, such as the general theory of Markov processes, branching and measure-valued 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

$\bullet$ Stochastic partial differential equations

$\bullet$ Dirichlet forms and stochastic analysis on metric measure spaces

$\bullet$ Stochastic processes on fractal sets and in random environment

$\bullet$ Jump type processes

Many natural, mechanical, chemical, and economic systems and processes can be described at the macroscopic level by a set of partial differential equations (PDEs) governing averaged quantities such as density, temperature, concentration, velocity, and so on. As Bernhard Riemann put it ``partial differential equations are the basis of all physical systems". However, in reality, the majority of physical systems are subject to the influence of internal, external or environmental noise. This calls for the study of PDEs with random coefficients and with random source terms. These PDEs are called ``stochastic'' partial differential equation (SPDEs). SPDEs also arise in research on deterministic models with random initial conditions, or as tractable approximations to complex deterministic systems. In many cases the presence of noise leads to new phenomena, not present in deterministic models---this sheds new light on various phenomena in physics, biology and economy. SPDEs have been an active research area in applied mathematics and engineering for many years. SPDEs are a rich and challenging field of research. It can be difficult to construct solutions, prove robustness of approximation schemes, and study properties like ergodicity and fluctuation statistics for most of the interesting and applicable SPDEs. A major difficulty encountered when dealing with stochastic PDEs is their lack of regularity. Recently, the field attracted a lot of attention in the mathematical community. One of the reasons is the spectacular progress in the field. Last year Martin Hairer was awarded a Fields Medal for his pioneering work on the theory of regularity structures for SPDEs and, in particular, for the study of the KPZ equation, a non-linear SPDE that can be used to describe random interface growth in physics. SPDE is a young and promising research direction. Given the recent advances, one can confidently predict that the field is going to flourish in the coming years and will become a significant research area in mathematics and mathematical physics with many important applications.

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.

We will also use the occasion of this conference to celebrate Krzysztof Burdzy' 60th birthday, who has made many important contributions to stochastic analysis and related fields. We expect this will help ensure a very high acceptance rate from the people we intend to invite. In fact, there are already 36 potential participants expressed interest to attend this workshop.