Probability measures in infinite dimensional spaces: random paths, random fields and random geometry

点击翻译按钮获取中文摘要

Infinite dimensional probability measures have been introduced and studied in various contexts, from the classical theory of diffusion processes and diffusion equations, to the more recent probabilistic models in geometry and statistical physics. This research program aims to explore some problems arising from these contexts. Gaussian free fields (GFF), which can be viewed as certain function (or distribution) spaces equipped with Gaussian measures, have played important roles in recent developments of various fields, such as the quantum field theory, statistical mechanics, conformal geometry and etc. I will continue studying geometrical properties of GFF in higher dimensions, and extend part of the existing literature on log-correlated GFF (i.e., the covariance of the GFF has log singularity in spatial variables), including both the theories and the techniques, to a more general class of Gaussian random fields. I will analyze various aspects of the geometry associated with these random fields, and explore potential applications of random geometry models in other subjects. On one hand, there are interesting problems arising from the comparison between the classical theory and its counterpart under the randomization; on the other hand, the study of random models provides new perspectives on problems in the classical setting.A rigorous mathematical interpretation of GFF is provided by the theory of Abstract Wiener Space (AWS), which concerns constructions and properties of general infinite dimensional Gaussian measures. I will review the structural properties of AWS through studying measure theoretical extensions of various functional equations on AWS. I hope to systematically identify the probabilistic adaptations of the functional equations which characterize homogeneous chaos on AWS. Potential findings could shed light on the fine structural features of AWS, provide further details of possible supporting spaces of the Gaussian measures, and reflect the distinct natures of the infinite dimensional Gaussian measures from those in finite dimensions.The AWS is the abstraction of the classical Wiener space, the space of the Brownian motion, which has long been the basis of the study of diffusion processes and more generally, probabilistic methods in partial differential equations. The Wright-Fisher (WF) equation - a diffusion equation degenerate at the boundary of a simplex domain - is commonly used in the population genetics to model the prevalence of certain alleles. I plan to use a probabilistic method to study the higher-dimensional WF equation, aiming at providing rigorous estimates for the fundamental solution near the boundary/corners. These estimates could be useful in simulating the solutions in practice. This work will also lead to studying in general the impact of the degeneracy at the boundary/corners on the regularity of the fundamental solution in higher dimensions.