Stochastic Partial Differential Equations, Gauge Theories, and Scaling Limits

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A core problem of probability is to extract important laws and information from complex and large random systems. Such random systems may be stocks with fluctuating prices in financial markets, cancer cells generated by random mutations in biological tissues, a large number of sub-atomic quantum interaction particles, or how diseases randomly spread among people. This project focuses on two types of models: motion of a large number of elementary particles under the interaction of random quantum fields, and the growth of polymers under random interference environments. Due to the generality of mathematical models, the research results of these two types of models will also be of great help in studying other systems, such as systems in materials science, finance, and biology. The main mathematical tool to be used is called stochastic partial differential equations.More specifically, the theory of stochastic partial differential equations will be used to study stochastic quantization of gauge theories, which are fundamental models in elementary particle physics. The PI will prove well-posedness of such singular equations, show that the solutions have infinite dimensional gauge symmetries, and expand the scope of regularity structure theory. The PI will also study stochastic partial differential equation scaling limits of discrete probabilistic models from statistical physics, such as directed polymers in an environment of interacting particle systems. This will allow to demonstrate universality of these stochastic partial differential equations. The main methods and techniques are the new theory of stochastic partial differential equations such as regularity structures, and their discrete versions. The study brings together tools from different areas, such as stochastic analysis, partial differential equations, quantum field theory, and statistical mechanics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.