Mathematicians are building a new framework to study random fields—the unpredictable, wiggly patterns seen in everything from quantum particle interactions to financial markets—by rewriting the rules of stochastic analysis. Current mathematical tools struggle with the roughest, most irregular random fields, particularly those that appear in quantum field theory, where fields are so erratic they can only be described as "distribution-valued" (like a spike that has no well-defined height at any single point). This project tackles that gap by developing a unified stochastic approach that treats these fields as transformations of simpler Gaussian randomness, guided by stochastic differential equations. The work bridges analysis, probability, and geometry. If successful, the research will give mathematicians and physicists a common language to solve long-standing problems in quantum gauge theories, supersymmetric models, and the two-dimensional sigma model. It could also forge unexpected connections to homogenization theory, geometric PDEs, optimal control, and the renormalization group—tools used across physics and engineering. This is fundamental science: there is no immediate practical application, but past advances in stochastic analysis have underpinned everything from weather forecasting to option pricing. A deeper understanding of random fields could eventually improve models of turbulent flow, quantum materials, or complex networks.
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The goal of this project is to develop a fully fledged stochastic analysis of Euclidean quantum field theories (QFTs). I will use it to generate progress towards a deep mathematical understanding of models like the two dimensional s-model, Euclidean quantum gauge theories and mixed Euclidean fermion/boson models or supersymmetric theories. I intend also to explore the use of stochastic methods in Minkowski quantum field theory. Stochastic analysis is here understood as a general approach to the study of random fields realized as push-forwards of suitable Gaussian reference measures under pathwise transformations given by stochastic (partial) differential equations. It extends in radically new ways the original ideas of Ito^ on Markovian diffusions to multidimensional local, nonlinear random fields whose sample paths are not continuous but only distribution-valued. The development of such stochastic analysis requires a tight interplay of analysis, probability and geometry. Euclidean QFTs provide a rich source of models which will be studied systematically with these new tools. Firstly, in order to test the new methods, by reproducing existing results in the language of stochastic analysis more familiar to probabilists. But more importantly, to attack new problems, spur new research directions and connect more tightly and organically with other branches of analysis, geometry and probability: e.g. homogenization theory, the theory of geometric PDEs, the geometry of fiber bundles, the theory of non commutative probability, the renormalization group, the theory of optimal control and the related functional inequalities. Euclidean QFTs are chosen as privileged subject of analysis both from their constructive interest in mathematical physics and because they are natural probabilistic objects which provide sources of new and deep concepts, ideas and tools of wide applicability in various other branches of mathematics.
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