StochFields

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.