Wick Renormalized Parabolic Stochastic Quantization Equations on Rough Metric Measure Spaces

TL;DR

This paper establishes conditions for solving Wick renormalized stochastic quantization equations on rough metric measure spaces, enabling rigorous analysis in non-integer dimensions relevant to quantum field theory.

Contribution

It provides new solvability criteria and constructs invariant measures for stochastic equations on fractal and rough spaces, expanding the mathematical framework for quantum field models.

Findings

Derived sufficient conditions based on Hausdorff and walk dimensions for local solutions.

Established global solution conditions and constructed invariant measures.

Applied results to fractals and product spaces, opening avenues in quantum physics.

Abstract

On metric measure spaces with sub-Gaussian heat kernel behavior in small time, we obtain a sufficient condition to solve Wick renormalized stochastic quantization equations with polynomial interaction. Given the power of the nonlinearity, the local solution condition depends on the Hausdorff dimension $d_h$, the walk dimension $d_w$, and the maximal spatial H\"older regularity of the heat kernel $\Theta$. A slightly more restrictive condition based on the same parameters is required for a global solution. For all global solutions, we construct an invariant measure for the Markov process defined by the solution. Our results apply to many rough spaces such as Barlow--Kigami type fractals as well as their Cartesian products and open up the possibility of making rigorous various structures in quantum field theory and statistical mechanics in non-integer dimensions. In the process, we build entirely from the short-time heat semigroup the necessary analytic framework that accommodates the issues which come with allowing rough local geometry.