Stochastic Conformal Flows in Even Dimensions

TL;DR

This paper introduces two stochastic versions of the Q-curvature flow on even-dimensional manifolds, constructs their weak solutions using Dirichlet forms, and explores their volume dynamics and connections to stochastic quantization.

Contribution

It defines and analyzes stochastic analogs of Q-curvature flows, demonstrating their volume evolution as well-known stochastic processes and linking one to stochastic quantization of Polyakov-Liouville measures.

Findings

Volume of NQF evolves as a square Bessel process.

Volume of LQF evolves as a CIR process.

LQF can serve as a stochastic quantization of Polyakov-Liouville measures.

Abstract

We define two stochastic analogs of a geometric flow on even-dimensional manifolds called $Q$-curvature flow, and use the theory of Dirichlet forms to construct weak solutions to both. The first of these flows, which we call the normalized $Q$ flow (NQF), preserves the intrinsic volume normalization from the deterministic setting. The second, which we call the Liouville $Q$ flow (LQF), has a different normalization motivated by a similar flow studied in arXiv:1904.10909. The volume dynamics of NQF and LQF are shown to evolve as square Bessel and CIR processes, respectively. We also show that under certain additional conditions, LQF is a stochastic quantization of the even-dimensional Polyakov-Liouville measures recently defined in arXiv:2105.13925.