Stochastic conformal flows in even dimensions

2025/06/01 by Jack Piazza, Piazza, Jack
Engineering · #FOS: Mathematics #Probability (math.PR) #Reservoir Engineering and Simulation Methods

paper · pdf · doi:10.48550/arxiv.2506.01217

openalex publication_date 2025/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Reconstruction of log-correlated fields from multiplicative chaos measures
• Conformally invariant random fields, quantum Liouville measures, and random Paneitz operators on Riemannian manifolds of even dimension
• Stochastic Ricci Flow on Compact Surfaces
• Decompositions of log-correlated fields with applications
• Liouville Quantum Gravity on the Riemann sphere
• On Gaussian multiplicative chaos
• A general convergence result for the Ricci flow in higher dimensions
• Global existence and convergence for a higher order flow in conformal geometry

Related

• Stochastic quantization of Liouville conformal field theory
• Stochastic Ricci Flow on Compact Surfaces
• The Prescribed Q-Curvature Flow for Arbitrary Even Dimension in a Critical Case
• The stochastic renormalized mean curvature flow for planar convex sets
• Some stochastic process without birth, linked to the mean curvature flow