Rayleigh–Type Renormalized Flows in Weighted Geometries Energy Dissipation, Invariance, and Applications to PDE Dynamics

This working paper develops a clean and self-contained framework for Rayleigh-type normalized flows in weighted geometries. The central idea is to encode heterogeneity directly at the differential level via a renormalized derivative, leading to a natural self-adjoint dissipation operator and a canonical normalization of the associated linear semigroup. The resulting flow preserves the L²(dμ) constraint, admits a sharp Lyapunov dissipation identity, and converges to eigenmodes under standard spectral gap assumptions. The emphasis is structural rather than problem-specific: the framework clarifies how weighted PDE dissipation, constrained gradient dynamics, and spectral extraction arise from a single geometric mechanism. This note is intended as a modular analytical tool for researchers working with weighted diffusion, spectral methods, or normalized evolution equations in heterogeneous media. It does not rely on probabilistic interpretations or optimal transport formulations, and can be read independently of other AGD-related works.