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Reweighting metric measure spaces and Onsager-Machlup

Zachary Selk

TL;DR

Derives a joint transformation formula for the Onsager–Machlup functional under simultaneous reweighting of both metric and measure on geodesic metric measure spaces. It shows that with a constant conformal factor $U$, $\\operatorname{OM}$ shifts by $V$; under a small-ball asymptotics $\\lim_{r\to0^+}\frac{\\mu_0(B_0(Cr,q))}{\\mu_0(B_0(r,q))}=C^p$, the shift is $-pU+V$, while nonconstant $U$ with a small-ball estimate can render $\\operatorname{OM}$ ill-defined. The results include corollaries such as a metric-based density-tuning construction, invariance under conformal metric changes when OM is defined, and a uniformization metric that forces $\\operatorname{OM}(x)=0$ for all $x$, with cartogram intuition highlighting geometry-driven normalization in infinite dimensions.

Abstract

Given a metric measure space $M:=(X,d,μ)$ the Onsager-Machlup (OM) functional is a real valued function that has been seen as a generalized notion of a probability density function. The effect of reweighting the measure on OM functionals has been studied, however analogous reweightings of the metric to the best of our knowledge remain open. In this short note, we prove a transformation formula for OM functionals on geodesic metric measure spaces under reweighting of both the metric and the measure.

Reweighting metric measure spaces and Onsager-Machlup

TL;DR

Derives a joint transformation formula for the Onsager–Machlup functional under simultaneous reweighting of both metric and measure on geodesic metric measure spaces. It shows that with a constant conformal factor , shifts by ; under a small-ball asymptotics , the shift is , while nonconstant with a small-ball estimate can render ill-defined. The results include corollaries such as a metric-based density-tuning construction, invariance under conformal metric changes when OM is defined, and a uniformization metric that forces for all , with cartogram intuition highlighting geometry-driven normalization in infinite dimensions.

Abstract

Given a metric measure space the Onsager-Machlup (OM) functional is a real valued function that has been seen as a generalized notion of a probability density function. The effect of reweighting the measure on OM functionals has been studied, however analogous reweightings of the metric to the best of our knowledge remain open. In this short note, we prove a transformation formula for OM functionals on geodesic metric measure spaces under reweighting of both the metric and the measure.

Paper Structure

This paper contains 3 sections, 5 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of main theorem
  3. Corollaries and discussion

Key Result

Theorem 1.2

Let $M_0=(X,d,\mu_0)$ be a metric measure space with OM function $\operatorname{OM}_0$ on $Z\subseteq X$. Suppose that $\mu=e^{-V}\mu_0$ for a locally uniformly continuous function $V:X\to \mathbb R$. Then the OM function of $M=(X,d,e^{-V}\mu_0)$ on $Z$ is $\operatorname{OM}=\operatorname{OM}_0+V$.

Figures (1)

  • Figure 1: A cartogram of Earth, reweighted by population, taken from Cartogram. By distorting the geometry of Earth, we can make population density uniform. This can be seen as analogous to Corollary \ref{['cor:cart']}.

Theorems & Definitions (10)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.4
  • proof
  • Corollary 3.1
  • Corollary 3.2
  • Corollary 3.3
  • proof