Entropic Measure on Multidimensional Spaces

Abstract

We construct the entropic measure ${\mathbb{P}}^{\beta }$ on compact manifolds of any dimension. It is defined as the push forward of the Dirichlet process (another random probability measure, well-known to exist on spaces of any dimension) under the {\em conjugation map} $$\Conj:\mathcal{P}(M)\to\mathcal{P}(M).$$ This conjugation map is a continuous involution. It can be regarded as the canonical extension to higher dimensional spaces of a map between probability measures on 1-dimensional spaces characterized by the fact that the distribution functions of $\mu$ and $\Conj(μ)$ are inverse to each other. We also present an heuristic interpretation of the entropic measure as $$d{\mathbb{P}}^{\beta }(\mu )=\frac{1}{Z}\exp (-\beta \cdot Ent(\mu |m))\cdot d{\mathbb{P}}^0(\mu ).$$

17 pages, 6 figures