Areas on the space of smooth probability density functions on $S^2$

TL;DR

This paper develops symbolic and numerical techniques to compute Poisson brackets on spaces of positive density measures on surfaces like the plane, torus, and sphere, enabling explicit calculations of symplectic areas.

Contribution

It introduces new methods for computing Poisson brackets on measure spaces on surfaces, with explicit applications to the 2-sphere and Gaussian measures.

Findings

Computed symplectic areas of finite regions on the 2-sphere.

Provided explicit examples for Gaussian measures.

Demonstrated effectiveness of symbolic and numerical methods.

Abstract

We present symbolic and numerical methods for computing Poisson brackets on the spaces of measures with positive densities of the plane, the 2-torus, and the 2-sphere. We apply our methods to compute symplectic areas of finite regions for the case of the 2-sphere, including an explicit example for Gaussian measures with positive densities.