Areas on the space of smooth probability density functions on $S^2$
This work addresses a mathematical problem in geometric mechanics and probability theory, but it appears incremental as it extends existing methods to specific manifolds.
The authors tackled the problem of computing Poisson brackets and symplectic areas for smooth probability density functions on spaces like the 2-sphere, developing symbolic and numerical methods and applying them to an explicit Gaussian example.
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.