Summary

Volume 1 was on establishing terminology and review of fundamental definitions from information theory, geometry, and probability theory on Euclidiean space, Volume 2 has focused on analogous concepts in the setting of Lie groups. A survey of problems that simultaneously involve Lie groups and information theory was provided, including the encoding/decoding of spatial pose (position and orientation). The physics that govern different kinds of communication systems gives rise to SDEs and their corresponding Fokker–Planck equations. In some instances, such as laser phase noise, these can be viewed as a probability flows on a group manifold. In other instances, such as the telegraph equation, Lie groups describe the symmetries of a PDE on Euclidean space. Stochastic models of phenomena such as the conformational fluctuations of DNA and the motions of robotic systems were examined. These lead to probability densities on the group of rigid-body motions, and properties of the corresponding conformational and parts entropy were studied. Numerical tools for solving Fokker–Planck equations on Lie groups such as the rotation group and group of rigid-body motions were reviewed.