Entropy Theory for Random Walks on Lie Groups

We develop entropy and variance results for the product of independent identically distributed random variables on Lie groups. Our results apply to the study of stationary measures in various contexts.

💡 Research Summary

The paper develops a comprehensive entropy‑theoretic framework for random walks on arbitrary real Lie groups. Starting from basic definitions, the authors treat both discrete and absolutely continuous probability measures on a Lie group (G) (with Haar measure (m_G)) and establish elementary entropy properties such as subadditivity under convolution and concavity‑based inequalities.

A central technical device is a family of smoothing distributions (s_{a,r}=\exp(\beta_{a,r})) where (\beta_{a,r}) has a compactly supported, approximately Gaussian density on the Lie algebra. For a (G)‑valued random variable (g) the “scale‑(r) entropy’’ is defined as
\