In the book "Stochastic Models, Information Theory, and Lie Groups, Volume 2" by Gregory S. Chirikjian, in constructing the Fokker-Planck equation on unimodular Lie groups the author considers a homogeneous process evolving on a group $G$, such that \begin{equation} \rho(g, t+\Delta t) = \rho(g,t) \ast \rho(g, \Delta t) = \int_G \rho(h,t) \rho(h^{-1} \circ g, \Delta t) dh\,, \end{equation} where $\rho(g,t)$ is a time-parametrized pdf on the group. The stochastic process is given by $$ d \mathbf{x} = \mathbf{h}(t) dt + H(t) d\mathbf{w}\,, $$ and the path $g(t) \in G$ is defined recursively as $$ g(t+dt) = g(t) \circ \exp \left( \sum_i dx_i E_i \right)\,, $$ with $g(0)$ being equal to the group identity and $E_i$ are the group generators.
What I fail to understand is why the assumption of homogeneity makes sense in this context.
Does homogeneity in group space follow from the stochastic process in consideration? Or is there some other way of justifying its use here?