The main new results in this paper are contained in the geometric Theorems 1 and~2 of Section~0.1 below and they are related to previous results of M.~Gromov and of myself (\cf\ \cite{1},~\cite{2}). These results are used to prove some general potential theoretic estimates on Lie groups (\cf\ Section~0.3) that are related to my previous work in the area (\cf\ \cite{3},~\cite{4}) and to some deep recent work of G.~Alexopoulos (\cf\ \cite{5},~\cite{21}).

MSC Classifications:

22E30, 43A80, 60J60, 60J65 show english descriptions Analysis on real and complex Lie groups [See also 33C80, 43-XX]
Analysis on other specific Lie groups [See also 22Exx]
Diffusion processes [See also 58J65]
Brownian motion [See also 58J65] 22E30 - Analysis on real and complex Lie groups [See also 33C80, 43-XX]
43A80 - Analysis on other specific Lie groups [See also 22Exx]
60J60 - Diffusion processes [See also 58J65]
60J65 - Brownian motion [See also 58J65]

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