We show that the logarithmic derivatives of the convolution
heat kernels on a uni-modular Lie group are exponentially integrable.
This result is then used to prove an “integrated” Harnack
inequality for these heat kernels. It is shown that this integrated
Harnack inequality is equivalent to a version of Wang’s Harnack
inequality. (A key feature of all of these inequalities is that they
are dimension independent.) Finally, we show these inequalities
imply quasi-invariance properties of heat kernel measures for two
classes of infinite dimensional “Lie” groups.