  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.
