  In this note, we derive a new logarithmic Sobolev inequality for the heat
kernel on the Heisenberg group. The proof is inspired from the historical
method of Leonard Gross with the Central Limit Theorem for a random walk. Here
the non commutative nature of the increments produces a new gradient which
naturally involves a Brownian bridge on the Heisenberg group. This new
inequality contains the optimal logarithmic Sobolev inequality for the Gaussian
distribution in two dimensions. We compare this new inequality with the
sub-elliptic logarithmic Sobolev inequality of Hong-Quan Li and with the more
recent inequality of Fabrice Baudoin and Nicola Garofalo obtained using a
generalized curvature criterion. Finally, we extend this inequality to the case
of homogeneous Carnot groups of rank two.
