  On the Euclidean space, we establish some Weighted Logarithmic Sobolev (WLS)
inequalities. We characterize a symmetry range in which optimal functions are
radially symmetric, and a symmetry breaking range. (WLS) inequalities are a
limit case for a family of subcritical Caffarelli-Kohn-Nirenberg (CKN)
inequalities with similar symmetry properties. A generalized carr\'e du champ
method applies not only to the optimal solution of the nonlinear elliptic
Euler-Lagrange equation and proves a rigidity result as for (CKN) inequalities,
but also to entropy type estimates, with the full strength of the carr\'e du
champ method in a parabolic setting. This is a significant improvement on known
results for (CKN). Finally, we briefly sketch some consequences of our results
for the weighted diffusion flow.
