Log-Harnack Inequality for Gruschin Type Semigroups
  By constructing a coupling in two steps and using the Girsanov theorem under a regular conditional probability, the log-Harnack inequality is established for a large class of Gruschin type semigroups whose generator might be both degenerate and non-Lipschitzian. 
Feng-Yu Wang and Lihu Xu
math.PR