L^2-Theory for non-symmetric Ornstein-Uhlenbeck semigroups on domains
  We present some new results on analytic Ornstein-Uhlenbeck semigroups and use them to extend recent work of Da Prato and Lunardi for Ornstein-Uhlenbeck semigroups on open domains O to the non-symmetric case. Denoting the generator of the semigroup by L_O, we obtain sufficient conditions in order that the domain Dom(\sqrt{-L_O}) be a first order Sobolev space. 
Joyce Assaad and Jan van Neerven
math.PR math.AP math.FA