Harnack and functional inequalities for generalized Mehler semigroups.
Let \(E\) be a real separable Hilbert space with a Borel \(\sigma\)-field \(L(E)\). Let \(T_t\), \(t\geq 0\), be a \(C_0\)-semigroup of bounded linear operators on \(E\). Let further \(\mu_t\), \(t\geq 0\), be a family of probability measures on \(E\) satisfying \[ \mu_{t+s}= (\mu_t\circ T^{-1}_s)* \mu_s,\quad t,s\geq 0. \] The generalized Mehler semigroup (also called generalized Ornstein--Uhlenbeck semigroup) associated with \(T_t\), \(\mu_t\) is defined by \[ P_tf(x)= \int_E f(T_tx+ y)\mu_t(dy), \text{ forall bounded }f\in L(E),\text{ and all }x\in E, \] (introduced by \textit{V. I. Bogachev, M. Röckner} and \textit{B. Schmuland} [Probab. Theory Relat. Fields 105, 193--225 (1996; Zbl 0849.60066)]). Under certain assumptions, the authors prove here Harnack type inequalities for such semigroups, which in particular imply upper bound estimates for the transition density. Poincaré and log-Sobolev inequalities are also proved in terms of estimates for the square field operators. The results are supported by concrete examples.
scientific article; zbMATH DE number 1992842
scientific article