$L^p$-$L^q$ estimates for transition semigroups associated to dissipative stochastic systems

In a separable Hilbert space, we study supercontractivity and ultracontractivity properties for a transition semigroups associated with a stochastic partial differential equations. This is done in terms of exponential integrability of Lipschitz functions and some logarithmic Sobolev-type inequalities with respect to invariant measures. The abstract characterization results concerning the improving of summability can be applied to transition semigroups associated to a stochastic reaction-diffusion equations.