Probabilistic representations and hyperbound estimates for semigroups

In this paper, we study lower order perturbations of a symmetric second-ord er differential operator generating a hypercontractive semigroup. We give a probabilistic representation of the, in general, not sub-Markovian semigro up associated with the perturbed operator and prove that the perturbed semi group is also hypercontractive under some exponential integrability conditi ons on the coefficients.