Regularity properties of Schrödinger and Dirichlet semigroups

First we compute Brownian motion expectations of some Kac's functionals. This allows a complete study of the semigroups generated by the formal differential operator $\text{H = ?}\text{1}\text{2}\text{Δ + V}$ on the various Lebesgue's spaces L^q=L^q$\text{R}$ⁿ, dx, whenever the negative part of V is in L^∞ + L^p for some $\text{p >}\text{max}\text{{1,}\text{n}\text{2}\text{}}$. Our approach is probabilistic and some of the proofs are surprisingly elementary. The negative infinitesimal generators of our semigroups are shown to be reasonable self-adjoint extensions of H. Under mild assumptions on V, H is unitary equivalent to the Dirichlet operator, say D, associated to its groundstate measure. We study regularity of the semigroups generated by D. We concentrate on hyper and supercontractivity and we give, using probabilistic techniques, new examples of potential functions V which give rise to hyper and supercontractive Dirichlet semigroups.