On the Schrodinger equation with potentials which are Laplace transforms of measures

We construct a pointwise solution for the time dependent Schrodinger equation on md with potentials and initial conditions which can grow exponentially at infinity and belong to the class of smooth Laplace transforms of complex measures on R-d. The methods used are both analytic and probabilistic and the result can be looked upon as an extension of rigorously defined Feynman path integrals to the case of potentials which can strongly grow at infinity. An appendix with the calculation of some Wiener integrals is also presented.