Measures not charging polar sets and Schrödinger equations in L p

Abstract

We study the Schrodinger equation (q − ℒ)u + µu = f, where ℒ is the generator of a Borel right process and µ is a signed measure on the state space. We prove the existence and uniqueness results in Lp, 1 ⩽ p < ∞. Since we consider measures µ charging no polar set, we have to use new tools: the Revuz formula with fine versions and the appropriate Revuz correspondence, the perturbation (subordination) operators (in the sense of G. Mokobodzki) induced by the regular strongly supermedian kernels. We extend the results on the Schrodinger equation to the case of a strongly continuous sub-Markovian resolvent of contractions on Lp. If the measure µ is positive then the perturbed process solves the martingale problem for ℒ − µ and its transition semigroup is given by the Feynman-Kac formula associated with the left continuous additive functional having µ as Revuz measure.