An Extended Generator and Schrödinger Equations

Abstract

The generator of a Borel right processis extended so that it maps functions to smooth measures. This extension may be defined either probabilistically using martingales or analytically in terms of certain kernels on the state space of the process. Then the associated Schrödinger equation with a (signed) measure serving as potential may be interpreted as an equation between measures. In this context general existence and uniqueness theorems for solutions are established. These are then specialized to obtain more concrete results in special situations.