Path Integration: Connecting Pure Jump and Wiener Processes

Abstract

The Feynman path integral is known to be a powerful tool in different domains of physics and various mathematical approaches to its construction were developed (see e.g. extensive reviews of the recent literature in [ABB], [SS], [K4], [K5]). However, most of them work only for very restrictive class of potentials. Moreover, they are often defined not as genuine integrals, but as some generalized functionals specified by some limiting procedure. In [K2], [K3] the author proposed a representation of the solutions to the Schrodinger equation in terms of the well defined infinite dimensional Feynman integral defined as a genuine integral over a bona fide σ-additive measure on an appropriate space of trajectories (usually the Cameron- Martin space). Thi s const ruct ion covers very general equations. In [K5] it is extend ed to the Schrodinger equations with magnetic fields with even singular vector potentials defined as Radon measures. The construction uses the idea of the regularization by means of the introduction of continuous quantum observations or complex times and extends the approach of Maslov-Chebotarev (see [MCh]) which was based on the pure jump processes that appear naturally in th e momentum representation of the Schrodinger equation, whose potential can be presented as a Fourier transform of a finite complex measure (Ito’s complex measure condition).