ADELIC PATH SPACE INTEGRALS

A framework for the study of path integrals on adelic spaces is developed, and it is shown that a family of path space measures on the localizations of an algebraic number held may, under certain conditions, be combined to form a global path space measure on its adele ring. An operator on the field of p-adic numbers analogous to the harmonic oscillator operator is then analyzed, and used to construct an Ornstein-Uhlenbeck type process on the adele ring of the rationals.