Periodic Gaussian Osterwalder-Schrader positive processes and the two-sided Markov property on the circle

Abel Klein and Lawrence J. Landau

Vol. 94 (1981), No. 2, 341–367

DOI: 10.2140/pjm.1981.94.341

Abstract

Gaussian processes in a class of stochastic processes associated with quantum systems at nonzero temperature (the periodic stochastic processes satisfying Osterwalder-Schrader (OS) positivity on the circle) are studied. A representation of the covariance function of a periodic Gaussian OS-positive process is obtained which gives a complete description of all such processes. The two-sided Markov property on the circle is studied and it is determined which periodic Gaussian OS-positive processes satisfy the two-sided Markov property on the circle. It is shown that every periodic Gaussian OS-positive process is the restriction of a periodic Gaussian two-sided Markov process. For nonperiodic Gaussian OS-positive processes it is shown that the two-sided Markov property is equivalent to the Markov property.

Mathematical Subject Classification 2000

Primary: 60G15

Secondary: 60J25

Milestones

Received: 29 November 1979

Published: 1 June 1981