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

1981; Mathematical Sciences Publishers; Volume: 94; Issue: 2 Linguagem: Inglês

10.2140/pjm.1981.94.341

1945-5844

Abel Klein, Lawrence J. Landau,

Quantum chaos and dynamical systems

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.!• Introduction.Certain stochastic processes are associated with quantum systems (e.g., Nelson [11, 12], Simon [14], Hoegh-Krohn [3], Albeverio and Hoegh-Krohn [1], Klein [5, 6], Driessler, Landau and Perez [2]).If the quantum system is at a nonzero temperature T then the associated stochastic process is periodic with period equal to the inverse temperature β = (&T)" 1 , where k is Boltzmann's constant (Hoegh-Krohn [3], Albeverio and Hoegh-Krohn [l], Driessler, Landau and Perez [2]).The simplest example is the Ornstein-Uhlenbeck process which is associated with the quantum mechanical harmonic oscillator.The Gaussian process X(t) indexed by the real line with mean zero and covariance E(X(t)X(s)) = (2m)" 1 exp( -\t -s\m) with m > 0 (the usual Ornstein-Uhlenbeck process) is associated with the one-dimensional harmonic oscillator with frequency m (i.e., with Hamiltonian H -Ij2(-d 2 /dx 2 + mVm)) at zero temperature (e.g., Simon [14]).If this harmonic oscillator is considered at a nonzero temperature T, then the associated stochastic process is the periodic Gaussian process X β (t) indexed by the real line with period β = (kT)-1 having mean zero and covariancefor \t -s\ £ β (Hoegh-Krohn [3]).We will call X β (t) the periodic Jo x (1 -exp(-/3i/-Λ + m*))]-ι [exp(-|ί -s\V-Δ + m 2 ) + exp(-(/3 -\t -s\)V-Δ + PERIODIC GAUSSIAN OSTERWALDER-SCHRADER POSITIVE PROCESSES 343for \t -s\ ^ β (at T > 0 we also needΓlnm-^m 2 ) < °o if d = 2, Γ m-'dpim 2 ) < oo if d = 1).