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We consider problems of integrating over the Bogoliubov measure in the space of continuous functions and obtain asymptotic formulas for one class of Laplace-type functional integrals with respect to the Bogoliubov measure. We also prove related asymptotic results concerning large deviations for the Bogoliubov measure. For the basic functional, we take the L p norm and establish that the Bogoliubov trajectories are Hölder-continuous of order γ < 1/2.

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…(1.22) Proposition 2 [3]. The distribution of the process η(t), t ∈ [0, β], coincides with the distribution of the Wiener process w(t), t ∈ [0, β].…”

Section: Remarkmentioning

confidence: 94%

See 1 more Smart Citation

Some asymptotic formulas for the Bogoliubov gaussian measure

Fatalov

2008

“…Investigating the functional integrals introduced in [18], [19], Sankovich proved [20]- [23] that these functional integrals are taken with respect to the special Gaussian measure called the Bogoliubov Gaussian measure in [21]. We describe this measure following [20]- [22].…”

Section: Introduction: Formulating the Main Resultsmentioning

confidence: 99%

“…The relation between the Wiener and Bogoliubov measures was described in [22], [30]. The Bogoliubov process turned out to be in the class of Wiener-type Gaussian processes in the sense that its trajectories have the same degree of regularity as those of the Wiener process: they are Hölder continuous of the order γ < 1/2 [30] and do not satisfy the Hölder condition of order γ > 1/2 (see [22] and [24]).…”

Section: Introduction: Formulating the Main Resultsmentioning

confidence: 99%

Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure

Fatalov

2011

We obtain new asymptotic formulas for two classes of Laplace-type functional integrals with the Bogoliubov measure. The principal functionals are the L p functionals with 0 < p < ∞ and two functionals of the exact-upper-bound type. In particular, we prove theorems on the Laplace-type asymptotic behavior for the moments of the L p norm of the Bogoliubov Gaussian process when the moment order becomes infinitely large. We establish the existence of the threshold value p0 = 2 + 4π 2 /β 2 ω 2 , where β > 0 is the inverse temperature and ω > 0 is the harmonic oscillator eigenfrequency. We prove that the asymptotic behavior under investigation differs for 0 < p < p0 and p > p0. We obtain similar asymptotic results for large deviations for the Bogoliubov measure. We establish the scaling property of the Bogoliubov process, which allows reducing the number of independent parameters.

Asymptotics of small deviations of the Bogoliubov processes with respect to a quadratic norm

Pusev

2010

We obtain results on small deviations of Bogoliubov's Gaussian measure occurring in the theory of the statistical equilibrium of quantum systems. For some random processes related to Bogoliubov processes, we find the exact asymptotic probability of their small deviations with respect to a Hilbert norm.The Bogoliubov measure, introduced in [1], [2], plays an important role in the theory of the statistical equilibrium of quantum systems. It occurs in representing Gibbs equilibrium means of Bose operators in the form of functional integrals using Bogoliubov's method of T -products (see [3]).We define the Bogoliubov process ξ(t), t ∈ [0, 1], as a Gaussian process with a zero mean and the covarianceIt follows from the general results of the theory of random processes that trajectories of a Bogoliubov process are almost certainly (a.c.) continuous. From definition (1), we obtain E ξ(1) − ξ(0) 2 = 0 and hencealmost all the trajectories of a Bogoliubov process belong to the space C 0 [0, 1] of continuous functions x(t) defined on the segment [0, 1] and satisfying the condition x(0) = x(1). The distribution of the process ξ(t) in the space C 0 [0, 1] endowed with a uniform metric is called the Bogoliubov measure μ B . Properties of the Bogoliubov measure, of functional integrals over this measure, and of the Bogoliubov process trajectories were studied in [2], [4], [5]. Approximate formulas constructed in [2] for functional integrals over a Bogoliubov measure are exact for functional polynomials. It was established in [4]that similarly to the case of Wiener processes, almost all trajectories of a Bogoliubov process do not satisfy the Hölder condition with the index γ > 1/2 and are therefore nondifferentiable. Some asymptotic formulas for functional integrals with respect to a Bogoliubov measure were obtained in [5], where the asymptotic form of the probability of large deviations of Bogoliubov processes in L p -norms, which is important in many problems in probability theory and mathematical physics, was also derived.Another fundamental characteristic of random processes (used, e.g., to estimate the accuracy of approximations of random process and to calculate the metric entropy of different sets) is the asymptotic form of their small deviations in one norm or another. The theory of small deviations for the norm of Gaussian processes has been rapidly developing in the last decade (see, e.g., [6], [7]). A connection was recently established between small deviations and problems in mathematical statistics: functional analysis of data and nonparametric Bayes estimates [8]- [10].In the majority of cases, it is possible to find a rough (logarithmic) asymptotic expression for the probabilities of small deviations for Gaussian processes. Calculating the exact asymptotic form is a much more complicated problem, whose solution is known for only a few processes [11]-[16].