Krein's spectral theory and the Paley-Wiener expansion for fractional Brownian motion

In this paper we develop the spectral theory of the fractional Brownian motion (fBm) using the ideas of Krein's work on continuous analogous of orthogonal polynomials on the unit circle. We exhibit the functions which are orthogonal with respect to the spectral measure of the fBm and obtain an explicit reproducing kernel in the frequency domain. We use these results to derive an extension of the classical Paley-Wiener expansion of the ordinary Brownian motion to the fractional case.Comment: Published at http://dx.doi.org/10.1214/009117904000000955 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A generalized notion of weak interpretability and the corresponding modal logic

AbstractDzhaparidze, G., A generalized notion of weak interpretability and the corresponding modal logic, Annals of Pure and Applied Logic 61 (1993) 113-160. A tree Tr(T1,...,Tn) of theories T1,...,Tn is called tolerant, if there are consistent extensions T+1,...,T+n of T1,...,Tn, where each T+i interprets its successors in the tree Tr(T+1,...,T+n). We consider a propositional language with the following modal formation rule: if Tr is a (finite) tree of formulas, then ♢Tr is a formula, and axiomatically define in this language the decidable logics TLR and TLRω. It is proved that TLR (resp. TLRω) yields exactly the schemata of PA-provable (resp. true) sentences, if ♢Tr(A1,...,An) is understood as (a formalization of) “Tr(PA + A1,...,PA + An) is tolerant”. In fact, TLR axiomatizes a considerable fragment of provability logic with quantifiers over ∑1-sentences, and many relations that have been studied in the literature can be expressed in terms of tolerance. We introduce and study two more relations between theories: cointerpretability and cotolerance which are, in a sense, dual to interpretability and tolerance. Cointerpretability is a characterization of ∑1-conservativity for essentially reflexive theories in terms of translations

Spin Supersolid in Anisotropic Spin-One Heisenberg Chain

We consider an S=1 Heisenberg chain with strong exchange (Delta) and single--ion uniaxial anisotropy (D) in a magnetic field (B) along the symmetry axis. The low energy spectrum is described by an effective S=1/2 XXZ model that acts on two different low energy sectors for a given window of fields. The vacuum of each sector exhibits Ising-like antiferromagnetic ordering that coexists with the finite spin stiffness obtained from the exact solution of the effective XXZ model. In this way, we demonstrate the existence of a spin supersolid phase. We also compute the full Delta-B quantum phase diagram by means of a quantum Monte Carlo simulation.Comment: 4+ pages, 2 fig

Representations of isotropic random fields with homogeneous increments, with applications to spacial fractional Brownian motion

This is a brief account of the current work by Dzhaparidze, van Zanten and Zareba, delivered as a lecture note at the conference Small deviations and related topics II held in St. Petersburg, September 12-19, 200

An optimal series expansion of the multiparameter fractional Brownian motion

We derive a series expansion for the multiparameter fractional Brownian motion. The derived expansion is proven to be rate optimal.Comment: 21 pages, no figures, final version, to appear in Journal of Theoretical Probabilit