23,987 research outputs found
Zamolodchikov relations and Liouville hierarchy in SL(2,R)_k WZNW model
We study the connection between Zamolodchikov operator-valued relations in
Liouville field theory and in the SL(2,R)_k WZNW model. In particular, the
classical relations in SL(2,R)_k can be formulated as a classical Liouville
hierarchy in terms of the isotopic coordinates, and their covariance is easily
understood in the framework of the AdS_3/CFT_2 correspondence. Conversely, we
find a closed expression for the classical Liouville decoupling operators in
terms of the so called uniformizing Schwarzian operators and show that the
associated uniformizing parameter plays the same role as the isotopic
coordinates in SL(2,R)_k. The solutions of the j-th classical decoupling
equation in the WZNW model span a spin j reducible representation of SL(2,R).
Likewise, we show that in Liouville theory solutions of the classical
decoupling equations span spin j representations of SL(2,R), which is
interpreted as the isometry group of the hyperbolic upper half-plane. We also
discuss the connection with the Hamiltonian reduction of SL(2,R)_k WZNW model
to Liouville theory.Comment: 49 p
Fractional-Parabolic Systems
We develop a theory of the Cauchy problem for linear evolution systems of
partial differential equations with the Caputo-Dzrbashyan fractional derivative
in the time variable . The class of systems considered in the paper is a
fractional extension of the class of systems of the first order in
satisfying the uniform strong parabolicity condition. We construct and
investigate the Green matrix of the Cauchy problem. While similar results for
the fractional diffusion equations were based on the H-function representation
of the Green matrix for equations with constant coefficients (not available in
the general situation), here we use, as a basic tool, the subordination
identity for a model homogeneous system. We also prove a uniqueness result
based on the reduction to an operator-differential equation.Comment: Version 3 contains corrections (pages 6 and 23) as compared with the
published tex
Stochastic Calculus for a Time-changed Semimartingale and the Associated Stochastic Differential Equations
It is shown that under a certain condition on a semimartingale and a
time-change, any stochastic integral driven by the time-changed semimartingale
is a time-changed stochastic integral driven by the original semimartingale. As
a direct consequence, a specialized form of the Ito formula is derived. When a
standard Brownian motion is the original semimartingale, classical Ito
stochastic differential equations driven by the Brownian motion with drift
extend to a larger class of stochastic differential equations involving a
time-change with continuous paths. A form of the general solution of linear
equations in this new class is established, followed by consideration of some
examples analogous to the classical equations. Through these examples, each
coefficient of the stochastic differential equations in the new class is given
meaning. The new feature is the coexistence of a usual drift term along with a
term related to the time-change.Comment: 27 pages; typos correcte
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