An extension of Wiener integration with the use of operator theory

With the use of tensor product of Hilbert space, and a diagonalization procedure from operator theory, we derive an approximation formula for a general class of stochastic integrals. Further we establish a generalized Fourier expansion for these stochastic integrals. In our extension, we circumvent some of the limitations of the more widely used stochastic integral due to Wiener and Ito, i.e., stochastic integration with respect to Brownian motion. Finally we discuss the connection between the two approaches, as well as a priori estimates and applications.Comment: 13 page

Non-holomorphic multi-matrix gauge invariant operators based on Brauer algebra

We present an orthogonal basis of gauge invariant operators constructed from some complex matrices for the free matrix field, where operators are expressed with the help of Brauer algebra. This is a generalisation of our previous work for a signle complex matrix. We also discuss the matrix quantum mechanics relevant to N=4 SYM on S^{3} times R. A commuting set of conserved operators whose eigenstates are given by the orthogonal basis is shown by using enhanced symmetries at zero coupling.Comment: 29 pages, typos corrected, references adde

A dual null formalism for the collapse of fluids in a cosmological background

In this work we revisit the definition of Matter Trapping Surfaces (MTS) introduced in previous investigations and show how it can be expressed in the so-called dual null formalism developed for Trapping Horizons (TH). With the aim of unifying both approaches, we construct a 2+2 threading from the 1+3 flow, and thus isolate one prefered spatial direction, that allows straightforward translation into a dual nul subbasis, and to deduce the geometric apparatus that follows. We remain as general as possible, reverting to spherical symmetry only when needed, and express the MTS conditions in terms of 2-expansion of the flow, then in purely geometric form of the dual null expansions. The Raychadhuri equations that describe both MTS and TH are written and interpreted using the previously defined gTOV (generalized Tolman-Oppenheimer-Volkov) functional introduced in previous work. Further using the Misner-Sharp mass and its previous perfect fluid definition, we relate the spatial 2-expansion to the fluid pressure, density and acceleration. The Raychaudhuri equations also allows us to define the MTS dynamic condition with first order differentials so the MTS conditions are now shown to be all first order differentials. This unified formalism allows one to realise that the MTS can only exist in normal regions, and so it can exist only between black hole horizons and cosmological horizons. Finally we obtain a relation yielding the sign, on a TH, of the non-vanishing null expansion which determines the nature of the TH from fluid content, and flow characteristics. The 2+2 unified formalism here investigated thus proves a powerful tool to reveal, in the future extensions, more of the very rich and subtle relations between MTS and TH.Comment: 10pp 1 fig. corrected for equation labels, cross listing correcte