Nonlinear transformations of the canonical gauss measure on Hilbert space and absolute continuity

The papers of R. Ramer and S. Kusuoka investigate conditions under which the probability measure induced by a nonlinear transformation on abstract Wiener space(,H,B) is absolutely continuous with respect to the abstract Wiener measure. These conditions reveal the importance of the underlying Hilbert spaceH but involve the spaceB in an essential way. The present paper gives conditions solely based onH and takes as its starting point, a nonlinear transformationT=I+F onH. New sufficient conditions for absolute continuity are given which do not seem easily comparable with those of Kusuoka or Ramer but are more general than those of Buckdahn and Enchev. The Ramer-Itô integral occurring in the expression for the Radon-Nikodym derivative is studied in some detail and, in the general context of white noise theory it is shown to be an anticipative stochastic integral which, under a stronger condition on the weak Gateaux derivative of F is directly related to the Ogawa integral.Research supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL 03 92 G 0008.     

High P–T Brillouin scattering study of H2O melting to 26 GPa

The properties of solid and liquid phases of H₂O at high pressure and temperature remain an active area of research. In this study, Brillouin spectroscopy has been used to determine the temperature dependence of sound velocities in H₂O as a function of pressure up to 26 GPa through the phase field of ice VII and into the liquid to a maximum temperature of 1200 K. The Brillouin shift of the quasi-longitudinal acoustic mode moves to lower frequencies upon melting at each pressure. As a test of the method, measurements of the melting of Ar by Brillouin scattering at several pressures show a similar behavior for the acoustic mode, and measured melting points are consistent with previous results. The results of H₂O melting are consistent with previously reported melting curves below 20 GPa. The data at higher pressure indicate that ice melts at a higher temperature than a number of previous studies have indicated.     

Monotonicity of the matrix geometric mean

An attractive candidate for the geometric mean of m positive definite matrices A ₁, . . . , A _m is their Riemannian barycentre G. One of its important operator theoretic properties, monotonicity in the m arguments, has been established recently by Lawson and Lim. We give an elementary proof of this property using standard matrix analysis and some counting arguments. We derive some new inequalities for G. One of these says that, for any unitarily invariant norm, ||| G ||| is not bigger than the geometric mean of |||A ₁|||, . . . , |||A _m |||.     

A general principle for limit theorems in finitely additive probability: The dependent case

In this paper we formulate and prove a general principle which enables us to deduce limit theorems for a sequence of random variables on a finitely additive probability space.     

Evolution equations for Markov processes: Application to the white-noise theory of filtering

LetX be a Markov process taking values in a complete, separable metric spaceE and characterized via a martingale problem for an operatorA. We develop a criterion for invariant measures when rangeA is a subset of continuous functions onE. Using this, uniqueness in the class of all positive finite measures of solutions to a (perturbed) measure-valued evolution equation is proved when the test functions are taken from the domain ofA. As a consequence, it is shown that in the characterization of the optimal filter (in the white-noise theory of filtering) as the unique solution to an analogue of Zakai (as well as Fujisaki-Kallianpur-Kunita) equation, it suffices to take domainA as the class of test functions where the signal process is the solution to the martingale problem forA.The research of A. G. Bhatt was supported by the National Board for Higher Mathematics, Bombay, India. Part of this work was done while R. L. Karandikar was visiting Erasmus University, Rotterdam, The Netherlands.     

On the Feynman-Kac formula and its applications to filtering theory

In this article the Feynman-Kac formula is obtained for a Markov process (X _t) whose transition probability function is not stationary. A converse to the Feynman-Kac formula is also obtained. This is used to prove the uniqueness of the solution to a measure-valued equation satisfied by the optimal filter in the white-noise approach to nonlinear filtering theory.Research partially supported by the Air Force Office of Scientific Research Contract No. F49620 85 C 0144 and by the Indian Statistical Institute.     

Post‐deposition annealing and crystallisation of Y-Ba-Cu-O thin films deposited on polycrystalline substrates

Superconducting thin films of Y-Ba-Cu-O have been prepared by 50 Hz ac sputtering on polycrystalline SrTiO₃, yttria-stabilized-ZrO₂, and MgO substrates. Films deposited on different substrate materials are required to be annealed at different temperatures to induce superconductivity. The optimum annealing temperature for a film depends on the reaction between the film and its substrate material. It appears that a correlation exists between the process of crystallisation and substrate-film reaction.     

Weak convergence to a Markov process: The martingale approach

Summary In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X _n } toX, whenX arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processesX _n ,X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.Research supported by National Board for Higher Mathematics, Bombay, IndiaPart of the work was done at University of California, Santa Barbara, USA