Stochastic integration w.r.t. continuous local martingales

In this note we develop the theory of stochastic integration w.r.t. continuous local martingales using a simple time change technique. We allow progressively measurable integrands.     

On universal Korovkin sets w.r.t. positive spectral contractions

We prove that there exists a finite universal Korovkin set with respect to positive spectral contraction operators for a Segal algebra on a locally compact abelian groupG if and only ifĜ is finite dimensional separable metrizable space.     

Fubini theorem w.r.t. stochastic measure on product measurable space

ＦＵＢＩＮＩＴＨＥＯＲＥＭｗ．ｒ．ｔ．ＳＴＯＣＨＡＳＴＩＣＭＥＡＳＵＲＥＯＮＰＲＯＤＵＣＴＭＥＡＳＵＲＡＢＬＥＳＰＡＣＥＣＨＥＮＰＥＩＤＥ（陈培德）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ,ｔｈｅＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳ...     

Spectral analysis of families of operator polynomials and a generalized Vandermonde matrix II: The infinite dimensional case

This paper deals with the problem of the existence of a common monic multiple for a given family of monic operator polynomials. In general such a multiple does not exist. Necessary and sufficient conditions for the existence are given in terms of the invertibility of a generalized Vandermonde operator matrix. The minimal possible degree of the multiple is determined and the role played by the spectral properties of the divisors is explained. The problems considered here have their origin in the spectral theory of non-self-adjoint operators.     

Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces

Journal of Optimization Theory and Applications - In vector optimization, it is of increasing interest to study problems where the image space (a real linear space) is preordered by a not...     

Talagrand''''s T_2-transportation Inequality w.r.t.a Uniform Metric for Diffusions

Using the method of Girsanov transformation,we establish the Talagrand's T_2-inequality for dif-fusion on the path space C([0,N],R~d) with respect to a uniform metric,with the constant independent of N.This improves the known results for the L~2-metric.     

On a class of operator equations arising in infinite dimensional replicator dynamics

We introduce a class of infinite dimensional replicator dynamics for the study of equilibrium selection in non-cooperative games and study their properties. Examples from economic theory are provided.     

Large deviations w.r.t. quasi-every starting point for symmetric right processes on general state spaces

Summary In the work of Donsker and Varadhan, Fukushima and Takeda and that of Deuschel and Stroock it has been shown, that the lower bound for the large deviations of the empirical distribution of an ergodic symmetric Markov process is given in terms of its Dirichlet form. We give a short proof generalizing this principle to general state spaces that include, in particular, infinite dimensional and non0metrizable examples. Our result holds w.r.t. quasi-every starting point of the Markov process. Moreover we show the corresponding weak upper bound w.r.t. quasi-every starting point.This research was supported by the Graduiertenkolleg Algebraische, analytische und geometrische Methoden und ihre Wechselwirkung in der modernen Mathematik, Bonn     

有界可料过程关于集值平方可积鞅的集值随机积分

本文定义了一类有界可料过程关于集值平方可积鞅的集值随机积分,并研究了集植随机积分的性质。此为建立集值随机分析的理论奠定了基础。     

The unitary group preserving maps (the infinite dimensional case)

Marcus's theorem on the unitary group preserving map is extended to the infinite dimensional case. The method of proof uses the Jordan automorphisms of the algebra of hermitian operators. A connection with the group of (linear bijective) isometries is established.     

An extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system

This paper proposes an extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system. Chaitin's Ω is defined as the probability that the universal self‐delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program‐size complexity H (s) of a given finite binary string s. In the standard way, H (s) is defined as the length of the shortest input string for U to output s. In the other way, the so‐called universal probability m is introduced first, and then H (s) is defined as –log₂ m (s) without reference to the concept of program‐size. Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator‐valued measure (POVM) in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour‐El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi‐POVM. We also give another characterization of Chaitin's Ω numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ω as a sum of the POVM elements of a universal semi‐POVM. The validity of this definition is discussed. In what follows, we introduce an operator version (s) of H (s) in a Hilbert space of infinite dimension using a universal semi‐POVM, and study its properties. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Cauchy problem for evolution equations with the Bessel operator of infinite order. I

We establish necessary and sufficient conditions under which the Bessel operator of infinite order is bounded in certain spaces. We study properties of the Bessel transformations of distributions from these spaces, those of convolutions, convolutors, and multiplicators.     

The cauchy problem for evolution equations with the Bessel operator of infinite order. II

We prove the correct solvability of the Cauchy problem for singular evolution equations of infinite order in classes of initial conditions that are generalized functions like ultra-distributions (analytic functionals).     

Polynomial operator matrices as generators: The general case

Of concern are systems of linear evolution equations