Operator self-similar processes on Banach spaces

[摘要] Operator self-similar (OSS) stochastic processes on arbitraryBanach spaces are considered. If the family of expectations ofsuch a process is a spanning subset of the space, it is provedthat the scaling family of operators of the process underconsideration is a uniquely determined multiplicative group ofoperators. If the expectation-function of the process iscontinuous, it is proved that the expectations of the process havepower-growth with exponent greater than or equal to0, that is, their norm is less than a nonnegative constant times such apower-function, provided that the linear space spanned by theexpectations has category 2 (in the sense of Baire) in itsclosure. It is shown that OSS processes whose expectation-functionis differentiable on an interval(s0,∞), for somes0≥1, have a unique scaling family of operators of the form{sH:s>0}, if the expectations of the process span a denselinear subspace of category2. The existence of a scaling familyof the form{sH:s>0}is proved for proper Hilbert spaceOSS processes with an Abelian scaling family of positiveoperators.

[学科分类] 应用数学