Operator Self-Similar Processes on Banach Spaces
Copyright © 2006 M. T. Matache and V. Matache. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are considered. If the family of expectations of such a process is a spanning subset of the space, it is proved that the scaling family of operators of the process under consideration is a uniquely determinedmultiplicative group of operators. If the expectation-function of the process is continuous, it is proved that the expectations of the process have power-growth with exponent greater than or equal to 0, that is, their norm is less than a nonnegative constant times such a power-function, provided that the linear space spanned by the expectations has category 2 (in the sense of Baire) in its closure. It is shown that OSS processes whose expectation-function is differentiable on an interval (s0,∞), for some s0 ≥ 1, have a unique scaling family of operators of the form {sH : s > 0}, if the expectations of the process span a dense linear subspace of category 2. The existence of a scaling family of the form {sH : s > 0} is proved for proper Hilbert space OSS processes with an Abelian scaling family of positive operators.