Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 82838, 18 pages
doi:10.1155/JAMSA/2006/82838
Operator self-similar processes on Banach spaces
Department of Mathematics, University of Nebraska, Omaha 68182-0243, NE, USA
Received 16 March 2005; Revised 21 July 2005; Accepted 30 July 2005
Copyright © 2006 Mihaela T. Matache and Valentin 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 determined multiplicative 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.