On constructions of isometric copies of $L^p(0,1)$ spaces $(0p\le 2)$ by stochastic $p$-stable processes

Jolanta Grala-Michalak, and Artur Michalak. "On constructions of isometric copies of $L^p (0, 1)$ spaces $(0 p \le 2)$ by stochastic $p$-stable processes." Commentationes Mathematicae 48.1 (2008): null. <http://eudml.org/doc/291382>.

@article{JolantaGrala2008,
abstract = {Let $S^p = \lbrace S_t^p : t = \frac\{k\}\{2^n\},\ 0 \le k \le 2^n,\ n \in \mathbb \{N\}\rbrace $ be a stochastic process on a probability space $(\Omega , \Sigma , P)$ with independent and time homogeneous increments such that $S_t^p - S_u^p$ is identically distributed as $(t- u)^\{1/p\} Z_p$ for each $0 \le u t \le 1$ where $Z_p$ is a given symmetric $p$-stable distribution. We show that the closed linear hull of $S^p$ forms an isometric copy of the real Lebesgue space $L^p (0, 1)$ in any quasi-Banach space $X$ consisting of $P$-a.e. equivalence classes of $\Sigma $-measurable real functions on $\Omega $ equipped with a rearrangement invariant quasi-norm which contains $S^p$ as a subset. It is possible to construct processes $S^p$ for $0 p \le 2$ on $[0, 1]$ with the Lebesgue measure. We show also a complex version of the result.},
author = {Jolanta Grala-Michalak, Artur Michalak},
journal = {Commentationes Mathematicae},
keywords = {$L^p$-spaces},
language = {eng},
number = {1},
pages = {null},
title = {On constructions of isometric copies of $L^p (0, 1)$ spaces $(0 p \le 2)$ by stochastic $p$-stable processes},
url = {http://eudml.org/doc/291382},
volume = {48},
year = {2008},
}

TY - JOUR
AU - Jolanta Grala-Michalak
AU - Artur Michalak
TI - On constructions of isometric copies of $L^p (0, 1)$ spaces $(0 p \le 2)$ by stochastic $p$-stable processes
JO - Commentationes Mathematicae
PY - 2008
VL - 48
IS - 1
SP - null
AB - Let $S^p = \lbrace S_t^p : t = \frac{k}{2^n},\ 0 \le k \le 2^n,\ n \in \mathbb {N}\rbrace $ be a stochastic process on a probability space $(\Omega , \Sigma , P)$ with independent and time homogeneous increments such that $S_t^p - S_u^p$ is identically distributed as $(t- u)^{1/p} Z_p$ for each $0 \le u t \le 1$ where $Z_p$ is a given symmetric $p$-stable distribution. We show that the closed linear hull of $S^p$ forms an isometric copy of the real Lebesgue space $L^p (0, 1)$ in any quasi-Banach space $X$ consisting of $P$-a.e. equivalence classes of $\Sigma $-measurable real functions on $\Omega $ equipped with a rearrangement invariant quasi-norm which contains $S^p$ as a subset. It is possible to construct processes $S^p$ for $0 p \le 2$ on $[0, 1]$ with the Lebesgue measure. We show also a complex version of the result.
LA - eng
KW - $L^p$-spaces
UR - http://eudml.org/doc/291382
ER -