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

Let \(S^p = \{S_t^p : t = \frac{k}{2^n},\ 0 \leq k \leq 2^n,\ n \in\mathbb{N}\}\) 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 \(0 \leq u \lt t \leq 1\) 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 \(0 \lt p \leq 2\) on \([0, 1]\) with the Lebesgue measure. We show also a complex version of the result.