S\(\alpha\)S \(M(t)\)-processes and their canonical representations (Q1310530)

An S\(\alpha\)S (= symmetric-\(\alpha\)-stable) random field \(\{X^ \alpha_ n({\mathbf t})\); \({\mathbf t}\in\mathbb{R}^ n\}\) \((0<\alpha\leq 2\), \(n\geq 1)\) is called the multi-parameter S\(\alpha\)S (Lévy) motion if \[ X^ \alpha_ n(\text{\textbf{0}})=0\text{ and }E\exp[iz(X^ \alpha_ n({\mathbf t})-X^ \alpha_ n({\mathbf s}))]=\exp(-c|{\mathbf t}-{\mathbf s}|\;| z|^ \alpha),\quad z\in\mathbb{R}, \] for any \({\mathbf t},{\mathbf s}\in\mathbb{R}^ n\) with a constant \(c>0\). We construct this random field by using integral geometry. Then the S\(\alpha\)S process \(\{M^ \alpha_{n,l,m}(t)=\int_{\xi\in S^{n-1}}X^ \alpha_ n(t\xi)v^ n_{l,m}(\xi)d\xi; t\geq 0\}\) \((0<\alpha\leq 2)\), where \(v^ n_{l,m}\) is a spherical harmonic function, is the S\(\alpha\)S analogue of Lévy- McKean's \(M(t)\)-process. And through this construction we have a causal stochastic integral representation \(M^ \alpha_{n,l,m}(t)=\int^ t_ 0 F_{n,l}(t,u)dZ^ \alpha_{n,l,m}(u)\), \(t\geq 0\), of this process. We show that in non-Gaussian case \((\alpha\neq 2)\) for any \(n\), \(l\) this representation is canonical, that is, the \(\sigma\)-field \(\sigma\{M^ \alpha_{n,l,m}(s); s\leq t\}\) is equal to the \(\sigma\)-field \(\sigma\{Z^ \alpha_{n,l,m}(s); s\leq t\}\) for any \(t\geq 0\), while in Gaussian case \((\alpha=2)\) this representation is not canonical if \(l\geq 3\).