Let \(E\) be real \(d\times d\) matrix whose eigenvalues have positive real parts. A scalar-valued random field \(\{X(x), x\in \mathbb{R}^d\}\) is called operator scaling for \(E\) and \(H>0\) if \[ \forall c>0, \{X(c^E(x)); x\in \mathbb{R}^d\}\mathop{=}\limits^{{\mathrm (fdd)}}\{c^HX(x); x\in\mathbb{R}^d\} \] where \(\mathop{=}\limits^{{\mathrm (fdd)}}\) means the equality of finite dimensional distributions. The sample path regularity of operator scaling \(\alpha\)-stable random fields is investigated. One of the tools is the change of polar coordinates with respect to the matrix \(E\). This notion is recalled in Section 3. For a Gaussian operator scaling random field \(X\) with stationary increments it gives \[ EX^2(x)=\tau^{2H}{E^{(x)}}E(X^2(l_E(x))) \] where \(\tau_E(x)\) is the radial part of \(X\) with respect to \(E\) and \(l_E(x)\) is its polar part. The Hölder regularity of sample paths of \(X\) follows from estimates of \(\tau_E(x)\) compared to \(\|x\|\). They are presented in section 3 as well. Section 2 discusses the harmonizable operator scaling random fields. Let \(\psi :\mathbb{R}^d\rightarrow[0,\infty)\) be a continuous function such that \[ \psi (c^{E^t}x)=c\psi (x) \quad \text{for \;all} \;c>0 \;\text{and}\;x\in \mathbb{R}^d. \] Let \(0<\alpha\leq 2\) and \(W_\alpha (d\xi)\) be a complex isotropic \(\alpha\)-stable random measure on \(\mathbb{R}^d\) with Lebesgue control measure. The random field \[ X_\alpha (x) =\mathfrak{R}\int_{\mathbb{R}^d}(e^{i(x,\xi)}-1)\psi (\xi)^{-1-q/\alpha}W_\alpha(d\xi), \quad x\in\mathbb{R}^d, q=\text{trace}(E), \] is called the harmonizable operator scaling stable random field. Some examples are presented.
In Section 4, a representation as LePage series of stochastic integrals with respect to an \(\alpha\)-stable random measure of \(\Lambda\) is presented if the control measure of \(\Lambda\) is finite, \(0<\alpha<2\). It is generalized in case of complex isotropic \(\alpha\)-stable random measure \(W_\alpha,\;0<\alpha<2\) with Lebesgue control measure.
By help of LePage representation of harmonizable field \(X_\alpha, (0<\alpha\leq 2)\) an upper bound for the modulus of continuity of \(X_\alpha\) and the critical Hölder regularity of its sample paths are considered in Section 5. In particular, if for eigenvector \(\Theta_j\) related to eigenvalue \(a_j\) of matrix \(E\) the critical Hölder exponent in direction \(\Theta_j\) is \(H_j=\frac{1}{a_j}\). Estimates of the Hausdorff and box dimensions for the graph of the field over a compact \(K\) are obtained as well.
In Section 6 the moving average operator scaling stable random field \[ Z_\alpha(x) = \int_{\mathbb{R}^d}\big(\varphi(x-y)^{1-q/\alpha}-\varphi(-y)^{1-q/\alpha}\big) M_\alpha(dy), \quad x\in \mathbb{R}^d \] is considered where \(q=\text{trace}(E), \varphi :\mathbb{R}^d\rightarrow[0,\infty)\) is a continuous \(E\)-homogeneous function and \(M_\alpha(dy)\) is an independently scattered \(S\alpha S\) random measure on \(\mathbb{R}^d\) with Lebesgue control measure. It is proved that sample paths of random field \(Z_\alpha, \;0<\alpha <2\) are almost surely unbounded on every open ball.
The Appendix contains the proofs of lemmas and propositions applied above.
The introduction contains short exact remarks on previous investigations in this field on various Gaussian and stable random fields including even applications in other sciences.