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Many phenomena in physics are described by a vector-valued homogeneous and isotropic random function in two or three space variables, or a random field. On the one hand, such a field can be uniquely represented as a sum of two components: a solenoidal one without divergence, and an irrotational one without curl. On the other hand, it may be represented as a sum of a longitudinal part, parallel to a fixed space direction, and a lateral part, orthogonal to that direction. If the longitudinal part of the solenoidal component has certain fractal and memory properties, what are the corresponding properties of its lateral part? Similarly, if the lateral part of the irrotational component has certain fractal and memory properties, what are the corresponding properties of its longitudinal part? We give an answer to those questions using well-known tools of classical real analysis. © 2025 Society for Industrial and Applied Mathematics.