On the spectral representation of symmetric stable processes

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The so-called spectral representation theorem for stable processes linearly imbeds each symmetric stable process of index p into L p (0 < p ≤ 2). We use the theory of L p isometries for 0 < p < 2 to study the uniqueness of this representation for the non-Gaussian stable processes. We also determine the form of this representation for stationary processes and for substable processes. Complex stable processes are defined, and a complex version of the spectral representation theorem is proved. As a corollary to the complex theory we exhibit an imbedding of complex L q into real or complex L p for 0 < p < q ≤ 2.