Symmetric Stable Processes and Fubini ' s Theorem

In the first half of this paper, a Fubini type identity in law which was previously developed by two of the authors between quadratic functionals of Brownian motion is extended in two directions: an analogue of this identity in law holds when Brownian motion is replaced by a symmetric stable process of any order a E (0,2), provided the function: x -* x2 is replaced by: x -> I x la; such Fubini type identities in law yield, as a particular case, an identity in law which resembles the integration by parts formula; as a consequence, some extensions of the Ciesielski-Taylor identities in law are obtained. The second half of the paper is devoted to showing that such Fubini type identities in law "nearly" characterize the symmetric stable processes. A characterization result of lesser scope is obtained for two particular classes of processes which satisfy the integration by parts identity in law: the class of Gaussian processes on one hand, and the class of squares of Gaussian processes on the other hand.