The General Product Moment Formula

  Returning to the set-up of Kac's moment formula, or, more generally, the setting of Subsection (3.1), by iterated application of the occupation measure identity there is the following generalization of (22) and (4) to a product of n additive functionals , :

 

where the sum extends over all permutations of . Theorem 5.2 of Dynkin [19] is this result in a slightly different framework. Dynkin assumes a symmetric potential density g(x,y), but (27) applies nonetheless without such symmetry, and without assuming the existence of a potential density provided (21) is valid. Dynkin has shown how in the symmetric case the moment formula (27) underlies a far-reaching isomorphism between the distribution of functionals of the occupation field of a symmetric Markov process, and the distribution of the square of a Gaussian field with covariance derived from the positive definite kernel g(x,y). See [17, 20, 43, 42, 41, 45] for further developments, and Rogers and Williams [57] I.27 for an elementary proof of Dynkin's isomorphism formula for a Markov chain, which is closely related to the discussion in Section 5 below.

Question. Is there any interesting connection between the occupation field of a Markov process that is not necessarily symmetric and the Gaussian process with covariance structure defined by the non-negative definite function (24)? It seems not, since it is not this positive kernel but the one derived more directly from g(x,y) that works in the symmetric case.