For positive measurable functions v and w, applying Kac's formula to , and and examining the result yields
where the terms in the decomposition can be understood using (16) and the occupation measure identity:
These formulae are related to an energy form associated with v and w [35, 27, 64]. The covariance of and is
For each initial distribution , (24) is a symmetric bilinear non-negative definite function of pairs of non-negative functions chosen from . Similar remarks apply to the more general CAFs considered in Subsection 3.2.