, which
is the value at time t=T of the additive functional
. In the abstract, a positive continuous
additive functional (PCAF) is an 
-adapted family 
of positive finite random variables satisfying the additivity
condition
We can define an operator 
analogous to the operator 
by
the formula
for positive 
-measurable f. Note that 
when 
. The validity of the analog of
(13), namely
requires a mild additional hypothesis. For instance, if
E is a complete separable metric space (with Borel
-algebra ) and X has
right-continuous sample paths, then (21) is valid with f and 
as
for (13). The proof of this assertion involves Ray-Knight
compactification methods found in [59], and is well beyond the scope
of this article.
Assuming the validity of (21), we can repeat the earlier argument to show
that
Kac's moment formula (4) and the F-K formula
(7) hold for any PCAF A, provided the operator is substituted
for . Ray [55], Section 2, used this version of the
F-K formula for finite linear combinations of local times.
