Suppose as in [8] (3.41) that X with state space E admits a
jointly
measurable local time process
relative to a reference measure dx on E,
so that for 
For example, X could be a Markov chain with countable state space,
with 
,
or a one-dimensional diffusion [56].
Fix a Markov killing time T for X.
Then 
serves as a density for the Green's kernel
of X killed at time T. From (22),
for all 
there is the formula
So for each a such that 
the covariance
is a symmetric non-negative definite function of 
.
Suppose now that a finite subset F of
E is such that 
, for all 
.
The results of Subsection 3.6 show that for any initial
distribution
on F the 
joint distribution of
is
determined by the values of the Green's function g(x,y) for
.
In particular,
all product moments
for non-negative
integer n(y)
have finite values which can be read from (27).
And the
joint moment-generating function of the
converges in a neighborhood of the origin and is given there by the
formula
where 
.
Put another way, (57) states that for v in a neighborhood of
, the function
is the unique solution f of the system of equations
For X a one-dimensional diffusion,
Ray [55] (2.1) derived this system of equations
for an h-process obtained conditioning on 
,
and went on to show that these equations imply the Ray-Knight descriptions for
the distribution of local times of one-dimensional diffusions stopped at
a Markov killing time.
See also Sheppard [61] who recovered most of Ray's results
with the help of Dynkin's isomorphism theorem, and [53] for
further discussion.
Note that the class of possible finite-dimensional
distributions for as above
is precisely the class
of joint distributions of total occupation times of
various states in a finite state Markov chain.
This can be understood by considering the time-changed Markov chain
where 
is the inverse of 
.
Williams [65, 66] used a similar time change argument to
derive variations
of formula (55) for local time processes associated with both
ordinary and fictitious states of a countable state Markov chain.
See Theorem 6.1 of [66].
An altogether different application of the F-K formula to the local times
of one-dimensional Lévy processes can be found in [6].
This work concerns the law of the Hilbert transform of 
with respect to x (for certain random T); it extends and simplifies
[23], in which Kac's moment formula (4) is used.
See also [31] regarding connections between the F-K formula and
path decompositions for one-dimensional BM.
Acknowledgement. We are grateful to Jay Rosen for his illuminating comments on the manuscript.
