by
P. J. Fitzsimmons and
Jim Pitman
May 29, 1998
Mark Kac
introduced a method for
calculating the distribution
of the integral 
for a function v
of a Markov process 
and a suitable random time T,
which yields the Feynman-Kac formula for the moment-generating
function of 
.
We review Kac's method, with emphasis on
an aspect often overlooked.
This is Kac's formula for moments of , which may
be stated as follows.
For any random time T such that the killed process
is Markov with substochastic semi-group
,
any non-negative measurable function v,
and any initial distribution 
, the 
moment of
is 
where
is the Green's operator of the killed process,
is the operator of multiplication by v,
and 
is the function that is identically 1.
