To recover more standard expressions of the F-K formula,
as presented in Section III.19 of Rogers-Williams [57],
let 
be the semigroup derived from 
by killing X
with state-dependent rate v(x).
So if T is the associated Markov killing time then
Summing formula (32) weighted by 
yields
an expression for the resolvent of this semigroup:
where 
is the
resolvent of the semi-group of X. Some mild regularity on
f and v are required to make sense of the second equality in
(34),
but when rearranged as
the formula holds as an identity of bounded
positive kernels for arbitrary 
and non-negative measurable
v.
This is formula (III.19.5) of
[57], which is one of the three forms
of the F-K formula presented by Rogers-Williams.
As they remark,
and as shown by the above argument,
(35) is a robust form of the F-K formula which
is valid with no hypotheses on the underlying Markov process X
beyond jointly-measurable paths.
To interpret the F-K formula as a statement relating the infinitesimal
generators of
and , let us recall that the weak
infinitesimal generator of (say) is the operator 
defined by
on the domain 
comprising those functions f for which the
pointwise limit
indicated in (36) exists and
. See
[15], where it is shown that for each , the resolvent
operator
is an injective mapping of 
(the class of bounded
-measurable
functions) onto , and 
. Briefly,
.
Viewed in these terms,
(35) amounts to the most common presentation of the F-K formula:
the domain 
consisting of those functions 
for
which uv is a
bounded function. In the context of symmetric Markov processes,
(37) can be reformulated in terms of ``Dirichlet forms''; see Section
6.1 of [24].
A third form of the F-K formula noted in [57] (III.19.7) is the following variant of (35):
which can be understood with almost no calculation due to the following
Probabilistic interpretation of (38).
Consider 
, the minimum of T and an independent
exponential time with rate 
. From (34),
the Green's operator for X killed at time is 
.
Since killing at time is the same as killing with
rate function 
, the 
distribution of
on 
is found by an easy variation of the ``last exit'' formula (31):
Thus (38) comes from integrating with respect
to the decomposition
for an arbitrary additive functional 
.
Another probabilistic interpretation of (38).
Multiplication of both sides of (38) by yields an
identity of Markov kernels which may be understood in another way.
By the companion of (39) with 
instead of
,
and by (39) and the memoryless property of 
,
Adding these two distributions we arrive at 
,
which is the
distribution of 
.
Remarks. For a constant function v, say 
for
all x, (38) reduces to the resolvent identity:
So the above arguments give two simple probabilistic interpretations
of this identity involving
the minimum 
of
two independent exponential variables and 
and
analysis of 
according to whether
or 
.
Since it is obvious that 
,
the resolvent identity also yields (35) in this special case.
For a general v, comparison of (35) and (38) establishes
the identity of kernels
where the first identity was interpreted probabilistically above.
There is also a probabilistic interpretation of the second identity in (44), involving the idea of resurrection of X after the killing time T, as considered in [46, 22]. Note that (44) is the Laplace transformed version of the following identity:
and that if 
is an initial distribution, then
for any measurable 
.
The first equality in (45) is evident from evaluation of this expectation
by conditioning on 
.
The second equality in (45) is obtained by the following
construction.
Assume for simplicity that v is bounded.
Given X, let 
where
are the points of a Poisson process on
with
intensity 
. The right hand expression in (45)
arises from
evaluating the expectation
(46) by conditioning on 
, where
Replacing the fixed time t by
gives a similar interpretation of the second equality in (44). From
this perspective, the middle and right hand expressions in
(44) and (45) are seen to be typical ``first entrance'' and
``last exit'' decompositions.
