Let
X
=
(
X
t
,
P
x
)
X = ({X_t},\,{P^x})
be a right Markov process and let
m
m
be an excessive measure for
X
X
. Associated with the pair
(
X
,
m
)
(X,\,m)
is a stationary strong Markov process
(
Y
t
,
Q
m
)
({Y_t},\,{Q_m})
with random times of birth and death, with the same transition function as
X
X
, and with
m
m
as one dimensional distribution. We use
(
Y
t
,
Q
m
)
({Y_t},\,{Q_m})
to study the cone of excessive measures for
X
X
. A "weak order" is defined on this cone: an excessive measure
ξ
\xi
is weakly dominated by
m
m
if and only if there is a suitable homogeneous random measure
κ
\kappa
such that
(
Y
t
,
Q
ξ
)
({Y_t},\,{Q_\xi })
is obtained by "birthing"
(
Y
t
,
Q
m
)
({Y_t},\,{Q_m})
, birth in
[
t
,
t
+
d
t
]
[t,\,t + dt]
occurring at rate
κ
(
d
t
)
\kappa (dt)
. Random measures such as
κ
\kappa
are studied through the use of Palm measures. We also develop aspects of the "general theory of processes" over
(
Y
t
,
Q
m
)
({Y_t},\,{Q_m})
, including the moderate Markov property of
(
Y
t
,
Q
m
)
({Y_t},\,{Q_m})
when the arrow of time is reversed. Applications to balayage and capacity are suggested.