Let
(
X
(
t
)
,
P
x
)
(X(t),{P^x})
and
(
Y
(
t
)
,
Q
x
)
(Y(t),{Q^x})
be transient Hunt processes on a state space
E
E
satisfying the hypothesis of absolute continuity (Meyer’s hypothesis (L)). Let
T
(
K
)
T(K)
be the first entrance time into a set
K
K
, and assume
P
x
(
T
(
K
)
>
∞
)
=
Q
x
(
T
(
K
)
>
∞
)
{P^x}(T(K) > \infty ) = {Q^x}(T(K) > \infty )
for all compact sets
K
⊆
E
K \subseteq E
. There exists a strictly increasing continuous additive functional of
X
(
t
)
,
A
(
t
)
X(t),A(t)
, so that if
T
(
t
)
=
inf
{
s
:
A
(
s
)
>
t
}
T(t) = {\text {inf}}\{s:A(s) > t\}
, then
(
X
(
T
(
t
)
)
,
P
x
)
(X(T(t)),{P^x})
and
(
Y
(
t
)
,
Q
x
)
(Y(t),{Q^x})
have the same joint distributions. An analogous result is stated if
X
X
and
Y
Y
are right processes (with an additional hypothesis). These theorems generalize the Blumenthal-Getoor-McKean Theorem and have interpretations in terms of potential theory.