If
X
t
{X_t}
is a continuous Markov process with infinitesimal generator A, if n is a kernel satisfying certain conditions, and if B is an operator given by
\[
B
g
(
x
)
=
∫
[
g
(
y
)
−
g
(
x
)
]
n
(
x
,
d
y
)
,
Bg(x)\, = \,\int {[ {g( y)\, - \,g(x)}]} \,n({x,\,dy}),
\]
then
A
+
B
A\, + \,B
will be the generator of a Markov process that has Lévy system
(
n
,
d
t
)
(n,\,dt)
. Conversely, if
X
t
{X_t}
has Lévy system
(
n
,
d
t
)
(n,\,dt)
, n satisfies certain conditions, and B is defined as above, then
A
−
B
A\, - \,B
will be the generator of a continuous Markov process.