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d
)
for each 0
R
≤
≤
∞
(i)
T
s,t
is a linear operator on
B
b
(
s
t<
.
(ii)
T
s,s
=
≥
I
for each
s
0.
(iii)
T
r,s
T
s,t
=
≤
≤
≤
∞
T
r,t
, whenever 0
r
s
t<
.
d
)
.
(iv)
f
≥
0 implies
T
s,t
f
≥
0 for all 0
≤
s
≤
t<
∞
and
f
∈
B
b
(
R
(v)
T
s,t
≤
1 for each 0
≤
s
≤
t<
∞
,i.e.
T
s,t
is a contraction.
(vi)
T
s,t
(
1
)
=
1 for all
t
≥
0.
d
)
. If we restrict ourselves to time-
homogeneous Markov processes satisfying (
16.1
), we obtain directly from the above
properties that the family of operators
T
t
:=
Here,
I
denotes the identity operator on
B
b
(
R
T
0
,t
forms a positivity preserving con-
traction semigroup. The
infinitesimal generator
)
of such a
process
X
with semigroup
(T
t
)
t
≥
0
is defined by the strong pointwise limit
A
with domain
D
(
A
1
t
A
u
:=
lim
t
→
(T
t
u
−
u)
(16.2)
0
+
d
)
for which the limit (
16.2
) exists with respect to
for all functions
u
∈
D
(
A
)
⊂
B
b
(
R
the sup-norm. We call
(
))
the
generator
of
X
. Generators of normal Markov
processes admit the
positive maximum principle
,i.e.
A
,
D
(
A
if
u
∈
D
(
A
)
and
sup
x
∈R
u(x)
=
u(x
0
)>
0
,
then
(
A
u)(x
0
)
≤
0
.
(16.3)
d
Furthermore, they admit a pseudodifferential representation (e.g. [22, 44, 94, 95]):
D
(
A
)
,
where C
0
d
)
⊂
D
(
A
)
Theorem 16.1.1
Let
A
be an operator with domain
(
R
A
(C
0
d
))
⊂
C(
R
)
,
where C
0
d
) denotes the space of smooth functions
and
(
R
(
R
d
.
Then
,
R
A
|
C
0
(
R
with support compactly contained in
d
)
is a pseudodifferential
operator
,
(
2
π)
−
d/
2
u(ξ )e
i
x,ξ
d
ξ,
(
A
u)(x)
:= −
ψ(x,ξ)
ˆ
(16.4)
d
R
C
0
d
) and
d
where u
∈
(
R
u is the Fourier transform of u
.
The symbol ψ(x,ξ)
ˆ
: R
×
d
R
→ C
is locally bounded in (x, ξ)
.
The function ψ(
·
,ξ) is measurable for every
ξ and ψ(x,
) is a negative definite function
,
see
[94,
Definition
3.6.5],
for every x
,
which admits the
Lévy-Khintchine representation
·
1
2
ψ(x,ξ)
=
c(x)
−
i
b(x), ξ
+
ξ,Q(x)ξ
1
ν(x,
d
z).
i
z, ξ
e
i
z,ξ
+
+
−
(16.5)
2
1
+|
z
|
d
0
=
z
∈R
d
d
d
d
×
d
Here
,
c
: R
→ R
,
b
: R
→ R
and Q
: R
→ R
are functions and ν(x,
·
) is a
d
measure on
R
for fixed x
∈ R
with
d
2
)ν(x,
d
z)
R
x
→
(
1
∧ |
z
|
(16.6)
z
=
0
continuous and bounded
.
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