In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process.
Let
E
be a locally compact, separable, metric space. We denote by
E
the Borel subsets of
E
. Let
Ω
be the space of right continuous maps from
[
0
,
∞
)
to
E
that have left limits in
E
, and for each
t
∈
[
0
,
∞
)
, denote by
X
t
the coordinate map at
t
; for each
ω
∈
Ω
,
X
t
(
ω
)
∈
E
is the value of
ω
at
t
. We denote the universal completion of
E
by
E
∗
. For each
t
∈
[
0
,
∞
)
, let
F
t
=
σ
{
X
s
−
1
(
B
)
:
s
∈
[
0
,
t
]
,
B
∈
E
}
,
F
t
∗
=
σ
{
X
s
−
1
(
B
)
:
s
∈
[
0
,
t
]
,
B
∈
E
∗
}
,
and then, let
F
∞
=
σ
{
X
s
−
1
(
B
)
:
s
∈
[
0
,
∞
)
,
B
∈
E
}
,
F
∞
∗
=
σ
{
X
s
−
1
(
B
)
:
s
∈
[
0
,
∞
)
,
B
∈
E
∗
}
.
For each Borel measurable function
f
on
E
, define, for each
x
∈
E
,
U
α
f
(
x
)
=
E
x
[
∫
0
∞
e
−
α
t
f
(
X
t
)
d
t
]
.
Since
P
t
f
(
x
)
=
E
x
[
f
(
X
t
)
]
and the mapping given by
t
→
X
t
is right continuous, we see that for any uniformly continuous function
f
, we have the mapping given by
t
→
P
t
f
(
x
)
is right continuous.
Therefore, together with the monotone class theorem, for any universally measurable function
f
, the mapping given by
(
t
,
x
)
→
P
t
f
(
x
)
, is jointly measurable, that is,
B
(
[
0
,
∞
)
)
⊗
E
∗
measurable, and subsequently, the mapping is also
(
B
(
[
0
,
∞
)
)
⊗
E
∗
)
λ
⊗
μ
-measurable for all finite measures
λ
on
B
(
[
0
,
∞
)
)
and
μ
on
E
∗
. Here,
(
B
(
[
0
,
∞
)
)
⊗
E
∗
)
λ
⊗
μ
is the completion of
B
(
[
0
,
∞
)
)
⊗
E
∗
with respect to the product measure
λ
⊗
μ
. Thus, for any bounded universally measurable function
f
on
E
, the mapping
t
→
P
t
f
(
x
)
is Lebeague measurable, and hence, for each
α
∈
[
0
,
∞
)
, one can define
U
α
f
(
x
)
=
∫
0
∞
e
−
α
t
P
t
f
(
x
)
d
t
.
There is enough joint measurability to check that
{
U
α
:
α
∈
(
0
,
∞
)
}
is a Markov resolvent on
(
E
,
E
∗
)
, which uniquely associated with the Markovian semigroup
{
P
t
:
t
∈
[
0
,
∞
)
}
. Consequently, one may apply Fubini's theorem to see that
U
α
f
(
x
)
=
E
x
[
∫
0
∞
e
−
α
t
f
(
X
t
)
d
t
]
.
The followings are the defining properties of Borel right processes:
Hypothesis Droite 1:
For each probability measure
μ
on
(
E
,
E
)
, there exists a probability measure
P
μ
on
(
Ω
,
F
∗
)
such that
(
X
t
,
F
t
∗
,
P
μ
)
is a Markov process with initial measure
μ
and transition semigroup
{
P
t
:
t
∈
[
0
,
∞
)
}
.
Hypothesis Droite 2:
Let
f
be
α
-excessive for the resolvent on
(
E
,
E
∗
)
. Then, for each probability measure
μ
on
(
E
,
E
)
, a mapping given by
t
→
f
(
X
t
)
is
P
μ
almost surely right continuous on
[
0
,
∞
)
.
