Borel right process explained

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

lE

the Borel subsets of

E

.Let

\Omega

be the space of right continuous maps from

[0,infty)

to

E

that have left limits in

E

,and for each

t\in[0,infty)

, denote by

Xt

the coordinate map at

t

; foreach

\omega\in\Omega

,

Xt(\omega)\inE

is the value of

\omega

at

t

. We denote the universal completion of

lE

by

lE*

.For each

t\in[0,infty)

, let

lFt=\sigma\left\{

-1
X
s

(B):s\in[0,t],B\inlE\right\},

*
lF
t

=\sigma\left\{

-1
X
s

(B):s\in[0,t],B\inlE*\right\},

and then, let

lFinfty=\sigma\left\{

-1
X
s

(B):s\in[0,infty),B\inlE\right\},

*
lF
infty

=\sigma\left\{

-1
X
s

(B):s\in[0,infty),B\inlE*\right\}.

For each Borel measurable function

f

on

E

, define, for each

x\inE

,

U\alphaf(x)=Ex\left[

infty
\int
0

e-\alphaf(Xt)dt\right].

Since

Ptf(x)=

x\left[f(X
E
t)\right]
and the mapping given by

tXt

is right continuous, we see that for any uniformly continuous function

f

, we have the mapping given by

tPtf(x)

is right continuous.

Therefore, together with the monotone class theorem, for any universally measurable function

f

, the mapping given by

(t,x)Ptf(x)

, is jointly measurable, that is,

lB([0,infty))lE*

measurable, and subsequently, the mapping is also

\left(lB([0,infty))lE*\right)λ ⊗

-measurable for all finite measures

λ

on

lB([0,infty))

and

\mu

on

lE*

.Here,

\left(lB([0,infty))lE*\right)λ ⊗

is the completion of

lB([0,infty))lE*

with respectto the product measure

λ\mu

. Thus, for any bounded universally measurable function

f

on

E

,the mapping

tPtf(x)

is Lebeague measurable, and hence, for each

\alpha\in[0,infty)

, one can define

U\alphaf(x)=

infty
\int
0

e-\alphaPtf(x)dt.

There is enough joint measurability to check that

\{U\alpha:\alpha\in(0,infty) \}

is a Markov resolvent on

(E,lE*)

,which uniquely associated with the Markovian semigroup

\{Pt:t\in[0,infty)\}

. Consequently, one may apply Fubini's theorem to see that

U\alphaf(x)=Ex\left[

infty
\int
0

e-\alphaf(Xt)dt\right].

The following are the defining properties of Borel right processes:

For each probability measure

\mu

on

(E,lE)

, there exists a probability measure

P\mu

on

(\Omega,lF*)

such that

(Xt,

*,
lF
t

P\mu)

is a Markov process with initial measure

\mu

and transition semigroup

\{Pt:t\in[0,infty)\}

.

Let

f

be

\alpha

-excessive for the resolvent on

(E,lE*)

. Then, for each probability measure

\mu

on

(E,lE)

, a mapping given by

tf(Xt)

is

P\mu

almost surely right continuous on

[0,infty)