In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process.
Let
be a locally compact, separable, metric space.
We denote by
the Borel subsets of
.
Let
be the space of right continuous maps from
to
that have left limits in
,
and for each
, denote by
the coordinate map at
; for
each
,
is the value of
at
.
We denote the universal completion of
by
.
For each
, let
![{\displaystyle {\mathcal {F}}_{t}=\sigma \left\{X_{s}^{-1}(B):s\in [0,t],B\in {\mathcal {E}}\right\},}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle {\mathcal {F}}_{t}^{*}=\sigma \left\{X_{s}^{-1}(B):s\in [0,t],B\in {\mathcal {E}}^{*}\right\},}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and then, let
![{\displaystyle {\mathcal {F}}_{\infty }=\sigma \left\{X_{s}^{-1}(B):s\in [0,\infty ),B\in {\mathcal {E}}\right\},}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle {\mathcal {F}}_{\infty }^{*}=\sigma \left\{X_{s}^{-1}(B):s\in [0,\infty ),B\in {\mathcal {E}}^{*}\right\}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
For each Borel measurable function
on
, define, for each
,
![{\displaystyle U^{\alpha }f(x)=\mathbf {E} ^{x}\left[\int _{0}^{\infty }e^{-\alpha t}f(X_{t})\,dt\right].}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Since
and the mapping given by
is right continuous, we see that
for any uniformly continuous function
, we have the mapping given by
is right continuous.
Therefore, together with the monotone class theorem, for any universally measurable function
, the mapping given by
, is jointly measurable, that is,
measurable, and subsequently, the mapping is also
-measurable for all finite measures
on
and
on
.
Here,
is the completion of
with respect
to the product measure
.
Thus, for any bounded universally measurable function
on
,
the mapping
is Lebeague measurable, and hence,
for each
, one can define
![{\displaystyle U^{\alpha }f(x)=\int _{0}^{\infty }e^{-\alpha t}P_{t}f(x)dt.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
There is enough joint measurability to check that
is a Markov resolvent on
,
which uniquely associated with the Markovian semigroup
.
Consequently, one may apply Fubini's theorem to see that
![{\displaystyle U^{\alpha }f(x)=\mathbf {E} ^{x}\left[\int _{0}^{\infty }e^{-\alpha t}f(X_{t})dt\right].}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The following are the defining properties of Borel right processes:[1]
- For each probability measure
on
, there exists a probability measure
on
such that
is a Markov process with initial measure
and transition semigroup
.
- Let
be
-excessive for the resolvent on
. Then, for each probability measure
on
, a mapping given by
is
almost surely right continuous on
.
Notes
References
- Sharpe, Michael (1988), General Theory of Markov Processes, ISBN 0126390606