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\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d71e1ce2ca2a456b13dd259d338eeadc41c30ab2)
![{\displaystyle {\mathcal {F}}_{t}^{*}=\sigma \left\{X_{s}^{-1}(B):s\in [0,t],B\in {\mathcal {E}}^{*}\right\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ad0db6b3f6501bed46d401cc7bd4ff7f5b9a5e)
and then, let
![{\displaystyle {\mathcal {F}}_{\infty }=\sigma \left\{X_{s}^{-1}(B):s\in [0,\infty ),B\in {\mathcal {E}}\right\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f3a5e966c84709be6a806a820b389a7ff5663d)
![{\displaystyle {\mathcal {F}}_{\infty }^{*}=\sigma \left\{X_{s}^{-1}(B):s\in [0,\infty ),B\in {\mathcal {E}}^{*}\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd7325599674444efd0961ca5c19d18ce3c52f4b)
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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbcb6bcdfd5b5f17f331d6e0d2f04446c526720e)
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/164e4a7d46b41f6ef9de7247e83ae0833014de37)
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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab11a46f95ce2f2dd0a999488d848b26c8827892)
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