1. Notes

2. References

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

and then, let

For each Borel measurable function on , define, for each ,

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

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

The following are the defining properties of Borel right processes:^[1]

• Hypothesis Droite 1:

For each probability measure on , there exists a probability measure on such that is a Markov process with initial measure and transition semigroup .

• Hypothesis Droite 2:

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

• {{citation|last=Sharpe|first=Michael|author-link=|title=General Theory of Markov Processes|edition=|year=1988|isbn=0126390606}}

• Cherry-pick
• Cherry picking (basketball)
• Cherry picking (disambiguation)
• Cherry picking tax avoidance
• Cherry Pie Picache
• Cherry pitter
• Cherry Plain
• Cherry Plain, New York
• Cherry Point, Alberta
• Cherry Point Marine Corps Air Station
• Cherrypoppers
• Cherry Poppin' Daddies discography
• Cherry Popsicle
• Cherry Pottage
• Cherry R. C.
• Cherry R C
• Cherry RC
• Cherry Red (Law & Order: Criminal Intent episode)
• Cherry-red spot
• Cherry red spots
• Cherry red TV
• Cherry Ridge Township, Pennsylvania
• Cherrys
• Cherry Saku Yuki!!
• Cherry Saku Yuuki!!