Borel right process (original) (raw)
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 ; foreach , 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 , The following are the defining properties of Borel right processes:
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| dbo:abstract | 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 ; foreach , 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 respectto 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: * 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 . (en) |
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| rdfs:comment | 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 ; foreach , 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 , The following are the defining properties of Borel right processes: (en) |
| rdfs:label | Borel right process (en) |
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