Continuous flows driving Markov processes and multiplicative $L^p$-semigroups
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917837500579840 |
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| author | Beznea, Lucian Bezzarga, Mounir Cimpean, Iulian |
| author_facet | Beznea, Lucian Bezzarga, Mounir Cimpean, Iulian |
| contents | We develop a method of driving a Markov processes through a continuous flow. In particular, at the level of the transition functions we investigate an approach of adding a first order operator to the generator of a Markov process, when the two generators commute. A relevant example is a measure-valued superprocess having a continuous flow as spatial motion and a branching mechanism which does not depend on the spatial variable. We prove that any flow is actually continuous in a convenient topology and we show that a Markovian multiplicative semigroup on an Lp space is generated by a continuous flow, completing the answer to the question whether it is enough to have a measurable structure, like a C0-semigroup of Markovian contractions on an $L^p$-space with no fixed topology, in order to ensure the existence of a right Markov process associated to the given semigroup. We extend from bounded to unbounded functions the weak generator (in the sense of Dynkin) and the corresponding martingale problem |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09407 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Continuous flows driving Markov processes and multiplicative $L^p$-semigroups Beznea, Lucian Bezzarga, Mounir Cimpean, Iulian Probability We develop a method of driving a Markov processes through a continuous flow. In particular, at the level of the transition functions we investigate an approach of adding a first order operator to the generator of a Markov process, when the two generators commute. A relevant example is a measure-valued superprocess having a continuous flow as spatial motion and a branching mechanism which does not depend on the spatial variable. We prove that any flow is actually continuous in a convenient topology and we show that a Markovian multiplicative semigroup on an Lp space is generated by a continuous flow, completing the answer to the question whether it is enough to have a measurable structure, like a C0-semigroup of Markovian contractions on an $L^p$-space with no fixed topology, in order to ensure the existence of a right Markov process associated to the given semigroup. We extend from bounded to unbounded functions the weak generator (in the sense of Dynkin) and the corresponding martingale problem |
| title | Continuous flows driving Markov processes and multiplicative $L^p$-semigroups |
| topic | Probability |
| url | https://arxiv.org/abs/2411.09407 |