Pitman's discrete $2M-X$ theorem for arbitrary initial laws and continuous time limits
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914998777806848 |
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| author | Bryc, Wlodzimierz Wesolowski, Jacek |
| author_facet | Bryc, Wlodzimierz Wesolowski, Jacek |
| contents | We discuss Pitman's representation of a Markov process, which serves as a discrete analog to the Bessel 3D process starting at time 0 from an arbitrary initial law. This representation involves maxima of lazy simple random walks and an auxiliary independent random variable.
The law of the auxiliary random variable is explicitly related to the initial law of the Markov process.
The proof is kept at an elementary level and relies on a reconstruction formula for the generalized Pitman transform.
We then use continuous-time limits to shed additional light on the relation between two representations of the Bessel 3D process that appeared in the description of the stationary measure of the KPZ fixed point on the half-line, as proposed by Barraquand and Le Doussal (2022). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_07144 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pitman's discrete $2M-X$ theorem for arbitrary initial laws and continuous time limits Bryc, Wlodzimierz Wesolowski, Jacek Probability We discuss Pitman's representation of a Markov process, which serves as a discrete analog to the Bessel 3D process starting at time 0 from an arbitrary initial law. This representation involves maxima of lazy simple random walks and an auxiliary independent random variable. The law of the auxiliary random variable is explicitly related to the initial law of the Markov process. The proof is kept at an elementary level and relies on a reconstruction formula for the generalized Pitman transform. We then use continuous-time limits to shed additional light on the relation between two representations of the Bessel 3D process that appeared in the description of the stationary measure of the KPZ fixed point on the half-line, as proposed by Barraquand and Le Doussal (2022). |
| title | Pitman's discrete $2M-X$ theorem for arbitrary initial laws and continuous time limits |
| topic | Probability |
| url | https://arxiv.org/abs/2304.07144 |