A reverse ergodic theorem for inhomogeneous killed Markov chains and application to a new uniqueness result for reflecting diffusions
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| Format: | Preprint |
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2021
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| _version_ | 1866910565177229312 |
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| author | Costantini, Cristina Kurtz, Thomas G. |
| author_facet | Costantini, Cristina Kurtz, Thomas G. |
| contents | Bass and Pardoux (1987) deduce from the Krein-Rutman theorem a reverse ergodic theorem for a sub-probability transition function, which turns out to be a key tool in proving uniqueness of reflecting Brownian Motion in cones in Kwon and Williams (1991) and Taylor and Williams (1993). By a different approach, we are able to prove an analogous reverse ergodic theorem for a family of inhomogeneous sub-probability transition functions. This allows us to prove existence and uniqueness for a semimartingale diffusion process with varying, oblique direction of reflection, in a domain with one singular point that can be approximated, near the singular point, by a smooth cone, under natural, easily verifiable geometric conditions.
Along the way we also show that if the reflecting Brownian motion in a smooth cone is a semimartingale then the parameter $α$ of Kwon and Williams (1991) is strictly less than 1, thus partially extending the results of Williams (1985) to higher dimension. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2106_07208 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A reverse ergodic theorem for inhomogeneous killed Markov chains and application to a new uniqueness result for reflecting diffusions Costantini, Cristina Kurtz, Thomas G. Probability 60J60, 60H10, 60J55, 60G17 Bass and Pardoux (1987) deduce from the Krein-Rutman theorem a reverse ergodic theorem for a sub-probability transition function, which turns out to be a key tool in proving uniqueness of reflecting Brownian Motion in cones in Kwon and Williams (1991) and Taylor and Williams (1993). By a different approach, we are able to prove an analogous reverse ergodic theorem for a family of inhomogeneous sub-probability transition functions. This allows us to prove existence and uniqueness for a semimartingale diffusion process with varying, oblique direction of reflection, in a domain with one singular point that can be approximated, near the singular point, by a smooth cone, under natural, easily verifiable geometric conditions. Along the way we also show that if the reflecting Brownian motion in a smooth cone is a semimartingale then the parameter $α$ of Kwon and Williams (1991) is strictly less than 1, thus partially extending the results of Williams (1985) to higher dimension. |
| title | A reverse ergodic theorem for inhomogeneous killed Markov chains and application to a new uniqueness result for reflecting diffusions |
| topic | Probability 60J60, 60H10, 60J55, 60G17 |
| url | https://arxiv.org/abs/2106.07208 |