Limit theorems for the generator of a symmetric Levy process with the delta potential
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913643082285056 |
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| author | Abildaev, Temirlan |
| author_facet | Abildaev, Temirlan |
| contents | We consider a one-dimensional symmetric Levy process that has local time. In the first part, we construct a self-adjoint extension of the generator of the process so that the constructed operator corresponds to the generator with the delta potential. Using the constructed operator, we extend the Feynman-Kac formula to the case of delta function-type potentials and prove a limit theorem for an operator semigroup corresponding to this formula. In the second part, we construct a one-parameter family of distributions that attract the sample paths of the process to a given point. We show that this family weakly converges to the distribution of a Feller process and prove a limit theorem for the distribution of a point where an attracted sample path comes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05522 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limit theorems for the generator of a symmetric Levy process with the delta potential Abildaev, Temirlan Probability 60G51, 60J55, 60F05, 47D08 (Primary) 60J35 (Secondary) We consider a one-dimensional symmetric Levy process that has local time. In the first part, we construct a self-adjoint extension of the generator of the process so that the constructed operator corresponds to the generator with the delta potential. Using the constructed operator, we extend the Feynman-Kac formula to the case of delta function-type potentials and prove a limit theorem for an operator semigroup corresponding to this formula. In the second part, we construct a one-parameter family of distributions that attract the sample paths of the process to a given point. We show that this family weakly converges to the distribution of a Feller process and prove a limit theorem for the distribution of a point where an attracted sample path comes. |
| title | Limit theorems for the generator of a symmetric Levy process with the delta potential |
| topic | Probability 60G51, 60J55, 60F05, 47D08 (Primary) 60J35 (Secondary) |
| url | https://arxiv.org/abs/2501.05522 |