On the exponential integrability of the derivative of intersection and self-intersection local time for fractional Brownian motion and a limit theorem related to the self-intersection local time for fractional Brownian motion
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914085080137728 |
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| author | Das, Kaustav Markowsky, Gregory Wu, Binghao Yu, Qian |
| author_facet | Das, Kaustav Markowsky, Gregory Wu, Binghao Yu, Qian |
| contents | We give the correct condition for existence of the $k$-th derivative of the intersection local time for fractional Brownian motion, which was originally discussed in [Guo, J., Hu, Y., and Xiao, Y., Higher-order derivative of intersection local time for two independent fractional Brownian motions, Journal of Theoretical Probability 32, (2019), pp. 1190-1201]. We also show that the $k$-th derivative of the intersection and self-intersection local times of fractional Brownian motion are exponentially integrable for certain parameter values. In addition, we show convergence in distribution when the existence condition is violated for the $k$-th derivative of self-intersection local time of fractional Brownian motion under scaling. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the exponential integrability of the derivative of intersection and self-intersection local time for fractional Brownian motion and a limit theorem related to the self-intersection local time for fractional Brownian motion Das, Kaustav Markowsky, Gregory Wu, Binghao Yu, Qian Probability 60J55, 60G22 We give the correct condition for existence of the $k$-th derivative of the intersection local time for fractional Brownian motion, which was originally discussed in [Guo, J., Hu, Y., and Xiao, Y., Higher-order derivative of intersection local time for two independent fractional Brownian motions, Journal of Theoretical Probability 32, (2019), pp. 1190-1201]. We also show that the $k$-th derivative of the intersection and self-intersection local times of fractional Brownian motion are exponentially integrable for certain parameter values. In addition, we show convergence in distribution when the existence condition is violated for the $k$-th derivative of self-intersection local time of fractional Brownian motion under scaling. |
| title | On the exponential integrability of the derivative of intersection and self-intersection local time for fractional Brownian motion and a limit theorem related to the self-intersection local time for fractional Brownian motion |
| topic | Probability 60J55, 60G22 |
| url | https://arxiv.org/abs/2510.00456 |