Stochastic equations with singular drift driven by fractional Brownian motion
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909861586927616 |
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| author | Butkovsky, Oleg Lê, Khoa Mytnik, Leonid |
| author_facet | Butkovsky, Oleg Lê, Khoa Mytnik, Leonid |
| contents | We consider stochastic differential equation $$ d X_t=b(X_t) dt +d W_t^H, $$ where the drift $b$ is either a measure or an integrable function, and $W^H$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1)$, $d\in\mathbb{N}$. For the case where $b\in L_p(\mathbb{R}^d)$, $p\in[1,\infty]$ we show weak existence of solutions to this equation under the condition $$ \frac{d}p<\frac1H-1, $$ which is an extension of the Krylov-Röckner condition (2005) to the fractional case. We construct a counter-example showing optimality of this condition. If $b$ is a Radon measure, particularly the delta measure, we prove weak existence of solutions to this equation under the optimal condition $H<\frac1{d+1}$. We also show strong well-posedness of solutions to this equation under certain conditions. To establish these results, we utilize the stochastic sewing technique and develop a new version of the stochastic sewing lemma. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_11937 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stochastic equations with singular drift driven by fractional Brownian motion Butkovsky, Oleg Lê, Khoa Mytnik, Leonid Probability 60H50, 60H10, 60G22 We consider stochastic differential equation $$ d X_t=b(X_t) dt +d W_t^H, $$ where the drift $b$ is either a measure or an integrable function, and $W^H$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1)$, $d\in\mathbb{N}$. For the case where $b\in L_p(\mathbb{R}^d)$, $p\in[1,\infty]$ we show weak existence of solutions to this equation under the condition $$ \frac{d}p<\frac1H-1, $$ which is an extension of the Krylov-Röckner condition (2005) to the fractional case. We construct a counter-example showing optimality of this condition. If $b$ is a Radon measure, particularly the delta measure, we prove weak existence of solutions to this equation under the optimal condition $H<\frac1{d+1}$. We also show strong well-posedness of solutions to this equation under certain conditions. To establish these results, we utilize the stochastic sewing technique and develop a new version of the stochastic sewing lemma. |
| title | Stochastic equations with singular drift driven by fractional Brownian motion |
| topic | Probability 60H50, 60H10, 60G22 |
| url | https://arxiv.org/abs/2302.11937 |