Space Regularity of Evolution Equations Driven by Rough Paths
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910412059967488 |
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| author | Addona, Davide Lorenzi, Luca Tessitore, Gianmario |
| author_facet | Addona, Davide Lorenzi, Luca Tessitore, Gianmario |
| contents | In this paper, we consider the linear evolution equation $dy(t)=Ay(t)dt+Gy(t)dx(t)$, where $A$ is a closed operator, associated to a semigroup, with good smoothing effects in a Banach space $E$, $x$ is a nonsmooth path, which is $η$-Hölder continuous for some $η\in (1/3,1/2]$, and $G$ is a non-smoothing linear operator on $E$. We prove that the Cauchy problem associated with the previous equation admits a unique mild solution and we also show that the solution increases the regularity of the initial datum as soon as time evolves. Then, we show that the mild solution is also an integral solution and this allows us to prove a Itô formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10650 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Space Regularity of Evolution Equations Driven by Rough Paths Addona, Davide Lorenzi, Luca Tessitore, Gianmario Analysis of PDEs Primary: 60L50, Secondary: 60H05, 60H15, 47D06, 35R60 In this paper, we consider the linear evolution equation $dy(t)=Ay(t)dt+Gy(t)dx(t)$, where $A$ is a closed operator, associated to a semigroup, with good smoothing effects in a Banach space $E$, $x$ is a nonsmooth path, which is $η$-Hölder continuous for some $η\in (1/3,1/2]$, and $G$ is a non-smoothing linear operator on $E$. We prove that the Cauchy problem associated with the previous equation admits a unique mild solution and we also show that the solution increases the regularity of the initial datum as soon as time evolves. Then, we show that the mild solution is also an integral solution and this allows us to prove a Itô formula. |
| title | Space Regularity of Evolution Equations Driven by Rough Paths |
| topic | Analysis of PDEs Primary: 60L50, Secondary: 60H05, 60H15, 47D06, 35R60 |
| url | https://arxiv.org/abs/2404.10650 |