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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2411.13266 |
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| _version_ | 1866929599086067712 |
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| author | Wei, Jinlong Wang, Wei Lv, Guangying Duan, Jinqiao |
| author_facet | Wei, Jinlong Wang, Wei Lv, Guangying Duan, Jinqiao |
| contents | We introduce the Lebesgue--Hölder--Dini and Lebesgue--Hölder spaces $L^p(\mathbb{R};{\mathcal C}_{\vartheta,ς}^{α,ρ}({\mathbb R}^n))$ ($\vartheta\in \{l,b\}, \, ς\in \{d,s,c,w\}$, $p\in (1,+\infty]$ and $α\in [0,1)$), and then use a vector-valued Calderón--Zygmund theorem to establish the maximal Lebesgue--Hölder--Dini and Lebesgue--Hölder regularity for a class of parabolic equations. As an application, we obtain the unique strong solvability of the following stochastic differential equation \begin{eqnarray*} X_{s,t}(x)=x+\int\limits_s^tb(r,X_{s,r}(x))dr+W_t-W_{s}, \ \ t\in [s,T], \ x\in \mathbb{R}^n, \ s\in [0,T], \end{eqnarray*} for the low regularity growing drift in critical Lebesgue--Hölder--Dini spaces $L^p([0,T];{\mathcal C}^{\frac{2}{p}-1,ρ}_{l,d}({\mathbb R}^n;{\mathbb R}^n))$ ($p\in (1,2]$), where $\{W_t\}_{0\leq t\leq T}$ is a $n$-dimensional standard Wiener process. In particular, when $p=2$ we give a partially affirmative answer to a longstanding open problem, which was proposed by Krylov and Röckner for $b\in L^2([0,T];L^\infty({\mathbb R}^n;{\mathbb R}^n))$ based upon their work ({\em Probab. Theory Relat. Fields 131(2): 154--196, 2005}). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13266 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A new maximal regularity for parabolic equations and an application Wei, Jinlong Wang, Wei Lv, Guangying Duan, Jinqiao Probability We introduce the Lebesgue--Hölder--Dini and Lebesgue--Hölder spaces $L^p(\mathbb{R};{\mathcal C}_{\vartheta,ς}^{α,ρ}({\mathbb R}^n))$ ($\vartheta\in \{l,b\}, \, ς\in \{d,s,c,w\}$, $p\in (1,+\infty]$ and $α\in [0,1)$), and then use a vector-valued Calderón--Zygmund theorem to establish the maximal Lebesgue--Hölder--Dini and Lebesgue--Hölder regularity for a class of parabolic equations. As an application, we obtain the unique strong solvability of the following stochastic differential equation \begin{eqnarray*} X_{s,t}(x)=x+\int\limits_s^tb(r,X_{s,r}(x))dr+W_t-W_{s}, \ \ t\in [s,T], \ x\in \mathbb{R}^n, \ s\in [0,T], \end{eqnarray*} for the low regularity growing drift in critical Lebesgue--Hölder--Dini spaces $L^p([0,T];{\mathcal C}^{\frac{2}{p}-1,ρ}_{l,d}({\mathbb R}^n;{\mathbb R}^n))$ ($p\in (1,2]$), where $\{W_t\}_{0\leq t\leq T}$ is a $n$-dimensional standard Wiener process. In particular, when $p=2$ we give a partially affirmative answer to a longstanding open problem, which was proposed by Krylov and Röckner for $b\in L^2([0,T];L^\infty({\mathbb R}^n;{\mathbb R}^n))$ based upon their work ({\em Probab. Theory Relat. Fields 131(2): 154--196, 2005}). |
| title | A new maximal regularity for parabolic equations and an application |
| topic | Probability |
| url | https://arxiv.org/abs/2411.13266 |