Universal limit theorem for rough differential equations driven by controlled rough paths
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866915997115482112 |
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| author | Li, Nannan Gao, Xing |
| author_facet | Li, Nannan Gao, Xing |
| contents | We study rough differential equations driven by controlled rough paths in the level-$2$ regime $1/3<α\le 1/2$. Given a reference rough path $\mathbf X=(1,X,\mathbb X)$ and an $\mathbf X$-controlled driver $\mathbf Z=(Z,Z')$, we first give a point-removal construction of the controlled rough integral $
\int_s^t Y_r\,d\mathbf Z_r $
and prove the corresponding remainder estimates. We then establish local and global well-posedness for the controlled-driven rough differential equation $
dY_t=F(Y_t)\,d\mathbf Z_t. $
A key structural result is the canonical lift of the controlled driver: from the controlled data $(\mathbf X,\mathbf Z)$ we construct a level-$2$ rough path \[
\widehat{\mathbf Z}=(1,Z,\mathbb Z),
\qquad
\mathbb Z_{s,t}:=\int_s^t Z_{s,u}\otimes dZ_u, \] and show that the controlled-driven equation is equivalent to the classical rough differential equation driven by $\widehat{\mathbf Z}$. This equivalence shows compatibility with classical rough path theory, while the controlled formulation keeps track of the dependence of the effective driver $Z$ on the reference rough path $\X$. Finally, we prove a universal limit theorem for the solution map $
(\mathbf X,\mathbf Z,Y_0)\longmapsto Y, $
which gives stability with respect to perturbations of the initial condition, the reference rough path, and the controlled driver. These results provide a natural framework for layered rough systems and equations driven by transformed or previously evolved rough signals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_09158 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Universal limit theorem for rough differential equations driven by controlled rough paths Li, Nannan Gao, Xing Probability 60L20, 60L50, 60H15, 60H17 We study rough differential equations driven by controlled rough paths in the level-$2$ regime $1/3<α\le 1/2$. Given a reference rough path $\mathbf X=(1,X,\mathbb X)$ and an $\mathbf X$-controlled driver $\mathbf Z=(Z,Z')$, we first give a point-removal construction of the controlled rough integral $ \int_s^t Y_r\,d\mathbf Z_r $ and prove the corresponding remainder estimates. We then establish local and global well-posedness for the controlled-driven rough differential equation $ dY_t=F(Y_t)\,d\mathbf Z_t. $ A key structural result is the canonical lift of the controlled driver: from the controlled data $(\mathbf X,\mathbf Z)$ we construct a level-$2$ rough path \[ \widehat{\mathbf Z}=(1,Z,\mathbb Z), \qquad \mathbb Z_{s,t}:=\int_s^t Z_{s,u}\otimes dZ_u, \] and show that the controlled-driven equation is equivalent to the classical rough differential equation driven by $\widehat{\mathbf Z}$. This equivalence shows compatibility with classical rough path theory, while the controlled formulation keeps track of the dependence of the effective driver $Z$ on the reference rough path $\X$. Finally, we prove a universal limit theorem for the solution map $ (\mathbf X,\mathbf Z,Y_0)\longmapsto Y, $ which gives stability with respect to perturbations of the initial condition, the reference rough path, and the controlled driver. These results provide a natural framework for layered rough systems and equations driven by transformed or previously evolved rough signals. |
| title | Universal limit theorem for rough differential equations driven by controlled rough paths |
| topic | Probability 60L20, 60L50, 60H15, 60H17 |
| url | https://arxiv.org/abs/2603.09158 |