Rough path metrics on a Besov–Nikolskii-type scale
Rough path metrics on a Besov–Nikolskii-type scale
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DOI：
10.1090/tran/7264
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发表时间：
2016-09
影响因子：
1.3
通讯作者：
P. Friz;David J. Promel
中科院分区：
文献类型：
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作者：
P. Friz;David J. Promel
这是众所周知的，因为开创性的工作[T。Lyons，粗糙信号驱动的微分方程，Rev. Mat. Iberoamericana, 14(1998)]，与受控微分方程相关的解映射在$q$ -变分响应中是局部Lipschitz连续的。$1/q$ -Hölder在粗糙路径的空间上键入度量，对于任何规律性$1/q \in (0,1]$。我们将其推广到一类新的besov - nikolskii型度量，具有任意正则性$1/q\in (0,1]$和可积性$p\in [ q,\infty ]$，其中情况$p\in \{ q,\infty \} $对应于已知情况。有趣的是，结果是通过已知的$q$ -变异粗略路径估计得到的。
It is known, since the seminal work [T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998)], that the solution map associated to a controlled differential equation is locally Lipschitz continuous in $q$-variation resp. $1/q$-H\"{o}lder type metrics on the space of rough paths, for any regularity $1/q \in (0,1]$. We extend this to a new class of Besov-Nikolskii-type metrics, with arbitrary regularity $1/q\in (0,1]$ and integrability $p\in [ q,\infty ]$, where the case $p\in \{ q,\infty \} $ corresponds to the known cases. Interestingly, the result is obtained as consequence of known $q$-variation rough path estimates.