论文标题
$ d $ d $ dimensional $ c^{2，α} $的预期签名，以$ d $ d $ -d $ -d $ c^{2，α} $ - 域在任何地方都有有限的收敛半径：$ 2 \ leq d \ leq 8 $
Expected signature of stopped Brownian motion on $d$-dimensional $C^{2, α}$-domains has finite radius of convergence everywhere: $2\leq d \leq 8$
论文作者
论文摘要
粗糙路径理论中的一个基本问题是，几何粗糙路径的预期特征是否完全决定了签名定律。一个足够的条件是预期的签名具有无限的收敛半径，这在固定时间间隔（包括布朗运动）上通过各种随机过程满足。相比之下，对于布朗尼的首次退出时间从有限的域$ω$停止了，只有知道在足够常规的$ω$上的预期签名半径的融合半径无处不在，并且当$ 2 $ - 2 $ dimemensional nimel dist dist $ω$时，收敛半径是有限的（[1]）。 在本文中，我们证明，在任何有限的$ c^{2，α} $ - 域$ω\ subset \ subset \ mathbb {r}^d $带有$ 2 \ leq d \ leq 8 $的$ 2 \ leq 8 $中，停止的布朗尼运动的预期标志是无处不在的有限radius。我们证明的关键要素是引入“域平均双曲线开发”（请参阅定义4.1），这使我们能够通过平均旋转域的平均pde系统来对称PDE系统，以实现预期签名的双曲线开发。
A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence, which is satisfied by various stochastic processes on a fixed time interval, including the Brownian motion. In contrast, for the Brownian motion stopped upon the first exit time from a bounded domain $Ω$, it is only known that the radius of convergence for the expected signature on sufficiently regular $Ω$ is strictly positive everywhere, and that the radius of convergence is finite at some point when $Ω$ is the $2$-dimensional unit disc ([1]). In this paper, we prove that on any bounded $C^{2,α}$-domain $Ω\subset \mathbb{R}^d$ with $2\leq d \leq 8$, the expected signature of the stopped Brownian motion has finite radius of convergence everywhere. A key ingredient of our proof is the introduction of a "domain-averaging hyperbolic development" (see Definition 4.1), which allows us to symmetrize the PDE system for the hyperbolic development of expected signature by averaging over rotated domains.
