The expected signature is an analogue of the Laplace transform for probability measures on rough paths. A key question in the area has been to identify a general condition to ensure that the expected signature uniquely determines the measures. A sufficient condition has recently been given by Chevyrev and Lyons and requires a strong upper bound on the expected signature. While the upper bound was verified for many well‐known processes up to a deterministic time, it was not known whether the required bound holds for random time. In fact, even the simplest case of Brownian motion up to the exit time of a planar disc was open. For this particular case we answer this question using a suitable hyperbolic projection of the expected signature. The projection satisfies a three‐dimensional system of linear PDEs, which (surprisingly) can be solved explicitly, and which allows us to show that the upper bound on the expected signature is not satisfied.

中文翻译：

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停在圆边界上的布朗运动的预期特征具有有限的会聚半径

预期签名是拉普拉斯变换的类似物，用于在粗糙路径上进行概率测度。该领域的一个关键问题是确定一般条件，以确保预期的签名唯一地确定措施。Chevyrev和Lyons最近给出了一个充分的条件，并要求在预期签名上有一个较高的上限。在确定性时间内已对许多知名过程的上限进行了验证，但尚不知道所需的上限是否适用于随机时间。实际上，即使是最简单的布朗运动（直到平面圆盘的出口时间）也已打开。对于这种特殊情况，我们使用期望签名的适当双曲投影来回答此问题。该投影满足线性PDE的三维系统，不能 满足。