平稳增量自相似过程局部时持续指数的普适性
Universality for persistence exponents of local times of self-similar processes with stationary increments
摘要：
我们证明了$\mathbb{P} ( \ell_X(0,T] \leq 1)=(c_X+o(1))T^{-(1-H)}$，其中$\ell_X$是任何常返$H$ -自相似实值过程$X$在$0$上的局部时间测度，其具有允许充分正则局部时间的平稳增量，并且$c_X$是仅依赖于$X$的常数。一个特殊的情况是高斯设置，即当底层过程是分数布朗运动，其中我们的结果解决了一个猜想Molchan [Commun. Math.Phys.205，97-111（1999）]，其获得了关于$\mathbb{P} ( \ell_X(0,T] \leq 1)$的衰变指数的上界$1-H$。我们的方法建立了一个新的连接持久性概率和Palm理论的自相似随机措施，从而提供了一个一般的框架，远远超出了高斯的情况下。
英文摘要：
We show that $\mathbb{P} ( \ell_X(0,T] \leq 1)=(c_X+o(1))T^{-(1-H)}$, where $\ell_X$ is the local time measure at $0$ of any recurrent $H$-self-similar real-valued process $X$ with stationary increments that admits a sufficiently regular local time and $c_X$ is some constant depending only on $X$. A special case is the Gaussian setting, i.e. when the underlying process is fractional Brownian motion, in which our result settles a conjecture by Molchan [Commun. Math. Phys. 205, 97-111 (1999)] who obtained the upper bound $1-H$ on the decay exponent of $\mathbb{P} ( \ell_X(0,T] \leq 1)$. Our approach establishes a new connection between persistence probabilities and Palm theory for self-similar random measures, thereby providing a general framework which extends far beyond the Gaussian case.
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