We show that \(\mathbb {P}( \ell _X(0,T] \le 1)=(c_X+o(1))T^{-(1-H)}\), 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 \(\mathbb {P}( \ell _X(0,T] \le 1)\). 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.

我们证明\（\ mathbb {P}（\ ell _X（0，T] \ le 1）=（c_X + o（1））T ^ {-（1-H）} \），其中\（\ ell _X \）是任何具有固定增量的递归H-自相似实值过程X在0处的局部时间度量，它允许有足够规则的局部时间，并且\（c_X \）是仅取决于X的某个常数。是高斯设置，即当基础过程是分数布朗运动时，在该过程中我们的结果解决了Molchan [Commun。Math。Phys。205，97-111（1999）]的猜想，该猜想获得了上界\（1-H \）关于\（\ mathbb {P}（\ ell _X（0，T] \ le 1）\）的衰减指数。我们的方法在持久性概率与Palm理论之间建立了新的联系，用于自相似随机度量，从而提供了远远超出高斯案例的一般框架。