Universality for Persistence Exponents of Local Times of Self-Similar Processes with Stationary Increments

We show that P(l(X)(0,T]<= 1)=(cX+o(1))(T-(1-H)), where l(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 cX 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 P(l(X)(0,T]<= 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.