Universality for persistence exponents of local times of self-similar processes with stationary increments

Abstract

We show that P(ℓ X (0, T ] ≤ 1) = (c X + 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 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 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.