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

2018/11/06 by Mönch, Christian
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1811.02417

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.

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