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

Abstract

AbstractWe show that $$\mathbb {P}( \ell _X(0,T] \le 1)=(c_X+o(1))T^{-(1-H)}$$ P ( ℓ X ( 0 , T ] ≤ 1 ) = ( c X + o ( 1 ) ) T - ( 1 - H ) , where $$\ell _X$$ 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 $$c_X$$ 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 $$1-H$$ 1 - H on the decay exponent of $$\mathbb {P}( \ell _X(0,T] \le 1)$$ 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.