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

Abstract

We 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.

Self-similar co-ascent processes and Palm calculus

“…Two-sided processes with stationary increments. In [19], massstationarity was used to derive the strong asymptotics of the quantity P( ((0, t]) ≤ 1) as t → ∞, where is the local time measure at 0 of an H-self-similar process X with stationary increments. The following related problem still remains open, see also [18,1,2]: Problem 8.…”

“…From the point of view of applications in the context of stochastic processes, Palm theory has proven very fruitful in tackling problems related to embedding distributions (of random variables or random functions) into Brownian paths, see [12,24,21,13], and also [20] for an application in discrete time. For non-Markovian processes, a related technique based on Zähle's approach in [29] has been employed in [19] to derive the persistence exponents of local times of self-similar processes with stationary increments.…”

Persistence probabilities of mixed FBM and other mixed processes

“…In the case that self-similarity is not available, the property of stationary increments turned out to be appropriate as another property that can be used to prove the existence of the persistence exponent, see [5]. Besides, one could derive persistence results even outside of the Gaussian setting if one assumes both self-similarity and stationary increments, see [7,38].…”