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.
Highlights
Persistence Probabilities for Fractional Brownian MotionWe study local times of stochastic processes from the point of view of persistence probabilities, i.e. the probabilities that a stochastic process remains inside a relatively small subset of its state space for a long time
The starting point of our investigation is Molchan’s celebrated and classical results [27] concerning the persistence of fractional Brownian motion on the real line, which we briefly summarise
We extend the definition of H -self-similar with stationary increments (H -sssi) to processes indexed by [0, ∞) by restricting the stationarity of increments to positive shifts only
Summary
We study local times of stochastic processes from the point of view of persistence probabilities, i.e. the probabilities that a stochastic process remains inside a relatively small subset of its state space for a long time. Molchan’s proofs require some technical tools, namely Slepian’s lemma and reproducing kernel Hilbert spaces, which are specific to the Gaussian setting His argument crucially relies on the connection of the persistence probability to a certain path integral functional and this relation is useful outside the FBM context, see, for example, [3]. One way of looking at our result is that it makes this connection rigorous; our method enables us to prove (4) using only the invariance properties of the underlying processes, without recurrence to specific distributional structures such as Gaussianity, the Markov or martingale property, etc It is immediate from (4) that we have κ = κ for B and the author believes that this is true in a more general context. We fix our notation and present our main result for the local time persistence probabilities in a general setting, Theorem 1 and in two special cases, namely for FBM and the Rosenblatt process. We provide some useful results from the literature as well as some auxiliary calculations for convenience of the reader
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