Christian Mönch, TU Darmstadt: Universality of persistence exponents for self-similar processes with stationary increments (Oberseminar Mathematische Stochastik)
Mittwoch, 22.05.2019 17:00 im Raum SRZ 117
In 1999, G. Molchan showed that for a centered fractional Brownian motion X on the real line \(Prob[X(t)< 1, 1< t< T]=T^(H-1+o(1))\), where $H$ is the index of self-similarity of X. Furthermore, he showed that the same tail exponent occurs for several other path functionals of X and conjectured that it also governs the tail of \(Prob[L(t)< 1, 1< t< T]\), where L is the local time at 0 of X, but was only able to prove a lower bound. In this talk, we present an entirely novel approach to persistence problems for self-similar processes with stationary increments based on Palm theory. Our technique is not limited to Gaussian processes, allows us to resolve Molchan's conjecture for ANY H-self-similar process with stationary increments that admits a sufficiently smooth local time and provides a better error estimate even for the known lower bound of \(Prob[L(t)<1, 1< t<T]\) in the fractional Brownian case. Furhermore, we recover in the more general setting the tail asymptotics of \(Prob[X(t)<1, 1<t<T]\) and some of the other path functionals orignially considered in Molchan's work, using our approach in combination with universal lower persistence bounds recently shown to hold by F. Aurzada, N. Guillotin-Plantard and F. Pène.