Inverse local times of fractional Brownian motion

In this paper, we study the inverse local time of a dd-dimensional fractional Brownian motion. We obtain that if L−1(t)L−1(t) is the inverse local time of fractional Brownian motion of Hurst index 0<β<10<β<1 with βd<1βd<1, then ∫0∞E(exp(−λL−1(t)))dt=Γ(1−βd)/λ1−βd(2π)d/2. This result raises the question whether the inverse local time of a fractional Brownian motion has a stable law of parameter 1−βd1−βd as it is the case for the standard one-dimensional Brownian motion.