Large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes

In this paper we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann–Liouville processes. We also show that a fractional Brownian motion and the related Riemann–Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of our large deviation estimates, we derive laws of iterated logarithm for the corresponding local times. The key points of our methods: (1) logarithmic superadditivity of a normalized sequence of moments of exponentially randomized local time of a fractional Brownian motion; (2) logarithmic subadditivity of a normalized sequence of moments of exponentially randomized intersection local time of Riemann–Liouville processes; (3) comparison of local and intersection local times based on embedding of a part of a fractional Brownian motion into the reproducing kernel Hilbert space of the Riemann–Liouville process. Key-words: local time, intersection local time, large deviations, fractional Brownian motion, Riemann–Liouville process, law of iterated logarithm. AMS subject classification (2010): 60G22, 60J55, 60F10, 60G15, 60G18. ∗Research partially supported by NSF grant DMS-0704024. †Research partially supported by NSF grant DMS–0805929. ‡Research partially supported by NSA grant MSPF-50G-049. §Research partially supported by Hong Kong RGC CERG 602206 and 602608. 1