Local times and sample path properties of the Rosenblatt process

Abstract

Let $Z=(Z_t{)}_{t\ge 0}$ be the Rosenblatt process with Hurst index $H\in (1/2,1)$. We prove joint continuity for the local time of $Z$, and establish Hölder conditions for the local time. These results are then used to study the irregularity of the sample paths of $Z$. Based on analogy with similar known results in the case of fractional Brownian motion, we believe our results are sharp. A main ingredient of our proof is a rather delicate spectral analysis of arbitrary linear combinations of integral operators, which arise from the representation of the Rosenblatt process as an element in the second chaos.

28 pages