Remark on logically constant self-similar processes

Abstract

Let X = {X(t), t∈ℝ+} be as self-similar processes with index α>0. We show that if X is locally constant and ℙ{X(1)=0}=0, then the law of X(t) is absolutely continuous. We discuss applicants of this result to homogeneous functions of a multidimensional fractional Brownian motion.