On locally constant self-similar processes

Let $X=\{X(t);\,t\in R_+\}$ be a real self-similar process. We say that $X$ is locally constant (LC) if for each $t>0$ with probability $1$ there exist $\epsilon>0$ such that $X(s)=X(t)$ for all $s\in(t-\epsilon,t+\epsilon)$.
We show that if a LC process $X$ is obtained by Lamperti transformation from an ergodic stationary process, then the law of $X(t)$ is absolutely continuous. Different examples are discussed. Special attention is payed to FBM and a large class of max-stable processes.