  Let $B^H$ be a fractional Brownian motion on $\mathbb{R}$ with Hurst
parameter $H\in(0,1)$, $F$ be its pathwise antiderivative with $F(0)=0$, and
let $B$ be a standard Brownian motion, independent of $B^H$. We show that the
zero energy part $A_t=F(B_t)-\int_0^t F'(B_s)dB_s$ of $F(B)$ has positive and
finite $p$-variation in a special sense for $p_0=\frac{2}{1+H}$. We also
present some simulation results about the zero energy part of a certain median
process which suggest that its $4/3$-variation is positive and finite.
