Let $B^H$ be a fractional Brownian motion on $\mathbb{R}$ with Hurst parameter $H\in (0,1)$ and let F be its pathwise antiderivative (so F is a differentiable random function such that $F^{\prime }(x)=B_x^H$) with $F(0)=0$. Let B be a standard Brownian motion, independent of $B^H$. We show that the zero energy part $A_t=F(B_t)-{\int }_0^t{F}^{\prime }(B_s)d{B}_s$ of $F(B)$ has positive and finite p-th 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 $4∕3$-th variation is positive and finite.