Example of a Dirichlet process whose zero energy part has finite p-variation

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 $A_t=F(B_t)-{\int }_0^t{F}^{\mathrm{\prime }}(B_s)d{B}_s$ 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 $4\mathrm{/}3$-variation is positive and finite.