Derivative for the intersection local time of fractional Brownian Motions

Let $B^{H_1}$ and $\tilde{B}^{H_2}$ be two independent fractional Brownian motions on ${\mathbb R}$ with respective indices $H_i\in (0,1)$ and $H_1\leq H_2$. In this paper, we consider their intersection local time $\ell_t(a)$. We show that $\ell_t(a)$ is differentiable in the spatial variable if $\frac1{H_1}+\frac1{H_2}>3$, and we introduce the so-called {\it hybrid quadratic covariation} $[f(B^{H_1}-\tilde{B}^{H_2}),B^{H_1}]^{(HC)}$. When $H_1<\frac12$, we construct a Banach space ${\mathscr H}$ of measurable functions such that the quadratic covariation exists in $L^2(\Omega)$ for all $f\in {\mathscr H}$, and the Bouleau-Yor type identity $$ [f(B^{H_1}-\tilde{B}^{H_2}),B^{H_1}]^{(HC)}_t=-\int_{\mathbb R}f(a)\ell_t(da) $$ holds. When $H_1\geq \frac12$, we show that the quadratic covariation exists also in $L^2(\Omega)$ and the above Bouleau-Yor type identity holds also for all H\"older functions $f$ of order $\nu>\frac{2H_1-1}{H_1}$.