On the differentiability of the local time of the ($1+\beta$)-stable super-Brownian motion

We consider the local time of the ($1+\beta$)-stable super-Brownian motion with $0<\beta <1$. It is shown by Mytnik and Perkins ({\em Ann. Probab.}, 31(3), 1413 – 1440, (2003)) that the local time, denoted by $L(t,x)$, is jointly continuous in $d=1$ while $L(t,x)$ is locally unbounded in $x$ in $d\ge 2$ where it exists. This paper shows that the local time is continuously differentiable in the spatial parameter $x$ in $d=1$. Moreover, we give a representation of the spatial derivative of the local time, denoted by $\frac{\mathrm{\partial }}{\mathrm{\partial }x}L(t,x)$, and further prove that the derivative is locally $\gamma$-Hölder continuous in $x$ with any index $\gamma \in (0,\frac{\beta }{1+\beta })$.