| Abstract: |
| We consider the local times of $(1 + \beta)$-stable super-Brownian motion with $0 < \beta < 1$. It is known from Sugitani (J. Math. Soc. Japan, 41(3), 437--462, 1989) that for $\beta = 1$, the local time is differentiable in $d=1$. For $0 < \beta < 1$, Mytnik and Perkins (Ann. Probab., 31(3), 1413--1440, 2003) proved that the local time, denoted by $L(t, x)$, is jointly continuous in $d = 1$ while it is locally unbounded in $x$ in $d \ge 2$ where it exists. This paper strengthens the results of Mytnik and Perkins for $d=1$ by showing that the local time $L(t, x)$ is continuously differentiable in the spatial parameter $x$. Moreover, we give a representation of the spatial derivative, denoted by $\frac{\partial}{\partial x}L(t, x)$, and further prove that the derivative is locally $\gamma$-H\older continuous in $x$ with any index $\gamma \in (0, \frac{\beta}{1+\beta})$. |
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