Derivative of intersection local time of independent symmetric stable motions

Let X and X̃ be two mutually independent symmetric stable motions in R1 with respective indices α and α̃. We show that the intersection local time βt(x) of X and X̃ is differentiable in the spatial variable if α+α̃>3, and moreover we have that the p-variation of the derivative βt′(0) is zero when p>2α∨α̃α∨α̃+α+α̃−3.