Continuous Differentiability of Renormalized Intersection Local Times in R^{1}

We study ${\gamma }_k(x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_k;t)$, the k-fold renormalized self-intersection local time for Brownian motion in $R^1$. Our main result says that ${\gamma }_k(x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_k;t)$ is continuously differentiable in the spatial variables, with probability 1.