Polar Sets and Multiple Points for Super-Brownian Motion

We study the closed support of the measure-valued diffusions of Watanabe and Dawson. When the spatial motion is Brownian, sufficient conditions involving capacity are given for a fixed set to be hit by the $k$-multiple points of the support process. The conditions are close to the necessary conditions found by Dawson, Iscoe and Perkins and lead to necessary and sufficient conditions for the existence of $k$-multiple points. When the spatial motion is a symmetric stable process of index $\alpha < 2$, the closed support is shown to be $\mathbb{R}^d$ or $\varnothing$.