On the connected components of the support of super Brownian motion and of its exit measure

Tribe proved in a previous paper that a typical point of the support of super Brownian motion considered at a fixed time is a.s. disconnected from the others when the space dimension is greater than or equal to 3. We give here a simpler proof of this result based on Le Gallʹs Brownian snake. This proof can then be adapted in order to obtain an analogous result for the support of the exit measure of the super Brownian motion from a smooth domain of Rd when d is greater than or equal to 4.