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We consider connectivity properties of the vacant set of (random) ensemblesof Wiener sausages in $\mathbb R^d$ in the transient dimensions $d \geq 3$. Weprove that the vacant set of Brownian interlacements contains at most oneinfinite connected component almost surely. For finite ensembles of Wienersausages, we provide sharp polynomial bounds on the probability that theirvacant set contains at least $2$ connected components in microscopic balls. Themain proof ingredient is a sharp polynomial bound on the probability thatseveral Brownian motions visit jointly all hemiballs of the unit ball whileavoiding a slightly smaller ball.

On questions of uniqueness for the vacant set of Wiener sausages and Brownian interlacements