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We consider a continuum percolation model on $\R^d$, $d\geq 1$.For$t,\lambda\in (0,\infty)$ and $d\in\{1,2,3\}$, the occupied set is given by theunion of independent Brownian paths running up to time $t$ whoseinitial pointsform a Poisson point process with intensity $\lambda\textgreater{}0$.When$d\geq 4$, the Brownian paths are replaced by Wiener sausageswith radius$r\textgreater{}0$.We establish that, for $d=1$ and all choices of $t$, nopercolation occurs,whereas for $d\geq 2$, there is a non-trivial percolationtransitionin $t$, provided $\lambda$ and $r$ are chosen properly.The laststatement means that $\lambda$ has to be chosen to be strictly smaller than thecritical percolation parameter for the occupied set at time zero(which isinfinite when $d\in\{2,3\}$, but finite and dependent on $r$ when $d\geq 4$).Wefurther show that for all $d\geq 2$, the unbounded cluster in the supercriticalphase is unique.Along the way a finite box criterion for non-percolation in theBoolean model is extended to radius distributions with an exponential tail.This may be of independent interest.The present paper settles the basicproperties of the model and should be viewed as a jumpboard for finer results.

Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster