  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 the
union of independent Brownian paths running up to time $t$ whoseinitial points
form 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 $t$, no
percolation occurs,whereas for $d\geq 2$, there is a non-trivial percolation
transitionin $t$, provided $\lambda$ and $r$ are chosen properly.The last
statement means that $\lambda$ has to be chosen to be strictly smaller than the
critical percolation parameter for the occupied set at time zero(which is
infinite when $d\in\{2,3\}$, but finite and dependent on $r$ when $d\geq 4$).We
further show that for all $d\geq 2$, the unbounded cluster in the supercritical
phase is unique.Along the way a finite box criterion for non-percolation in the
Boolean model is extended to radius distributions with an exponential tail.
This may be of independent interest.The present paper settles the basic
properties of the model and should be viewed as a jumpboard for finer results.
