  We present, for the first time, the result (from 2008) that (normal, strongly
continuous) Markov semigroups on $\mathscr{B}(G)$ ($G$ a separable Hilbert
space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle
perturbation of a noise) if and only if the Markov semigroup is spatial (that
is, if it dominates an elementary CP-semigroup). The proof is by general
abstract nonsense (taken from Arveson's classification of $E_0$-semigroups on
$\mathscr{B}(H)$ by Arveson systems up to cocycle conjugacy) and not, as usual,
by constructing the cocycle as a solution of a quantum stochastic differential
equation in the sense of Hudson and Parthasarathy. All other results that make
similar statements (especially, [Mem. Amer. Math. Soc. 240 (2016), vi+126
pages, arXiv:0901.1798]) for more general $C^*$-algebras) have been proved
later by suitable adaptations of the methods exposed here. (They use Hilbert
module techniques, which we carefully avoid here in order to make the result
available without any appeal to Hilbert modules.)
