  We study entanglement growth in a harmonic oscillator chain subjected to the
weak measurement of observables which have been smeared-out over a length scale
$R$. We find that entanglement grows diffusively ($S \sim t^{1/2}$) for a large
class of initial Gaussian states provided the measurement scale $R$ is
sufficiently large. At late times $t \gtrsim \mathcal{O}(L^{2})$ the entropy
relaxes towards an area-law value which we compute exactly. We propose a
modified quasi-particle picture which accounts for all of these main features
and agrees quantitatively well with our essentially exact numerical results.
The quasiparticles are associated with the modes of a non-Hermitian effective
Hamiltonian. At small wave-vector $k$, the quasiparticles transport entropy
with a finite velocity, but have a lifetime scaling as $1/k^2$; the concurrence
of these two conditions leads directly to the observed $t^{1/2}$ growth.
