  We consider the Rosenzweig-Porter model $H = V + \sqrt{T}\, \Phi$, where $V$
is a $N \times N$ diagonal matrix, $\Phi$ is drawn from the $N \times N$
Gaussian Orthogonal Ensemble, and $N^{-1} \ll T \ll 1$. We prove that the
eigenfunctions of $H$ are typically supported in a set of approximately $NT$
sites, thereby confirming the existence of a previously conjectured non-ergodic
delocalized phase. Our proof is based on martingale estimates along the
characteristic curves of the stochastic advection equation satisfied by the
local resolvent of the Brownian motion representation of $H$.
