  We introduce and study a model of active Brownian motion with multiplicative
noise describing fluctuations in the self-propulsion or activity. We find that
the standard picture of density accumulation in slow regions is qualitatively
modified by active fluctuations, as stationary density profiles are generally
not determined only by the mean self-propulsion speed landscape. As a result,
activity gradients generically correlate the particle self-propulsion speed and
orientation, leading to emergent polarization at interfaces pointing either
towards dense or dilute regions depending on the amount of noise in the system.
We discuss how active noise affects the emergence of motility induced phase
separation. Our work provides a foundation for systematic studies of active
matter self-organization in the presence of activity landscapes and active
fluctuations.
