The phenomenon of led-gentrification concerns many port cities around the world. Technical and urban constraints forced the port operations to leave their original site leaving large areas in central cities. The authorities then implemented major projects whose ambitions are to restore an identity based on a new report from the city to its waterfront, and rebuilding a centrality degraded by port activity. We propose to produce a measure of results by reading the movements of prices in surrounding spaces with a β-convergence analysis. This analysis is developed in the case of the city of Bordeaux, with the exhaustive database on real estate transactions. It appears that the new identity unifies home prices and generates a new centrality by a gentrification in which the speed and distribution are still dependent on the historical structure of prices.