We propose a dynamical process for network evolution, aiming  at explaining the emergence of the small world phenomenon, i.e., the statistical observation that any pair of individuals are linked by a short chain of acquaintances  computable by a simple decentralized routing algorithm, known as greedy routing. Previously proposed dynamical processes enabled to demonstrate experimentally (by simulations) that the small world phenomenon can emerge from local dynamics. However, the analysis of greedy routing using the  probability distributions arising from these dynamics is quite complex  because of mutual dependencies.  In contrast, our process enables  complete formal analysis. It is based on the combination of two simple  processes: a random walk process, and an harmonic forgetting process.  Both processes reflect natural behaviors of the individuals, viewed as  nodes in the network of inter-individual acquaintances. We prove that,  in $k$-dimensional lattices, the combination of these two processes generates long-range links mutually  independently distributed as a k-harmonic distribution. We analyze the performances  of greedy  routing  at the stationary regime  of our process, and prove that the  expected number of steps for routing from any source to any target in  any multidimensional lattice is a polylogarithmic function  of the  distance between the two nodes in the lattice. Up to our knowledge,  these results are the first formal proof that navigability in small  worlds can emerge from a dynamical process for network evolution. Our dynamical process can find practical applications to the design of spatial gossip and resource location protocols.