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We prove that the Ricci flow g(t) starting at any metric on the euclideanspace that is invariant by a transitive nilpotent Lie group N, can be obtainedby solving an ODE for a curve of nilpotent Lie brackets. By using that this ODEis the negative gradient flow of a homogeneous polynomial, we obtain that g(t)is type-III, and, up to pull-back by time-dependent diffeomorphisms, that g(t)converges to the flat metric, and the rescaling |R(g(t))|g(t) convergessmoothly to a Ricci soliton, uniformly on compact sets. The Ricci soliton limitis also invariant by some transitive nilpotent Lie group, though possiblynon-isomorphic to N.

The Ricci flow for simply connected nilmanifolds