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We prove uniqueness of a martingale problem with boundary conditions on asimplex associated to a differential operator with an unbounded drift. We showthat the solution of the martingale problem remains absorbed at the boundaryonce it attains it, and that, after hitting the boundary, it performs adiffusion on a lower dimensional simplex, similar to the original one. We alsoprove that in the diffusive time scale condensing zero-range processes evolveas this absorbed diffusion.

A martingale problem for an absorbed diffusion: the nucleation phase of condensing zero range processes