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

Abstract

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

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Acknowledgements

The authors wish to thank J. Farfan, T. Funaki and I. Karatzas for stimulating discussion, and the two referees for their careful reading and valuable remarks. The first author acknowledges financial support from the Vicerrectorado de investigación de la PUCP.

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Correspondence to C. Landim.

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Beltrán, J., Jara, M. & Landim, C. A martingale problem for an absorbed diffusion: the nucleation phase of condensing zero range processes . Probab. Theory Relat. Fields 169, 1169–1220 (2017). https://doi.org/10.1007/s00440-016-0749-6

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Keywords

  • Diffusion with boundary conditions
  • Martingale problem
  • Nucleation phase
  • Metastability
  • Condensing zero-range process

Mathematics Subject Classification

  • 60J60
  • 60K35
  • 82C22