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A phase transition for a stochastic PDE related to the contact process
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  • Published: June 1994

A phase transition for a stochastic PDE related to the contact process

  • Carl Mueller1 &
  • Roger Tribe2 

Probability Theory and Related Fields volume 100, pages 131–156 (1994)Cite this article

  • 284 Accesses

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Summary

We consider the one-dimensional heat equation, with a semilinear term and with a nonlinear white noise term. R. Durrett conjectured that this equation arises as a weak limit of the contact process with longrange interactions. We show that our equation possesses a phase transition. To be more precise, we assume that the initial function is nonnegative with bounded total mass. If a certain parameter in the equation is small enough, then the solution dies out to 0 in finite time, with probability 1. If this parameter is large enough, then the solution has a positive probability of never dying out to 0. This result answers a question of Durett.

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Authors and Affiliations

  1. Department of Mathematics, University of Rochester, 14627, Rochester, NY, USA

    Carl Mueller

  2. Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK

    Roger Tribe

Authors
  1. Carl Mueller
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  2. Roger Tribe
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Supported by an NSA grant, and by the Army's Mathematical Sciences Institute at Cornell

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Mueller, C., Tribe, R. A phase transition for a stochastic PDE related to the contact process. Probab. Th. Rel. Fields 100, 131–156 (1994). https://doi.org/10.1007/BF01199262

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  • Received: 07 June 1993

  • Revised: 01 April 1994

  • Issue Date: June 1994

  • DOI: https://doi.org/10.1007/BF01199262

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Mathematics Subject Classification (1991)

  • 60H15
  • 35R60
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