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