A PHASE-TRANSITION FOR A STOCHASTIC PDE RELATED TO THE CONTACT PROCESS

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 long-range 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