Triviality of the 2D stochastic Allen-Cahn equation. (English) Zbl 1245.60063

Hairer, Martin; Ryser, Marc Daniel; Weber, Hendrik

Electron. J. Probab. 17, Paper No. 39, 14 p. (2012).

Summary: We consider the stochastic Allen-Cahn equation driven by mollified space-time white noise. We show that, as the mollifier is removed, the solutions converge weakly to 0, independently of the initial condition. If the intensity of the noise simultaneously converges to 0 at a sufficiently fast rate, then the solutions converge to those of the deterministic equation. At the critical rate, the limiting solution is still deterministic, but it exhibits an additional damping term.

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
81T08	Constructive quantum field theory

Keywords:

SPDEs; Allen-Cahn equation; white noise; stochastic quantisation