Non-linear parabolic PDE arise in many physical and biological settings; we often need to incorporate the effects of additive white noise. The resultant stochastic partial differential equations are well-understood in 1D. In higher spatial dimensions, there is an interesting dichotomy: such models are popular in application, while mathematicians assume these models to be ill-posed. We investigate the specific case of the two dimensional Allen-Cahn equation driven by additive white noise. Without noise, the Allen-Cahn equation is 'pattern-forming'. Does the presence of noise affect this behaviour? The precise notion of a weak solution to this equation is unclear. Instead, we regularize the noise and introduce a family of approximations. We discuss the continuum limit of these approximations and show that it exhibits divergent behavior. Our results show that a series of published numerical studies are somewhat problematic: shrinking the mesh size in these simulations does not lead to the recovery of a physically meaningful limit. This is joint work with Marc Ryser and Paul Tupper.
Group for Research in Decision Analysis