Strong convergence of parabolic rate 1 of discretisations of stochastic Allen-Cahn-type equations

Abstract

Consider the approximation of stochastic Allen-Cahn-type equations (i.e. 1 + 1-dimensional space-time white noise-driven stochastic PDEs with polynomial nonlinearities F such that F(±∞) = ∓∞) by a fully discrete space-time explicit finite difference scheme. The consensus in literature, supported by rigorous lower bounds, is that strong convergence rate 1/2 with respect to the parabolic grid meshsize is expected to be optimal. We show that one can reach almost sure convergence rate arbitrarily close to (and no better than) 1 when measuring the error in appropriate negative Besov norms, by temporarily ‘pretending’ that the SPDE is singular.