Strong convergence of parabolic rate 1 of discretisations of stochastic Allen-Cahn-type equations
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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 1 (and no better) when measuring the error in appropriate negative Besov norms, by temporarily `pretending' that the SPDE is singular.
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