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 $F(\pm \infty )=\mp \infty$) 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.

arXiv comment34 pages, 3 figures

References in corpus (9)

• A theory of regularity structures
• Strong Approximation of Stochastic Allen-Cahn Equation with White Noise
• Analysis of Some Splitting Schemes for the Stochastic Allen-Cahn Equation
• Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations
• Optimal pointwise approximation of SDEs based on brownian motion at discrete points
• Finite difference schemes for stochastic partial differential equations in Sobolev spaces
• Optimal rate of convergence for stochastic Burgers-type equations
• Optimal rate of convergence for approximations of SPDEs with non-regular drift
• Convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations on 2D torus