Strong convergence of parabolic rate $1$ of discretisations of stochastic Allen-Cahn-type equations
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866929511250001920 |
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| author | Gerencsér, Máté Singh, Harprit |
| author_facet | Gerencsér, Máté Singh, Harprit |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_09222 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Strong convergence of parabolic rate $1$ of discretisations of stochastic Allen-Cahn-type equations Gerencsér, Máté Singh, Harprit Probability Numerical Analysis 60H35, 60H15 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. |
| title | Strong convergence of parabolic rate $1$ of discretisations of stochastic Allen-Cahn-type equations |
| topic | Probability Numerical Analysis 60H35, 60H15 |
| url | https://arxiv.org/abs/2209.09222 |