Numerical approximation of the stochastic heat equation with a distributional reaction term
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914866107777024 |
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| author | Goudenège, Ludovic Haress, El Mehdi Richard, Alexandre |
| author_facet | Goudenège, Ludovic Haress, El Mehdi Richard, Alexandre |
| contents | We study the numerical approximation of the stochastic heat equation with a distributional reaction term. Under a condition on the Besov regularity of the reaction term, it was proven recently that a strong solution exists and is unique in the pathwise sense, in a class of Hölder continuous processes. For a suitable choice of sequence $(b^k)_{k\in \mathbb{N}}$ approximating $b$, we prove that the error between the solution $u$ of the SPDE with reaction term $b$ and its tamed Euler finite-difference scheme with mollified drift $b^k$, converges to $0$ in $L^m(Ω)$ with a rate that depends on the Besov regularity of $b$. In particular, one can consider two interesting cases: first, even when $b$ is only a (finite) measure, a rate of convergence is obtained. On the other hand, when $b$ is a bounded measurable function, the (almost) optimal rate of convergence $(\frac{1}{2}-\varepsilon)$-in space and $(\frac{1}{4}-\varepsilon)$-in time is achieved. Stochastic sewing techniques are used in the proofs, in particular to deduce new regularising properties of the discrete Ornstein-Uhlenbeck process. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_08201 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerical approximation of the stochastic heat equation with a distributional reaction term Goudenège, Ludovic Haress, El Mehdi Richard, Alexandre Probability Numerical Analysis We study the numerical approximation of the stochastic heat equation with a distributional reaction term. Under a condition on the Besov regularity of the reaction term, it was proven recently that a strong solution exists and is unique in the pathwise sense, in a class of Hölder continuous processes. For a suitable choice of sequence $(b^k)_{k\in \mathbb{N}}$ approximating $b$, we prove that the error between the solution $u$ of the SPDE with reaction term $b$ and its tamed Euler finite-difference scheme with mollified drift $b^k$, converges to $0$ in $L^m(Ω)$ with a rate that depends on the Besov regularity of $b$. In particular, one can consider two interesting cases: first, even when $b$ is only a (finite) measure, a rate of convergence is obtained. On the other hand, when $b$ is a bounded measurable function, the (almost) optimal rate of convergence $(\frac{1}{2}-\varepsilon)$-in space and $(\frac{1}{4}-\varepsilon)$-in time is achieved. Stochastic sewing techniques are used in the proofs, in particular to deduce new regularising properties of the discrete Ornstein-Uhlenbeck process. |
| title | Numerical approximation of the stochastic heat equation with a distributional reaction term |
| topic | Probability Numerical Analysis |
| url | https://arxiv.org/abs/2405.08201 |