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Numerical approximation of the stochastic heat equation with a distributional reaction term

Ludovic Goudenège, El Mehdi Haress, Alexandre Richard

TL;DR

This work develops a rigorous numerical analysis for the stochastic heat equation with a distributional drift in Besov spaces, establishing strong convergence of a taming-Euler finite-difference scheme with mollified drift. By decomposing the error into stability and discretisation components and employing the Stochastic Sewing Lemma, the authors derive explicit convergence rates that depend on the Besov regularity of the drift, including sub-critical and critical (limit) regimes. A key novelty is the regularisation effect of both continuous and discrete Ornstein–Uhlenbeck processes, which provides quantitative control over singular integrals and enables sharp rates even when the drift is a distribution or a finite measure. The results are sharp in several regimes, and the framework unifies discrete and continuous regularisation, offering a pathway for weak-extensibility and potential extensions to implicit schemes and broader SPDEs.

Abstract

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.

Numerical approximation of the stochastic heat equation with a distributional reaction term

TL;DR

This work develops a rigorous numerical analysis for the stochastic heat equation with a distributional drift in Besov spaces, establishing strong convergence of a taming-Euler finite-difference scheme with mollified drift. By decomposing the error into stability and discretisation components and employing the Stochastic Sewing Lemma, the authors derive explicit convergence rates that depend on the Besov regularity of the drift, including sub-critical and critical (limit) regimes. A key novelty is the regularisation effect of both continuous and discrete Ornstein–Uhlenbeck processes, which provides quantitative control over singular integrals and enables sharp rates even when the drift is a distribution or a finite measure. The results are sharp in several regimes, and the framework unifies discrete and continuous regularisation, offering a pathway for weak-extensibility and potential extensions to implicit schemes and broader SPDEs.

Abstract

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 approximating , we prove that the error between the solution of the SPDE with reaction term and its tamed Euler finite-difference scheme with mollified drift , converges to in with a rate that depends on the Besov regularity of . In particular, one can consider two interesting cases: first, even when is only a (finite) measure, a rate of convergence is obtained. On the other hand, when is a bounded measurable function, the (almost) optimal rate of convergence -in space and -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.
Paper Structure (69 sections, 37 theorems, 300 equations)

This paper contains 69 sections, 37 theorems, 300 equations.

Table of Contents

  1. Introduction
  2. Organisation of the paper.
  3. Framework and results
  4. Notations and definitions
  5. Definition of solutions and the numerical scheme
  6. Main results
  7. Discussion on the optimality of the rate of convergence
  8. Convergence of the tamed Euler finite-difference scheme
  9. Organisation of the proofs of the main results
  10. Proof of Proposition \ref{['prop:main-SPDE']}
  11. Proof of \ref{['eq:unifschemev0']}, \ref{['eq:main-result-SPDE2']} and \ref{['eq:main-result-SPDE-critic2']}
  12. Bound on $V^k$.
  13. Bound on $\mathcal{E}^{1,n,k}$.
  14. Bounds on $\mathcal{E}^{2,n,k}$ and $\mathcal{E}^{3,n,k}$.
  15. Conclusion in the sub-critical case.
  16. ...and 54 more sections

Key Result

Lemma 2.5

For $t \in [0,1]$ and $x \in \mathbb{T}$, denote by $t_h$ and $y_n$ the leftmost gridpoint from $t$ in $\Lambda_h$ and from $y$ in $\mathbb{T}_n$ respectively. For $k,n\in\mathbb{N}$, the process $u^{n,k}$ defined in eq:numerical-scheme on the domain $\mathbb{T}_{n}\times \Lambda_{h}$ extends to $[0

Theorems & Definitions (74)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Theorem 2.7
  • Remark 2.8
  • Corollary 2.9
  • Corollary 2.10
  • ...and 64 more