Strong Convergence Rates for Stochastic Cahn–Hilliard Discretizations

Definitions and mathematical statement

Supplied mathematical target. Determine optimal strong convergence rates for temporal discretizations of the stochastic Cahn--Hilliard--Cook equation.

Meaning of a rate, conditional on a specified formulation. Once the stochastic equation, its solution, a temporal approximation, and a strong error functional have been specified, let $\mathcal E(\tau)\geq 0$ denote that error for time step $\tau>0$. An algebraic convergence rate $\alpha>0$ means that there exist constants $C<\infty$ and $\tau_0>0$, independent of $\tau$, such that \[ \mathcal E(\tau)\leq C\tau^\alpha \qquad\text{for every admissible }0<\tau\leq\tau_0. \] Calling a rate optimal additionally requires a specified comparison class and a precise sharpness criterion; a supremum of admissible exponents need not itself be attained.

Unresolved formulation. The supplied record does not state the stochastic differential equation or specify the spatial domain, dimension, boundary conditions, coefficients, nonlinear potential, driving noise and its covariance, initial data and their regularity, solution concept, time interval, temporal discretization scheme, or treatment of any spatial discretization. It also does not specify the norm, moment order, temporal evaluation convention, or uniformity class defining strong error. Consequently, $\mathcal E$ and the intended meaning of optimality cannot be instantiated from this record. The displayed bound explains rate terminology only; it does not supply missing hypotheses or assert a convergence result. The target remains a research question requiring these specifications before a precise mathematical proposition can be stated.

Short English statement

Determine the best possible strong convergence rates for time discretizations of the stochastic Cahn--Hilliard--Cook equation. The supplied record leaves the equation, assumptions, numerical methods, error criterion, and meaning of optimality unspecified, so it does not determine a single precise problem.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30003389$, exact source problem code catalogue entry, titled "Strong Convergence Rates for Stochastic Cahn--Hilliard Discretizations," in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2017.
• Supplied source citation: Mini-Workshop: Stochastic Differential Equations: Regularity and Numerical Analysis in Finite and Infinite Dimensions (2018), Oberwolfach Reports, pp. 453--498, DOI 10.4171/owr/2017/9. Supplied link: https://doi.org/10.4171/owr/2017/9. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No specific page, speaker, or passage identifying this problem is supplied.
• Supplied literature source, labeled "Oberwolfach report": https://ems.press/content/serial-article-files/46671.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the record as a workshop-level statement that several open problems were posed, without identifying one precise proposition. It therefore assigns no stronger mathematical resolution without additional context. This is supplied dated curation, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming self-containedness and preservation of the target. These are supplied verification claims, not independent confirmation of mathematical precision or correctness.