Optimal convergence under weakened coercivity for the Airy equation

Determine whether weakening the coercivity condition for the nonlinear stochastic Airy equation can yield the optimal temporal convergence rate one-half, rather than the rate one-third obtained under the current coercivity framework.

Background

The paper applies tamed exponential Euler schemes to the nonlinear stochastic Airy equation, whose leading operator is third order. With the available coercivity estimate in the space used for the analysis, the resulting convergence rate is one-third rather than the optimal one-half achieved for second-order hyperbolic problems. The authors explicitly identify a weakened coercivity condition as a possible route to recovering the optimal rate, but do not establish such a condition or prove the corresponding convergence result.

References

The resulting convergence rate is $\frac{1}{3}$, reflecting the coercivity available in $Y=H1$ paired with the equation's higher order. Future work might address whether a weakened coercivity condition allows for optimal convergence rates.

— Strong convergence rates of tamed exponential Euler schemes for superlinear hyperbolic SPDEs  (2609.08872 - Klioba, 8 Sep 2026) in Section 1, subsection “Main results”