Kernkonzepte
The author analyzes the long-time behavior of numerical schemes for monotone SPDEs driven by multiplicative noise, establishing exponential ergodicity and unique invariant measures.
Zusammenfassung
The content discusses the analysis of numerical schemes for monotone SPDEs driven by multiplicative noise, focusing on ergodicity and strong error estimates. It explores the application to the stochastic Allen–Cahn equation and addresses key theoretical questions in the field.
The study involves deriving uniform estimates, establishing exponential ergodicity, and addressing strong error analysis for numerical approximations. The content highlights advancements in numerical algorithms for nonlinear SPDE systems driven by multiplicative noise.
Key topics include temporal average convergence, invariant measures, ergodic limits, and strong approximation issues in infinite-dimensional stochastic systems. The paper provides insights into computational challenges and theoretical developments in analyzing SPDEs with additive and multiplicative noise.
Statistiken
L2 + L7/2 < λ1.
L1 + L6/2 < λ1.
(L1 + L6/2) ∨ (L2 + L7/2) < λ1.
Zitate
"Applying these results to the stochastic Allen–Cahn equation indicates that these schemes always have at least one invariant measure."
"As a significant asymptotic behavior, the ergodicity characterizes the case of temporal average coinciding with spatial average."
"We also show that these numerical invariant measures are exponentially ergodic."
"The authors first studied Galerkin-based linear implicit Euler scheme and high order integrator to approximate the invariant measures of a parabolic SPDE driven by additive white noise."