Invariant Measures for a Stochastic Kuramoto–Sivashinsky Equation

Abstract

For the one-dimensional Kuramoto–Sivashinsky equation with random forcing term, existence and uniqueness of solutions is proved. Then, the Markovian semigroup is well defined; its properties are analyzed in order to provide sufficient conditions for existence and uniqueness of invariant measures for this stochastic equation. Finally, regularity results are presented.

Keywords:

• Invariant measures
• Irreducibility
• Pathwise uniqueness
• Strongly Feller

AMS Subject Classification:

• Primary 60H15
• Secondary 35Q99

Notes

¹We should work first with the Galerkin approximation and then pass to limit as n → ∞. But we show the basic steps for .