Support set and unique ergodicity of stochastic KdV equation

This paper presents the support set and unique ergodicity of invariant measure given in Ekren (2018) of stochastic KdV equation, which also offers an answer to the question about support set raised in Wu (2023). To begin with, we investigate the exponential decay of the H2 norm of the solution. Subsequently, we employ the truncated method in conjunction with the dominated convergence theorem to prove the support set and regularity of the invariant measure. Finally, we establish the unique ergodicity with the help of the Birkhoff ergodic theorem, primarily relying on precise estimates of ||ux||L infinity.