Under controllability and noise-growth conditions, the KdVB equation with localized or multiplicative white noise has a unique invariant measure, with exponential mixing in the localized case.
On unique ergodicity in nonlinear stochastic partial differential equations[J]
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Ergodicity for the randomly forced Korteweg-de Vries-Burgers equation
Under controllability and noise-growth conditions, the KdVB equation with localized or multiplicative white noise has a unique invariant measure, with exponential mixing in the localized case.