In this talk, we study the long time behaviour of some stochastic partial differential equations (SPDEs). After introducing the notions of ergodicity, unique ergodicity and convergence to equilibrium, we will discuss how these have been proven for a very large class of parabolic SPDEs. We will then shift our attention to dispersive SPDEs, where the general strategy for the parabolic case fails. We will describe this failure for the wave equation on the 1-dimensional torus and present a recent result that settles unique ergodicity even in this case.

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