Ergodicity of the stochastic fractional reaction–diffusion equation

A stochastic reaction–diffusion equation with fractional dissipation is considered. The smaller the exponent of the equation is, the weaker the dissipation of the equation is. The equation is discussed in detail when the exponent changes. The aim is to prove the well-posedness, existence and uniqueness of an invariant measure as well as strong law of large numbers and convergence to equilibrium. Without the analytic property of the semigroup, some methods are used to overcome the difficulties to get the energy estimates. The results in this paper can be applied to the classic reaction–diffusion equation with Wiener noise.