Asymptotic behavior of stochastic fractional power dissipative equations on Rn

We study a reaction–fractional diffusion equation with additive noise on the entire space Rn with particular interest in the asymptotic behavior of solutions. We first transform the equation into a random equation whose solutions generate a random dynamical system. A priori estimates for solutions are derived when the nonlinearity satisfies certain growth conditions. Using estimates for far-field values of solutions and a cut-off technique, asymptotic compactness is proved. Thus, the existence of a random attractor in L2(Rn) is established.