Global random attractor of stochastic hydrodynamical equation for the Heisenberg paramagnet with multiplicative noise on ℝd

The stochastic hydrodynamical equation for the Heisenberg paramagnet with multiplicative noise defined on the entire ℝd is mainly investigated. The global random attractor for the random dynamical system associated with the equation is obtained. The method is to transform the stochastic equation into the corresponding partial differential equations with random coefficients by Ornstein–Uhlenbeck process. The uniform priori estimates for far-field values of solutions have been studied via a truncation function, and then the asymptotic compactness of the random dynamical system is established.