Large Deviations of Invariant Measures of Fractional Reaction-Diffusion Equations Driven by Multiplicative Noise on R^n

In this paper, we establish large deviations of invariant measures for fractional stochastic reaction-diffusion equations with polynomial reaction term driven by unbounded multiplicative noise on R^n. Under a dissipativity condition, we first prove the Freidlin-Wentzell large deviation principle of solution trajectories uniformly over all initial data in L^2(R^n). The key ingredient is a positive-time L^\infty-estimate for differences of controlled solutions which is independent of all initial data and hence yields uniform convergence of controlled stochastic solutions over the entire tphase space. Then using the uniform trajectory estimates, contraction and a second-moment estimate, we further prove the large deviations of invariant measures of the equation with a good quasipotential without introducing weighted spaces or establishing exponential tightness. The non-compactness of Sobolev embeddings in R^n is circumvented by uniform spatial tail-ends estimates. This is the first result on large deviations of invariant measures of the equation with multiplicative noise on unbounded domains.