Large deviations for invariant measure of stochastic Allen-Cahn equation with inhomogeneous boundary conditions and multiplicative noise

We prove the validity of a small noise large deviation principle for the family of invariant measures $\{{\mu }_{\epsilon }{\}}_{\epsilon >0}$ associated to the one dimensional stochastic Allen-Cahn equation with inhomogeneous Dirichlet boundary conditions, perturbed by unbounded multiplicative noise. The main difficulty is that the system is not strongly dissipative. Using L. Simon’s convergence theorem, we show that the dynamics of the noiseless system converge in large time to the minimizer of the Ginzburg-Landau energy functional, which is unique due to the boundary condition. We obtain an estimate of the invariant measure on the bounded set in the Sobolev space $W^{k^{\star },p^{\star }}$, where $k^{\star }{p}^{\star }>1$, and $p^{\star }$ is large. As a corollary of the main result, we show that ${\mu }_{\epsilon }$ concentrates around the unique minimizer with such boundary conditions exponentially fast when $\epsilon$ is sufficiently small.