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

Summary · qwen2.5:32b

The study validates a small noise large deviation principle for invariant measures of a one-dimensional stochastic Allen-Cahn equation under inhomogeneous Dirichlet boundary conditions and multiplicative noise, overcoming challenges due to weak dissipation. The dynamics converge to the unique minimizer of the Ginzburg-Landau energy functional, with the invariant measure $\mu_\epsilon$ concentrating exponentially around this minimizer as $\epsilon$ approaches zero. A key detail is the use of L. Simon's convergence theorem to establish these results in a Sobolev space framework.

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arXiv:2512.10536v3 Announce Type: replace Abstract: 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.