2025/12/11 by Rui Bai, Bai, Rui, Chunrong Feng +3
Computer Science · Economics, Econometrics and Finance · Materials Science · Mathematics · #math.AP #math.PR #msc:35R60 #msc:37A50 #msc:37L55 #msc:60F10 #msc:60H15
paper · pdf · doi:10.48550/arxiv.2512.10536
40 pages. Revised Section 4.1 and the exponential estimate for invariant measures. The noise coefficient is now assumed to have strictly sublinear growth
arxiv created 2026/07/28 · arxiv updated 2026/07/30
We establish a small-noise large deviation principle for the family of invariant measures \με\ε>0 associated with the one-dimensional stochastic Allen-Cahn equation, subject to inhomogeneous Dirichlet boundary conditions and driven by unbounded multiplicative noise. The main novelty is that the deterministic system is only weakly dissipative, while the noise coefficient is allowed to have strictly sublinear growth arbitrarily close to linear. Using L. Simon's convergence theorem, we prove that every trajectory of the corresponding noiseless equation converges, as time tends to infinity, to the unique minimiser of the Ginzburg-Landau energy functional determined by the boundary conditions. A key ingredient is an exponential estimate for the invariant measures outside bounded subsets of Wk^⋆,p^⋆, where k^⋆ p^⋆>1 and p^⋆ are sufficiently large; such subsets are compact in the underlying space of continuous functions. As a consequence of the large deviation principle, we show that, as ε→ 0, the invariant measures με concentrate exponentially fast around the unique minimiser.