Asymptotic Preservation of Probabilistic Behaviors of Stochastic Heat Equation

This chapter explores the asymptotic behavior of stochastic heat equations under discretization as the noise intensity varies, focusing on the impact of noise on the longterm dynamics of solutions. Specifically, we consider two types of asymptotic properties: large deviations of invariant measures in the small-noise limit and weak intermittency under high noise intensity. The asymptotic behavior of large deviation rate functions for invariant measures and of Lyapunov exponents is analyzed for both spatial semi-discretizations and full discretizations.