Central limit theorems for spatial averages of the stochastic heat equation via Malliavin–Stein’s method

Suppose that $$\{u(t, x)\}_{t >0, x \in {\mathbb {R}}^d}$$ is the solution to a d-dimensional stochastic heat equation driven by Gaussian noise white in time and has spatially homogeneous covariance satisfies Dalang’s condition. The purpose of this paper establish quantitative central limit theorems for spatial averages form $$N^{-d} \int _{[0,N]^d} g(u(t,x))\, \mathrm {d}x$$ , as $$N\rightarrow \infty $$ where g Lipschitz-continuous function or belongs class locally-Lipschitz functions, using combination Malliavin calculus Stein’s method normal approximations. Our results include theorem Hopf–Cole KPZ equation. We also functional these averages.