Limit theorems for time-dependent averages of nonlinear stochastic heat equations

We study limit theorems for time-dependent averages of the form $X_t:=\frac{1}{2L(t)}\int_{-L(t)}^{L(t)} u(t, x) \, dx$, as $t\to \infty$, where $L(t)=\exp(\lambda t)$ and $u(t, x)$ is the solution to a stochastic heat equation on $\mathbb{R}_+\times \mathbb{R}$ driven by space-time white noise with $u_0(x)=1$ for all $x\in \mathbb{R}$. We show that for $X_t$ (i) the weak law of large numbers holds when $\lambda>\lambda_1$, (ii) the strong law of large numbers holds when $\lambda>\lambda_2$, (iii) the central limit theorem holds when $\lambda>\lambda_3$, but fails when $\lambda <\lambda_4\leq \lambda_3$, (iv) the quantitative central limit theorem holds when $\lambda>\lambda_5$, where $\lambda_i$'s are positive constants depending on the moment Lyapunov exponents of $u(t, x)$.