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 $tto infty$, 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 $xin mathbb{R}$. 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 $lambda <lambda_4leq lambda_3$, (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 $u(t, x)$.