Spatial fluctuation for stochastic heat equation with Hölder coefficients

In this paper, we establish associativity, spatial ergodicity and a central limit theorem for certain nonnegative solutions to the stochastic heat equation ${\mathrm{\partial }}_{t}u=\frac{1}{2}{\mathrm{\partial }}_x^{2}u+u^{\gamma }\xi$ with $\gamma \in (0,1)$. When $\gamma =\frac{1}{2}$, we derive a limit for the moment generating function of the spatial integral and provide a lower bound on the spatial growth of the solution.