Spatial fluctuation for stochastic heat equation with Hölder coefficients

Abstract

In this paper, we establish associativity, spatial ergodicity and a central limit theorem for certain nonnegative solutions to the stochastic heat equation ${\partial }_{t}u=\frac{1}{2}{\partial }_x^{2}u+u^3\hat{I}¾$ with ${\hat{I}}^3\in (0,1)$. When ${\hat{I}}^3=\frac{1}{2}$, 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.