Let us consider a general SPDE of the form $$\partial_t h = F(h, \partial_x h, (\partial_x h)^2, \partial_x^2 h,..) + \eta,$$ where $\eta$ is a normal random variable in space and time with $$<\eta(x,t) \eta(x', t')> = 2D \delta(x-x')\delta(t-t').$$
If I integrate this SPDE in time over $[0s,500s]\times[1m,2^{14}m]$, and analyze the resulting function $h(x,t=500s)$, I see that $h(x,t=500s)$ has some interesting properties, which are irrelavent right now.
However, if I integrate the same SPDE over $x$ on the same domain, the resulting function $h(x,t=500s)$ is quite different from what I had obtained before.
Question:
Considering the fact that I'm numerically integrating the same SPDE, but just along different dimensions, I cannot understand why the resulting functions, namely $h(x,t=500s)$, are different, so why am I getting different results in these cases?