Typically, we see the notation $W(t)$ which represents an $\mathbb{R}^n$-dimensional Brownian motion (BM). What about $W(x)$? I've seen this in some papers where it is considered a space-dependent BM.
I don't understand how this can make sense. I know that $W(x,t)$ can be thought of as a separate BM at every point $x$. So perhaps $W(x)$ is a "slice" of $W(x,t)$ at a fixed time. Would this have the same properties as $W(t)$? If so, how can we interpret $W(x)$ without thinking of $x$ as a time variable? And if not, what would this thing be?