Consider a spatio-temporal Brownian motion/Wiener process $dW(x,t)$ for $x\in [0,1]$ and $t\geq 0$. I want to come up with an "$n$-sized" discretization of the form $dW_n(x_i,t)$ for $i=1,\dots, n$, and I'm not sure how to go about this.
I know that one can discretize in time as in this paper, where the discrete version is just a piecewise constant function that takes random values at each step.
For the spatial discretization, my first thought is that we can simply take a "slice" of $dW(x,t)$ at each point $x_i$, and define $dW_n(x_i,t)$ that way. Is that correct? And how would we go about proving convergence, i.e. something like $\mathbb{E} \|dW_n(x_i, t) - dW(x,t) \|^2_2 \rightarrow 0$ as $n \rightarrow \infty$?