Is there such a thing as Brownian motion with two indices? In particular, does something like this exist: $$W_t(s)= \epsilon_t \sqrt{s}\,,$$
where, for fixed $t$, the usual properties hold:
- $W_t(0)=0$.
- $s \to W_t(s)$ is continuous in $s$.
- $\{W_t(s)\}_{s\geq0}$ has independent stationary increments in $s$
- for any $s$, $W_t(s)$ is $N(0,1)$,
but where, additionally, for fixed $s$, $W_t(s)$ is a discrete-time normal random variable, following, say, an AR(1) process with respect to $t$?
The $s$-index would be spatial, for instance, and the $t$-index temporal.
How would one handle conditional expectations $\mathrm{E}_t[\cdot]$, $\mathrm{E}_s[\cdot]$, $\mathrm{E}_{s,t}[\cdot]$?
You might be interested in multivariate Brownian motion. That is, a Brownian motion that takes values in $\mathbb{R}^d$ rather than $\mathbb{R}$. You can then take $t=1,...,d$. If you choose an appropriate covariance matrix, this should result in a sample of length $d$ of an AR process, at each time point.
There are also generalizations to infinite dimensions.