Question

What is a zero dimensional stochastic system?

Answer 1

It just means that there is no space dimension in the equation. Quite often you'd write, for example, df(x,t)=g(x,t)dt+σ(x,t)dW(x,t) (and typically, say in a Ginzburg-Landau situation, g would actually be a functional of f, depending on t, x, f, and f's derivatives, making the whole thing "nonlocal" in some sense), but when it is zero dimensional, there is no dependence on x.