Edit: The following statement is an "answer" to the first part of the question and should be ignored, since the "real meat" is in the addendum :-)
This is a rather general question, let me point out two directions in which one could proceed:
What you describe is an example of a discrete approximation to a stochastic differential equation via an Euler schema, for more details have a look at this page on the wiki Azimuth: stochastic differential equation. It is possible to define stochastic differential equations in higher dimensions, of course, and on smooth real manifolds, for example. The discrete Euler schema for an equation with additive white noise adds a Gaussian random variable on every timestep, but it is possible to define equations with different noise/random processes, see for example stochastic integral on wikipedia.
You stick to discrete time equations and processes, in this case a good buzz word to look for are "random iterative models", see for example the book of the same name by Marie Duflo, (review in ZMATH here). I'm sure there are a lot more buzz words connected to your question :-)
About calculating the distribution: There are examples in continuous and in discrete time where it is possible to calculate the distribution exactly, or to prove certain concentration theorems. In continuous time, there are for example SDE where exact solutions are known, resp. associated Fokker-Planck equations where the solution is known exactly. An example is the Ornstein-Uhlenbeck process. There it is possible to explicitly state the distribution of $x(t)$ given $x_{t=0} := x_0$.
As for discrete times, there are several theorems about the rate of convergence to the stationary distribution for certain Markov chains in Duflo's book.
