The infinite random walk: two approaches leads to two different solutions. How to solve? The "random walk" process is well known for mathematicians, see for instance https://en.wikipedia.org/wiki/Random_walk . It is also known as "the drunk walk".
It is demonstrated that if you have a n x n board (or even n x n .. .x n in higher dim boards) and move infinite number of steps, every square on board will be visited an infinite number times.
Now suppose you have also an infinite "n" square board and perform an infinite number of steps on it. Will every square on this board receive a visit? how many visits?
The first approach to solve this is supposing you already have an "infinite speed walker" running over finite n x n board , and then try to extend n to be infinite. In this case, every square will be visited infinite times, and so limit to infinite board shows it will be completely filled infinite times for every size of n.
The second approach is to suppose you already have an "infinite n board", and begin running a finite speed walker on it every time faster, extending speed to infinity. In this case, board will never be filled, because for every size of speed, will be always squares not visited.
Then the question (for me) remains open: how will look the behavior of infinite size board with an infinite speed walker into? 
For me, this question is intriguing, because if this problem has no solution, then it shows that "time" can be created from "infinite speeds and sizes problems", since infinite speed walker will be always trying to solve the infinite size board, producing dynamic patterns into the board at every moment.
If you suppose a kind of mathematical "space" infinitely big and where things runs at random at infinite speed, very interesting things could appear into, for example "time", as happened in our universe.
Thanks for any idea
Miguel
 A: Your formalism is extremely messy compared to the usual formalism. In the more normal formalism, you just have a regular random walk on an infinite $n$-dimensional integer lattice. In an infinite number of time steps, every point is either visited finitely many (possibly $0$) times or infinitely many times, because the process is irreducible. Thus your question reduces to the question of whether the process is recurrent or transient. We actually know the answer: the process is recurrent when $n=1$ or $n=2$ and transient when $n \geq 3$. This is actually a deep fact which has ripples in seemingly unrelated subjects such as PDE involving the Laplacian.
Find a reference on infinite state space Markov chains for more details.
The rest of your question seems to really be more about scaling issues as you send the size of the lattice and the speed of the walker (or, equivalently, the time threshold for terminating the process) to infinity, rather than what happens when they are actually set equal to infinity. This is a considerably more complicated problem that's easier to address if you formulate it in a well-defined way (i.e. give a precise scaling relationship between, say, the size of the lattice and the termination time that you are interested in).
