Consider a one dimensional random walk. At each time step my walker will go left with probability , and right with probability . It stays where it is with probability . Furthermore these probabilities are dependent on the walker's position in space, so it's really and . I'm imagining I'm on a finite line of length, L, although it doesn't matter too much.
Now if , then we just have a normal random walker. In my problem I have the following setup: but . What does this mean? At any given point, x, my walker is more likely to go left than right. If it does go left it will come back with the same rate (although it's more likely to go left again).
So here's the question: If I leave this for a really long time, what is the equilibrium probability distribution for the walkers position, ?