What is the solution for a one-dimensional random walk in potentials?

[blue2script]

I have a problem concerning a one-dimensional random walk in potentials. Assume a one-dimensional space [0,1] and a probability distribution p(x). At every point x we have a probability p(x) to go left and 1-p(x) to go right. Assume some smooth distribution of p(x) with boundaries p(0) = 0 and p(1) = 1. Now begin a random walk at x=0.5 with some step-size dx (e.g. d = 0.01) and capture the position of the walker at every time-step t. The boundary constraint assures that the walker remains inside [0,1].

I would assume that after sufficient time steps I get a steady distribution of the position of the walker. This would be equivalent to the probability distribution of finding the walker at some point in the potential.

However, I have yet no idea how to calculate this distribution from some given p(x). I tried to set up a differential equation using the fact that in the steady case the flow from point x to x + dx and back must be zero. However, I would get a pole at x = 0.5 which is pretty useless. I can post the calculation if someone is interested.

I would be glad for every hint how one could solve this problem. Thanks in advance!
Blue2script

 

[Eynstone]

I have a problem concerning a one-dimensional random walk in potentials. Assume a one-dimensional space [0,1] and a probability distribution p(x). At every point x we have a probability p(x) to go left and 1-p(x) to go right. Assume some smooth distribution of p(x) with boundaries p(0) = 0 and p(1) = 1. Now begin a random walk at x=0.5 with some step-size dx (e.g. d = 0.01) and capture the position of the walker at every time-step t. The boundary constraint assures that the walker remains inside [0,1].

I would assume that after sufficient time steps I get a steady distribution of the position of the walker. This would be equivalent to the probability distribution of finding the walker at some point in the potential.

However, I have yet no idea how to calculate this distribution from some given p(x). I tried to set up a differential equation using the fact that in the steady case the flow from point x to x + dx and back must be zero. However, I would get a pole at x = 0.5 which is pretty useless. I can post the calculation if someone is interested.

I assume a fixed step size ; we can let it tend to zero later.
Firstly, the boundary conditions don't warrant that the walker won't trip at 0 or 1 ( for instance, take d=1/300). If you want that constraint with initial position =1/2, the step size must be 1/2n for some integer n , making the walk discrete.
In such a case, this can be modeled as a Markov chain with 2n+1 states ( x=1/2 & other 2n possible positions ;with transition probabilities p(xn)&c. at each state. The limit of the transition probability matrix yields the long run probabilities.
Finally,letting d->0 gives a continuous time Markov process.

P.S. : Is this a model of any physicsl phenomenon? I ask out of sheer curiosity.