# Non-uniform random walk

I'm searching a solution for this problem:

Given a segment of length $L$, from $0$ to $L$ divided in $N$ subsegments of the same length, a particle, starting from the subsegment in $x_k$ has a probability to jump to the subsegment in $x_{k+1}$ given by:

$$P(x_k\rightarrow x_{k+1})=\frac{1}{2}e^{-\alpha x_k^2}$$

and a probability to jump from $x_k$ to $x_{k-1}$ given by: $$P(x_k\rightarrow x_{k-1})=1-\frac{1}{2}e^{-\alpha x_k^2}$$

What is the probability $P_L(\alpha,L,T)$ to find the particle out from the segment after a time $T$ if the time interval between two jumps is one second?

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Is $x_k$ a number or an interval? –  Henry Feb 14 '12 at 17:25
xk is a number identifying the left side of the interval. So x0 is the origin and xL is the last interval in the segment –  Riccardo.Alestra Feb 14 '12 at 17:28
You can also identify an interval with a number xk, if this assumption meke the question easier –  Riccardo.Alestra Feb 14 '12 at 17:35
Your question is ill-defined. How do you identify an interval with a number (take the coordinates of the left end/middle/right end / the number of the interval when you count them from left to right / another way)? How do you define the transitions starting from the extremal sub-intervals (is there an "outside" for the particle to go)? Why does $P_L (\alpha,L,T)$ has two $L$ arguments, but does not depend on the starting point? What do you mean by the "the particle is out": does it hit the boundary before time $T$? Is there some kind of process which allows it to get back? –  D. Thomine Feb 14 '12 at 20:55
There is no 'reflection' from the boundaries. The particle jumps from an interval to the adiacent and the probability of the jump doesn't depend on the size of the interval. The starting point is some interval inside the segment. The particle is 'outside' when it's in a interval outside the segment, assuming all the line is divided into intervals of the same length. You are right: the probability of the process depend on the starting point. –  Riccardo.Alestra Feb 15 '12 at 8:19