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?