## Random walk with exponential probability

I am trying to solve the following problem:

On a segment there are $2N$ points from $-N$ to $N$ passing through zero.

A particle jumps from the position $k$ to a position $k+1$ or from a position $k$ to $k-1$ with probability given by $$P(\vert{k}\vert\rightarrow\vert{k+1}\vert)=\frac{1}{2}exp[-\alpha(\vert{N}\vert-\vert{k}\vert)]$$

and a probability from $\vert{k}\vert\rightarrow\vert{k-1}\vert$:

$$P(\vert{k}\vert\rightarrow\vert{k-1}\vert)=1-\frac{1}{2}exp[-\alpha(\vert{N}\vert-\vert{k}\vert)]$$

Starting from a point $k$, the particle begins to jump and the process ends when $\vert{k}\vert>\vert{N}\vert$

what is the probability for the process to end after a time $T$ starting from the point $k$?

Can someone help me to solve this problem? I posed this problem in a slightly different form also on 'mathematics stackexchange'.

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I posed this problem in a slightly different form also on 'mathematics stackexchange'... where some back and forth exchanges with some readers occurred, to try to reach a convincing formulation of the question (not reached yet). Speaking frankly, this makes me doubt that abandoning the question there and asking this one here was the right thing to do. – Didier Piau Feb 15 2012 at 14:32
Your convention for describing the transition probabilities is somewhat ambiguous. In particular, what happens for $k=0$? (I assume you want it to be 50:50, but that's not what the formulas give.) It sounds like you're setting up a random walk in an exponentially decaying "gravitational field" centered at the origin, with points near (but not at) the origin being strongly pulled back toward the origin. Is that really what you want? – Barry Cipra Feb 15 2012 at 17:32
@Barry Cipra: more or less your interpretation could be useful. – Riccardo.Alestra Feb 15 2012 at 18:47
Riccardo, it would help if you'd edit your question. I would suggest simplifying it to the equivalent random walk on $[0,N]$ with $P(0 \rightarrow 1)=1$ and $P(k \rightarrow k+1) = 1-P(k \rightarrow k-1) = (1/2)\exp(-\alpha(N-k))$ for $k>0$. At the same time, you might also take a look at Günther Rudolph's paper, "The Fundamental Matrix Of The General Random Walk With Absorbing Boundaries" at ls11-www.informatik.uni-dortmund.de/people/… and see if anything nice happens when you plug in your exponential expressions for Rudolph's $p_i$s and $q_i$s. – Barry Cipra Feb 15 2012 at 19:44