”Random walk” Let A, B, x be integers, \(A \leqslant x \leqslant B\) . Consider the particle that starts moving from the point x at time \(t = 0\) . At each step \(\Delta t = 1\) the particle can move left or right from its recent position with the probabilities p and \(q = 1 - p\) correspondingly. If at some step it reaches the points A or B, it stays there forever (see example figure below).
The source’s Figure 8 illustrates one sample trajectory with \(A=-B=5\) and \(x=2\); the graphical trajectory is not present in the deterministic extraction.
Assuming that the total number of steps approaches infinity, compute the probabilities \(\alpha ( x ) , \beta ( x )\) to find the particle at the points A and B correspondingly, as functions of the initial position x of the particle.
Find the mean time \(m ( x )\) of a random walk of the particle before it hits A or B. Assume that \(m ( x ) < \infty\) . Check that if \(p = q = \frac { 1 } { 2 }\) and \(A = - B\) , then \(m ( 0 ) = B ^ { 2 }\) . Hence the mean time of random walk is given by a square of distance traveled.