Stationary Probability and First-Passage Time of Biased Random Walk

In this paper, we consider the stationary probability and first-passage time of biased random walk on 1D chain, where at each step the walker moves to the left and right with probabilities p and q respectively（0 p, q 1,p + q = 1）. We derive exact analytical results for the stationary probability and first-passage time as a function of p and q for the first time. Our results suggest that the first-passage time shows a double power-law F;, where the exponent γ = 2 for N < |p-q|;and γ = 1 for N > |p-q|;. Our study sheds useful insights into the biased random-walk process.