Mean First Passage Time — Random Walk with Absorbing Barriers

Drift (p − q) 0.00

Start position x₀/N 0.50

Walker count 80

Theory MFPT—

Simulated mean—

Absorbed—

Survival fraction—

Physics

For a 1D biased random walk on {0,…,N} with absorbing barriers, step right with probability p = ½+d, left with q = ½−d. Starting at x₀:

Drift-free (d=0): T(x₀) = x₀(N−x₀)

With drift: T(x₀) = (N/2d)·[(1−ρˣ⁰)/(1−ρᴺ)] − x₀/(2d), where ρ = q/p.

MFPT is a fundamental quantity in chemical kinetics, neural firing (integrate-and-fire), and financial ruin problems. The distribution is the inverse Gaussian.