Solution and Analysis of a One-Dimensional First-Passage Problem with a Nonzero Halting Probability

Figure 2

Numerical results of 𝑃(𝑥,𝑡) as a function of 𝑡 (𝑥=50). The graphs are separated according to the value of Δ𝑝=𝑝_←−𝑝_→: (a) Δ𝑝=0.2, (b) Δ𝑝=0.1, and (c) Δ𝑝=0. The two curves in each panel represent 𝑝_→=0.1 and 0.3. Long-time behavior of 𝑃(𝑥,𝑡) in Δ𝑝=0 is shown in (d) on logarithmic scales, which suggests a power law 𝑃(𝑥,𝑡)∝𝑡^−3/2.

(a)

(b)

(c)

(d)