Diffusion and first-passage problems

Consider unbiased one-dimensional Itô diffusion $dx=\sqrt{2D}\,dW_t$ with constant positive D. Its density obeys $\partial_tp=D\partial_x^2p$. The same process has backward generator $\mathcal L=D\partial_x^2$, acting on functions of the starting position. For a first exit from the interval between zero and L, the right-exit probability h obeys $Dh''=0$, with h equal to zero at the left endpoint and one at the right. Integrating twice gives a linear function of the starting position.

This probability result is independent of D because multiplying the diffusion speed changes how fast a path explores the interval, but not the symmetry between left and right at a given starting point. It relies on zero drift and constant diffusion; a directional drift can change the hitting probabilities. Spatially varying diffusion additionally requires specifying its drift and stochastic interpretation before reusing the simple result. An absorbing boundary means the trajectory is stopped on its first encounter. Continuing to count paths after they cross a target would instead calculate ordinary transition probabilities, which include repeated returns and answer a different question.

Let T(x) be the mean remaining time before absorption at either endpoint. After a short interval dt, the process has spent dt and has a new starting position. Averaging the Taylor expansion of the remaining time gives $T(x)=dt+T(x)+DT''(x)dt$ to leading order. Cancelling the common term yields the backward Poisson equation $DT''=-1$. Its negative curvature reflects a longer mean wait when starting farther from either absorbing boundary. Setting T to zero at both endpoints fixes the two integration constants.

A reflecting endpoint has a different condition. No probability is lost there, and for constant-D unbiased diffusion the corresponding backward condition is zero normal derivative of T. With reflection at zero and absorption at L, use $T'(0)=0$ and $T(L)=0$. The resulting mean time is larger than when zero is absorbing, since trajectories visiting the left endpoint are retained instead of stopped. The boundary condition is part of the stochastic model, not a mathematical convenience that may be changed without changing the physical question.

The survival probability S(t) is the probability that absorption has not yet occurred. Its negative derivative gives the first-passage density, and the mean passage time equals the integral of S when that integral converges. In a finite interval with an absorbing endpoint, the survival tail decays sufficiently rapidly for the mean to be finite. On the infinite half-line, unbiased diffusion reaches a finite target with probability one, yet its mean hitting time diverges. Rare long excursions make probability-one arrival compatible with an infinite average wait.

This is why the scale $\ell^2/D$ should not automatically be called a mean first-passage time. It estimates when typical diffusive displacement becomes comparable to a distance ell. Actual means depend on domain size, drift, boundary type, and the tail of the stopping-time distribution. For narrow targets in higher dimensions, geometry matters as well. In experiments finite observation windows censor long trajectories, which can make a divergent or very large theoretical mean appear finite unless the unobserved survival tail is considered.