First-Passage Time Statistics

• First-passage time statistics are defined as the random times at which a stochastic process first reaches a specified threshold, with key observables like survival probability and first-passage density.
• They encompass a range of models—including Brownian motion, Lévy flights, and quantum trajectories—that illustrate the effects of geometry, memory, and confinement on process dynamics.
• The framework reveals that full distribution analysis provides deeper insights than mean values alone, highlighting aspects such as sample-to-sample variability and non-renewal behavior.

Searching arXiv for relevant first-passage time statistics papers to ground the article. First-passage time statistics concerns the random time $\tau$ at which a stochastic process first reaches a prescribed threshold, absorbing set, or exit boundary. In the formulations collected across diffusion, Lévy flights, non-Markovian dynamics, surface-constrained motion, active matter, and quantum trajectories, the central objects are the survival probability $S$ and the first-passage-time density $f$ or $\phi$, linked by $f(\tau)=-dS/d\tau$, together with derived quantities such as the mean first-passage time (MFPT), moments, leapover lengths, uniformity indices, and counting-threshold first-passage distributions (0706.3641, Urdapilleta, 2015, Gross et al., 2021, Ladenburger et al., 5 Nov 2025). A recurrent theme is that the full distribution often carries information that is not captured by the mean alone: universality can coexist with strong geometry dependence, memory can modify both tails and boundary layers, and confinement can introduce multiple competing time scales rather than a single characteristic one.

1. Core definitions and observables

In bounded-domain diffusion, the first-passage time $\tau$ is the random time when a trajectory first hits an absorbing part of the boundary. For Brownian motion in a domain $S$ with absorbing boundary $\partial S_a$ and reflecting boundary $\partial S_r$, the survival probability is

$S(t|r_0)=\int_S P(r,t|r_0)\,dr,$

the first-passage-time density is

$S$0

and the MFPT is

$S$1

The same logic appears in one-dimensional drift-diffusion, where $S$2, and in exit problems on surfaces, where $S$3 and $S$4 (Mattos et al., 2012, Urdapilleta, 2015, Gross et al., 2021).

A second class of observables tests trajectory-to-trajectory variability rather than only moments. Mattos et al. define the uniformity index

$S$5

for two independent realizations started from the same point. Values near $S$6 indicate similar arrival times, whereas values near $S$7 or $S$8 indicate strong desynchronization. The corresponding distribution

$S$9

is used as a diagnostic of whether the MFPT is representative of typical single-trajectory behavior (Mattos et al., 2012, Mattos et al., 2013).

In jump processes and quantum transport, first passage is defined with respect to a counting variable rather than a spatial coordinate. Ptaszyński considers the first time $f$0 that the net jump number reaches a threshold $f$1, while Rudge and Kosov define $f$2 as the first time the net number of electrons transferred to the drain reaches $f$3. In both settings, the first-passage distribution is extracted from tilted or $f$4-resolved generators, and the cumulants of $f$5 become a time-domain complement to full counting statistics (Ptaszynski, 2018, Rudge et al., 2019).

2. Backward equations, image constructions, and numerical solvers

A standard analytical route is the backward equation. For a Brownian particle in an arbitrary one-dimensional potential with an exponential temporally decaying superimposed field,

$f$6

the backward Fokker-Planck equation for the survival probability is

$f$7

A series ansatz

$f$8

leads, after Laplace transformation, to a recurrence with the shift $f$9. This structure yields explicit Wiener-process and Ornstein-Uhlenbeck results and provides moments through differentiation of the Laplace transform (Urdapilleta, 2015).

On surfaces of general shape, the same principle appears in geometric form. For the intrinsic Itô drift-diffusion process

$\phi$0

the mean first-passage time $\phi$1 solves the Backward-Kolmogorov boundary-value problem

$\phi$2

with

$\phi$3

The generalized moving least squares (GMLS) framework of Trask and collaborators discretizes this problem directly on a point cloud. The reported geometry reconstruction converges as $\phi$4 for $\phi$5th-order polynomials, while the surface operator approximation achieves empirical $\phi$6 convergence in $\phi$7, with sample rates $\phi$8, $\phi$9, and $f(\tau)=-dS/d\tau$0 for $f(\tau)=-dS/d\tau$1 (Gross et al., 2021).

A complementary route is the method of images. For a one-dimensional Brownian motion with purely time-dependent drift and diffusion,

$f(\tau)=-dS/d\tau$2

Molini et al. show that an exact image solution exists only when $f(\tau)=-dS/d\tau$3 and $f(\tau)=-dS/d\tau$4 are proportional functions of time, so that the ratio $f(\tau)=-dS/d\tau$5 remains constant. Under that condition, the transition density, survival probability, and first-passage density are obtained analytically for power-law and periodic drivers. Outside this proportionality class, the image construction is only approximate (Molini et al., 2010).

For discrete Markov networks, the corresponding operator framework is built from transition matrices $f(\tau)=-dS/d\tau$6 and the tilted generator $f(\tau)=-dS/d\tau$7. The Laplace-transformed first-passage distribution obeys the central relation

$f(\tau)=-dS/d\tau$8

which makes it possible to compute FPT distributions for arbitrary sets of counted transitions and to compare them directly with the scaled cumulant-generating function of full counting statistics (Ptaszynski, 2018).

3. Classical universality classes: Brownian motion, Lévy flights, and gated diffusion

For symmetric Lévy flights with stability index $f(\tau)=-dS/d\tau$9, Koren et al. derive exact first-passage and leapover statistics. The survival probability decays as $\tau$0, so the first-passage density follows the universal Sparre-Andersen law

$\tau$1

The exponent $\tau$2 is independent of $\tau$3, whereas the prefactor depends on $\tau$4, the generalized diffusion coefficient $\tau$5, and the threshold $\tau$6. In the same model, leapover lengths are asymptotically power-law distributed with index $\tau$7 for one-sided Lévy flights and, surprisingly, with index $\tau$8 for symmetric Lévy flights (0706.3641).

The one-sided Lévy-flight problem falls into a different universality class. Because jumps cannot go backward, the Laplace transform of the first-passage density is a Mittag-Leffler function, the exact density is expressed through the Wright $\tau$9-function, the tail is exponentially tempered, and all positive moments exist. In particular,

$S$0

This establishes a sharp contrast between symmetric Lévy flights, which have a universal $S$1 tail and divergent mean, and one-sided Lévy flights, which have narrow arrival-time laws with finite mean (0706.3641).

Related departures from the one-channel Brownian paradigm arise in gated diffusion. In two-channel Markov-additive diffusion in a three-dimensional spherical domain, the particle recognizes the target only in one internal mode. Grebenkov and collaborators prove that, despite the perfectly non-recurrent motion of two-channel Markov additive diffusion in $S$2 dimensions, the long-time first-passage statistics do not display Poisson-like behavior if none of the phases has a vanishing diffusion coefficient. In the intermittent immobilization limit $S$3, the long-time tail becomes Poissonian again, and the process reduces to an effective one-channel diffusion with a renormalized diffusion coefficient (Godec et al., 2016).

Non-linear diffusion generates yet another asymptotic class. For the non-linear diffusion equation with diffusivity $S$4, the free first-passage density on $S$5 is normalized to unity but decays as

$S$6

so the MFPT diverges for every $S$7. Under harmonic confinement, by contrast, the long-time tail is cut off and the MFPT becomes finite. The distinction is therefore not between linear and non-linear diffusion as such, but between unbounded and confined geometries (Chelminiak, 2022).

4. Geometry, confinement, and bi-scaling structure

Confinement introduces geometry-dependent mechanisms that can dominate first-passage statistics. On surfaces of general shape, the GMLS studies show that the MFPT is influenced by surface geometry, drift dynamics, and spatially dependent diffusivities. On a truncated torus with a double-well drift, the mean FPT from minima grows exponentially in the barrier height and numerically recovers $S$8 for $S$9. With a diffusivity field $\partial S_a$0, a dead zone of low diffusivity traps trajectories and dramatically increases $\partial S_a$1 as $\partial S_a$2 or as the slow-region radius grows. On necked surfaces of revolution, the projected free energy $\partial S_a$3 acts as an entropic barrier and $\partial S_a$4 diverges as $\partial S_a$5, with power-law scaling (Gross et al., 2021).

For confined compact processes, Bar-Avrahami and Barkai show that the first-passage-time density generically splits into two regimes: a short-time part that is indistinguishable from the free-space solution and a long-time part controlled by the finite extent or confining force of the domain. Their bi-scaling theory uses two scaling functions, one inherited from the infinite system and one sensitive to finite size effects. In large systems this challenges the use of a single time scale, because the full density across all times is assembled from both contributions rather than from one collapsed curve (Baravi et al., 2023).

Baravi, Kessler, and Barkai sharpen this picture for recurrent confined processes by identifying a transition in the moment spectrum at

$\partial S_a$6

where $\partial S_a$7 is the persistence exponent. For $\partial S_a$8, the moments converge to those of the infinite system; for $\partial S_a$9, they are governed by a non-normalizable infinite density and scale with the macroscopic confinement time. The same framework extends to diffusion in a confining potential in the high-temperature limit, with the potential strength replacing the system size as the relevant scale (Baravi et al., 20 Mar 2025).

Aging modifies these confinement scalings in a precise way. For general non-Markovian scale-invariant processes in arbitrary dimension, all FPT moments obey universal scalings with the confining volume and with non-trivial exponents. In this framework, linear scaling $\partial S_r$0 is universal for non-aging, scale-invariant processes, whereas a non-linear power law $\partial S_r$1 with $\partial S_r$2 is the hallmark of aging (Levernier et al., 2017).

5. When the mean first-passage time is meaningful

A central issue in bounded domains is whether the MFPT is a representative time scale. Mattos et al. address this by analyzing the distribution $\partial S_r$3 of the uniformity index. For a generic two-cutoff first-passage law

$\partial S_r$4

they obtain a closed form for $\partial S_r$5 in terms of the modified Bessel function $\partial S_r$6. The persistence exponent $\partial S_r$7 controls the shape: for $\partial S_r$8, $\partial S_r$9 is always unimodal with a peak at $S(t|r_0)=\int_S P(r,t|r_0)\,dr,$0; for $S(t|r_0)=\int_S P(r,t|r_0)\,dr,$1, it is nearly flat; and for $S(t|r_0)=\int_S P(r,t|r_0)\,dr,$2, there is a critical ratio $S(t|r_0)=\int_S P(r,t|r_0)\,dr,$3 beyond which $S(t|r_0)=\int_S P(r,t|r_0)\,dr,$4 becomes bimodal, with a minimum at $S(t|r_0)=\int_S P(r,t|r_0)\,dr,$5 and maxima near $S(t|r_0)=\int_S P(r,t|r_0)\,dr,$6 and $S(t|r_0)=\int_S P(r,t|r_0)\,dr,$7 (Mattos et al., 2012).

This produces an operational criterion for the usefulness of the MFPT. If $S(t|r_0)=\int_S P(r,t|r_0)\,dr,$8 is unimodal, most realizations lie close to $S(t|r_0)=\int_S P(r,t|r_0)\,dr,$9 and the MFPT is a good typical measure. If $S$00 is bimodal, sample-to-sample fluctuations are large and the MFPT is a poor predictor of a single-trajectory outcome. A related quantitative indicator is the relative error

$S$01

with $S$02 correlating with bell-shaped $S$03 and $S$04 with M-shaped $S$05 (Mattos et al., 2012).

The phase boundary depends sensitively on geometry and starting point. In finite wedges, circular disks with a narrow aperture, and fully absorbing triangular domains, Mattos et al. construct charts that separate bell-shaped from M-shaped regions. In another set of bounded-domain examples, they report explicit thresholds such as $S$06 in a finite interval, $S$07 in a two-dimensional disk, and $S$08 in a three-dimensional sphere. These thresholds show that all ordinary moments can be finite while the full first-passage statistics still exhibit strong trajectory-to-trajectory fluctuations (Mattos et al., 2013).

A plausible implication is that “MFPT” and “representative time scale” are distinct notions. The mean can exist, and even be easy to compute, while the underlying distribution remains broad enough that two nominally identical realizations are likely to arrive at very different times.

6. Memory, non-renewal, and first passage beyond Markovianity

For non-Markovian walkers, the event of first contact builds up correlations that feed back on both the FPT statistics and the absorbing-wall probability density. Using a generalized Langevin equation and Onsager’s regression hypothesis, Park et al. derive an exact Volterra integral equation for the FPT density of subdiffusive fractional Brownian motion and obtain the closed-form interpolation

$S$09

This reproduces the Markovian limit $S$10, where $S$11, and yields the non-trivial long-time scaling

$S$12

for $S$13 (Sakamoto et al., 2023).

The same analysis predicts a non-standard boundary layer. For large $S$14 and $S$15,

$S$16

so the surviving density vanishes at the boundary with $S$17 for all $S$18. This contrasts with the linear boundary layer of ordinary diffusion and directly signals the breakdown of the method of images in the presence of memory (Sakamoto et al., 2023).

In discrete Markov networks, non-renewal appears in a different form. If successive first-passage intervals are independent, then

$S$19

and there are strict relations between the cumulants of the full counting statistics and the first-passage distribution. When correlations are present, the FPT distribution carries additional information about internal dynamics, such as switching between dynamical states, and the breaking of the fluctuation theorem for first-passage times may reveal the multicyclic nature of the Markovian network (Ptaszynski, 2018).

Quantum transport provides an explicit application of this distinction. In molecule-based transport with backtunneling, cotunneling, and electron-phonon coupling, Rudge and Kosov show that waiting-time statistics do not correctly predict renewal and non-renewal behavior, whereas the first-passage-time distribution does. Because the FPT formalism is built from the full $S$20-resolved master equation, it treats forward and backward transitions on equal footing and remains applicable in bidirectional regimes where standard waiting-time constructions fail (Rudge et al., 2019).

7. Quantum, active, and extreme-event extensions

In continuously monitored quantum systems, first passage can be defined in Hilbert-space probability rather than in configuration space. For the overlap

$S$21

with a decoherence-free subspace $S$22, the quantum first-passage time is

$S$23

Under homodyne monitoring, $S$24 obeys the drift-free Itô SDE

$S$25

and the exact first-passage-time density follows from a spectral expansion of the associated Fokker-Planck operator. All Hamiltonian and measurement-operator details enter only through the single scale parameter $S$26, which is why the resulting distribution is described as universal (Ladenburger et al., 5 Nov 2025).

Large deviations of first-passage statistics in open quantum systems can also be computed within a semi-Markov-process framework. Liu’s core object is the equation of poles

$S$27

which determines the scaled cumulant-generating functions of counting statistics and, for simple counting variables, the scaled generating functions of first-passage times. For current-like variables the root method generally fails unless the pole equation reduces to a quadratic form, because the nonuniqueness between the roots and the region of convergence obstructs a direct identification of the relevant branch (Liu et al., 2024).

Active matter brings orientation as an explicit control parameter. For an active Brownian particle in two dimensions approaching an absorbing wall, a perturbative expansion in small Péclet number gives

$S$28

with $S$29. The zeroth-order density is the half-line Brownian first-passage law, while the first-order correction generates anisotropic arrival statistics. The median $S$30 becomes smaller for particles initially pointing toward the wall and larger for those pointing away, and the anisotropy is maximal when the initial distance is of order the persistence length (Baouche et al., 7 Mar 2025).

Extreme first-passage statistics introduce another layer of structure. For the fastest first-passage time under time-dependent particle injection, Grebenkov et al. derive the full density $S$31 for arbitrary injection intensity and show that extended injection can markedly alter the large-$S$32 asymptotics of the mean fastest arrival time. They also show that convergence to the Gumbel limit can be considerably slowed, so that the asymptotic Gumbel law may be inapplicable for biologically relevant particle numbers (Grebenkov et al., 24 Mar 2025). At the opposite extreme, the slowest first-passage time in the thermodynamic limit has a density-controlled crossover: in one-dimensional boxes the limiting law interpolates between Fréchet at low density and Gumbel at high density, while in compact fractal media the walk dimension $S$33 and fractal dimension $S$34 enter through the single-particle survival scaling function (Baravi et al., 7 Sep 2025).

Taken together, these results show that first-passage time statistics is not a single theory but a family of exact, asymptotic, and numerical frameworks organized by symmetry, memory, confinement, geometry, internal state structure, and observation protocol. The shared objects are simple—survival probabilities, hitting distributions, and moments—but the resulting phenomenology ranges from universal $S$35 tails and Mittag-Leffler laws to bi-scaling, non-renewal, non-Poissonian long-time behavior, and Hilbert-space first passage.