First-Passage Decomposition

• First-passage decomposition is defined as partitioning a system's dynamics into sectors based on the first time a threshold or barrier is crossed, integrating probabilistic and thermodynamic insights.
• It employs methods such as Laplace transforms, Volterra convolutions, and quantum amplitude decompositions to yield exact or asymptotic representations in diverse settings.
• Applications span Markov processes, quantum measurements, reliability analysis, and disordered media, enabling a refined analysis of hitting times and dynamic responses.

First-passage decomposition denotes a class of exact or asymptotic representations in which a stochastic, thermodynamic, quantum, or reliability-theoretic problem is partitioned according to the first time a threshold, state, current level, or barrier is reached. In the supplied literature, the decomposition appears in several mathematically distinct forms: factorization of generalized partition functions into equilibrium and first-passage sectors, Volterra convolution formulas relating threshold-conditioned propagators to first-passage densities, Hilbert-space expansions of quantum evolution into no-crossing and first-crossing amplitudes, shell- and slab-wise decompositions of passage times in disordered media, and pathwise classifications of barrier crossing modes with different predictability properties [(Ryazanov, 2021); (Ptaszynski, 2018); (Sokolovski, 2012); (Guillaume, 3 Apr 2026)].

1. Core formulations and canonical objects

A first-passage time is, in its most basic form, a stopping time. For a process $X(t)$ and threshold $a$, one representative definition is

$T_\gamma^x=\inf\{t>0:X(t)=a\},\qquad X(0)=x>0.$

In finite-state Markov current problems, the corresponding object is the density $F(N|\tau)$ for the first time a counting variable reaches threshold $N$. For a càdlàg adapted process crossing a continuous barrier $b(t)$, the stopping time is

$T=\inf\{t\ge 0:X_t\ge b(t)\}.$

In quantum settings, the relevant event can instead be the first departure from a projected subspace or the first localization of a continuously monitored state near a decoherence-free sector (Ryazanov, 2021, Ptaszynski, 2018, Guillaume, 3 Apr 2026, Ladenburger et al., 5 Nov 2025).

These formulations support different decomposition principles.

Setting	Object decomposed	Canonical representation
Generalized thermodynamics	Partition function and entropy	$Z(\beta,\gamma)=Z_\beta Z_\gamma$, $s=s_\beta+s_\gamma=s_\beta-\Delta$
Markov current statistics	Threshold-conditioned propagator	Volterra convolution between $T(N|t)$, $a$0, and $a$1
Quantum first crossing	Unitary evolution operator	$a$2-evolution plus an integral over first-departure amplitudes
Barrier crossing for càdlàg processes	Event $a$3	$a$4
Dynamic reliability sensitivity	System limit-state hypersurface	Sum of surface integrals over constrained component hypersurfaces

A plausible implication is that “first-passage decomposition” is not a single formalism but a recurrent structural motif: the full dynamics is rewritten in terms of first-hit sectors, first-hit times, or first-hit boundary pieces, with each sector carrying distinct probabilistic or thermodynamic meaning.

2. Renewal, convolution, and propagator decompositions in Markov systems

For continuous-time Markov chains on finite networks, one exact formulation decomposes threshold-conditioned probabilities by the first time a current-like counting variable reaches a prescribed value. With $a$5 and threshold $a$6, the probability to be in state $a$7 at time $a$8 with net count $a$9 satisfies a Volterra convolution: $T_\gamma^x=\inf\{t>0:X(t)=a\},\qquad X(0)=x>0.$0 After Laplace transformation, this yields

$T_\gamma^x=\inf\{t>0:X(t)=a\},\qquad X(0)=x>0.$1

and hence a closed expression for the Laplace transform of the first-passage-time distribution. In renewal systems, the decomposition becomes multiplicative,

$T_\gamma^x=\inf\{t>0:X(t)=a\},\qquad X(0)=x>0.$2

which is equivalent to independent first-passage increments and implies linear scaling of FPT cumulants with $T_\gamma^x=\inf\{t>0:X(t)=a\},\qquad X(0)=x>0.$3. In the single-reset class, this yields exact algebraic bridges between FPT cumulants and full-counting-statistics cumulants, such as

$T_\gamma^x=\inf\{t>0:X(t)=a\},\qquad X(0)=x>0.$4

When renewal fails, the decomposition still exists at the level of convolution, but the factorization $T_\gamma^x=\inf\{t>0:X(t)=a\},\qquad X(0)=x>0.$5 breaks down; the paper on renewal and nonrenewal systems uses coupled quantum dots to show that correlations between successive FPTs then expose telegraphic switching and multicyclic structure not encoded in full counting statistics alone (Ptaszynski, 2018).

A related but more directly probabilistic formulation treats first passage as probability flow into an absorbing sink. In Sekimoto’s discrete-network construction, one removes outgoing edges from the goal set $T_\gamma^x=\inf\{t>0:X(t)=a\},\qquad X(0)=x>0.$6, keeps the reduced generator $T_\gamma^x=\inf\{t>0:X(t)=a\},\qquad X(0)=x>0.$7 on $T_\gamma^x=\inf\{t>0:X(t)=a\},\qquad X(0)=x>0.$8, and writes

$T_\gamma^x=\inf\{t>0:X(t)=a\},\qquad X(0)=x>0.$9

The first-passage density is then

$F(N|\tau)$0

so first passage is decomposed into edgewise flux contributions into the absorbing set. The same logic yields the discrete-time formula

$F(N|\tau)$1

for transient submatrix $F(N|\tau)$2 (Sekimoto, 2021).

For discrete first-passage time in Markov networks, an exact analytic construction replaces each edge by a cascade of $F(N|\tau)$3 fictitious unidirectional edges, computes the continuous first-passage law, and then takes the limit $F(N|\tau)$4 with transition rates set to $F(N|\tau)$5. In that limit, each hop time converges to a delta function at one unit of time, and the DFPT becomes a matrix-power decomposition in the effective propagator $F(N|\tau)$6. The resulting probability mass function

$F(N|\tau)$7

and the joint law

$F(N|\tau)$8

decompose first passage simultaneously over hop number, predecessor node, and last absorbing edge (Albert, 2024).

3. Thermodynamic and ensemble-theoretic decomposition

A distinct meaning of first-passage decomposition arises when first-passage time is promoted to a thermodynamic variable. In the generalized Gibbs formalism of “First passage time and change of entropy,” the relevant distribution is

$F(N|\tau)$9

where $N$0 and $N$1 are conjugate multipliers for internal energy and first-passage time. The generalized partition function is

$N$2

Under the assumption

$N$3

the partition function factorizes: $N$4 Here $N$5 is the usual equilibrium partition function, while $N$6 is the Laplace transform of the FPT density $N$7. This is the paper’s explicit first-passage decomposition: equilibrium statistics plus a first-passage sector controlled by the conjugate parameter $N$8 (Ryazanov, 2021).

The same factorization induces an entropy decomposition. The local specific entropy

$N$9

satisfies

$b(t)$0

The paper rewrites this as

$b(t)$1

so the deviation of entropy from equilibrium is entirely attributable to the first-passage sector. Since

$b(t)$2

the chain

$b(t)$3

expresses all FPT moments in terms of entropy deviation and external forces. The paper explicitly states: “Thus, all moments of the distribution of the first passage time are expressed in terms of the deviation of the entropy from its equilibrium value and the external forces acting on the system” (Ryazanov, 2021).

The formalism is general with respect to the underlying random process. The paper inserts known first-passage laws from drift–diffusion, current-threshold inverse Gaussian distributions, and the Feller process into the thermodynamic sector through their Laplace transforms. A plausible implication is that this framework separates the determination of $b(t)$4 from its thermodynamic embedding: stochastic dynamics determines the first-passage law, while the decomposition $b(t)$5 determines how that law enters entropy, forces, and response.

4. Quantum first-crossing decompositions

In quantum mechanics, first-passage decomposition can refer to a decomposition of amplitudes rather than probabilities. For discontinuous states in one-dimensional Schrödinger dynamics, Sokolovski uses a projector $b(t)$6 onto a subspace $b(t)$7 and writes the exact identity

$b(t)$8

The first term is evolution that has not yet left $b(t)$9; the integral term collects amplitudes that first leave at time $T=\inf\{t\ge 0:X_t\ge b(t)\}.$0. Applied to a wavefunction split at a discontinuity, this yields an emitted localized wave $T=\inf\{t\ge 0:X_t\ge b(t)\}.$1 originating from the boundary. The current

$T=\inf\{t\ge 0:X_t\ge b(t)\}.$2

is then determined by interference between the initial state and this first-passage-induced localized wave. Depending on the discontinuity, the short-time current behaves as $T=\inf\{t\ge 0:X_t\ge b(t)\}.$3, $T=\inf\{t\ge 0:X_t\ge b(t)\}.$4, or $T=\inf\{t\ge 0:X_t\ge b(t)\}.$5. In this setting, first-passage decomposition is a first-crossing decomposition of the propagator into “no crossing” plus “first crossing at $T=\inf\{t\ge 0:X_t\ge b(t)\}.$6” contributions (Sokolovski, 2012).

A second quantum realization appears in continuously monitored quantum diffusion. There, the object that undergoes first passage is the overlap of a stochastic quantum trajectory with a decoherence-free subspace. For two decoherence-free sectors $T=\inf\{t\ge 0:X_t\ge b(t)\}.$7 and $T=\inf\{t\ge 0:X_t\ge b(t)\}.$8, the overlap

$T=\inf\{t\ge 0:X_t\ge b(t)\}.$9

obeys the scalar Itô diffusion

$Z(\beta,\gamma)=Z_\beta Z_\gamma$0

with diffusion coefficient

$Z(\beta,\gamma)=Z_\beta Z_\gamma$1

The associated Fokker–Planck equation is solved with absorbing thresholds at $Z(\beta,\gamma)=Z_\beta Z_\gamma$2 and $Z(\beta,\gamma)=Z_\beta Z_\gamma$3, and the first-passage densities to the two sectors admit exact eigenfunction expansions,

$Z(\beta,\gamma)=Z_\beta Z_\gamma$4

$Z(\beta,\gamma)=Z_\beta Z_\gamma$5

The paper further gives universal formulas for the mean and variance of the threshold-hitting time in terms of

$Z(\beta,\gamma)=Z_\beta Z_\gamma$6

Its central claim is that the form of the FPT distribution does not depend on the system Hamiltonian or on the measurement operator, apart from the effective parameter $Z(\beta,\gamma)=Z_\beta Z_\gamma$7; this is a quantum trajectory analogue of spectral first-passage decomposition in classical diffusion (Ladenburger et al., 5 Nov 2025).

5. Spatial heterogeneity, disordered media, and geometric path decompositions

In lattice first-passage percolation, one decomposition strategy is geometric rather than probabilistic. Gouéré partitions $Z(\beta,\gamma)=Z_\beta Z_\gamma$8 into slabs via horizontal edge sets $Z(\beta,\gamma)=Z_\beta Z_\gamma$9 and transverse edge sets $s=s_\beta+s_\gamma=s_\beta-\Delta$0, then defines modified passage times $s=s_\beta+s_\gamma=s_\beta-\Delta$1 by making edges in $s=s_\beta+s_\gamma=s_\beta-\Delta$2 free and edges in $s=s_\beta+s_\gamma=s_\beta-\Delta$3 forbidden. The key identity

$s=s_\beta+s_\gamma=s_\beta-\Delta$4

turns the monotonicity problem for

$s=s_\beta+s_\gamma=s_\beta-\Delta$5

into a comparison between the original geodesic and a family of slab-collapsed environments. Under

$s=s_\beta+s_\gamma=s_\beta-\Delta$6

the paper proves that $s=s_\beta+s_\gamma=s_\beta-\Delta$7 is non-decreasing. Here first-passage decomposition means decomposing the path cost into contributions from successive coordinate slabs and controlling each contribution through modified environments (Gouéré, 2012).

A more explicit environment decomposition appears in Brochette first-passage percolation, where all edges lying on the same integer line have equal passage time. This line-wise dependence yields several exact decompositions: subadditive decomposition for point-to-point limits, shell decompositions in dimension two,

$s=s_\beta+s_\gamma=s_\beta-\Delta$8

and envelope decompositions used in the shape theorem. The resulting time constant is

$s=s_\beta+s_\gamma=s_\beta-\Delta$9

and the limiting shape is the $T(N|t)$0 diamond. The paper’s terminology makes clear that the line structure is itself the decomposition principle: independent line variables generate correlated path costs that remain tractable under slab, shell, and chemical-distance arguments (Marivain, 20 Jan 2025).

For heterogeneous diffusion in bounded domains, first-passage decomposition becomes a decomposition of the full density into path classes with different time scales. In a spherically symmetric domain with inner absorbing target, outer reflecting boundary, and piecewise diffusivity ratio $T(N|t)$1, the full FPT distribution exhibits three regimes: a short-time Lévy–Smirnov form associated with direct trajectories, a long-time exponential tail associated with indirect trajectories that reach the reflecting boundary, and a third, intermediate regime corresponding to brief excursions away from the target. The intermediate regime exists only for

$T(N|t)$2

and the paper emphasizes that for $T(N|t)$3 it includes the most likely FPTs. It also shows that the MFPT is dominated by the long-time exponential tail and can differ from the typical FPT by orders of magnitude (Godec et al., 2015).

A further disordered-interval variant uses the backward equation and probability generating functions to obtain all moments and the full FPT distribution for nearest-neighbor hopping with site-dependent disorder, and finds that the FPT distribution can be bimodal for certain realizations of the hopping rates. This suggests that decomposition by disorder realization can be as important as decomposition by asymptotic time regime in finite one-dimensional media (Holehouse et al., 2023).

6. Pathwise stopping-time structure, reliability surfaces, and branching renewal equations

For a càdlàg adapted process $T(N|t)$4 crossing a continuous barrier $T(N|t)$5, the most refined pathwise first-passage decomposition in the supplied material is the fourfold partition

$T(N|t)$6

where $T(N|t)$7 is continuous contact, $T(N|t)$8 is contact from the left followed by an upward jump, $T(N|t)$9 is exact hit by jump, and $a$00 is strict overshoot by jump from below. The paper shows that the left-contact component $a$01 is always accessible and becomes predictable under a no-premature-left-contact condition, with the canonical announcing sequence built from the running supremum

$a$02

On the gap side, under a structural exclusion of predictable gap-crossings, the restricted time $a$03 is totally inaccessible. In the semimartingale setting, the same decomposition yields compensator formulas for the jump-driven modes and a decomposition of the compensator of the default indicator into its predictable jump part and continuous part (Guillaume, 3 Apr 2026).

In first-passage dynamic reliability of linear systems under Gaussian excitation, decomposition is transferred from trajectories to hypersurfaces. With system limit state

$a$04

the sensitivity of the first-passage failure probability is written as

$a$05

where each $a$06 is a surface integral over the constrained component hypersurface $a$07. Because for linear systems

$a$08

the component hypersurfaces are hyperplanes and the integrands are explicit. The paper introduces importance sampling over component indices and reports that the number of function evaluations required for the sensitivity analysis is typically on the order of $a$09 to $a$10, with reuse of function evaluations across design parameters (Xian et al., 26 Oct 2025).

At the population level, branching under first-passage resetting turns first passage into the renewal kernel of a branching process. With first-passage density $a$11 to threshold $a$12 and fixed offspring number $a$13, the expected population density satisfies the exact renewal equation

$a$14

The growth rate $a$15 is then determined by the generalized Euler–Lotka relation

$a$16

or, with time-dependent yield $a$17,

$a$18

For fixed offspring number and fixed mean replication time, stochastic timing fluctuations necessarily enhance growth relative to a deterministic clock. When offspring yield depends on the first-passage time, the same decomposition exposes a yield–delay trade-off and permits analytic optimization, including a bacteriophage lysis application consistent with empirical data (Kumar et al., 15 May 2026).

Taken together, these developments show that first-passage decomposition extends beyond the computation of hitting-time laws. It organizes equilibrium and nonequilibrium ensembles, renewal and nonrenewal current statistics, quantum crossing amplitudes, pathwise stopping-time taxonomy, dynamic-reliability sensitivities, and branching growth laws into mathematically explicit sectors whose separation is often more informative than the aggregate first-passage distribution itself.