Zero-Law for Stochastic Resetting

• The paper establishes that stochastic resetting in CTRWs follows an exact threshold where jump probability dominates reset probability, ensuring meaningful long-time displacement.
• It unifies various reset protocols—from complete to partial and dichotomous resetting—and connects them with zero-current and invariance principles in nonequilibrium systems.
• The study highlights robust criteria for reset-induced ordering and universal behavior while identifying limitations, especially the absence of a universal thermodynamic zeroth-law.

“Zero-law for stochastic resetting” denotes a family of law-like statements about reset-driven stochastic processes rather than a single universally accepted theorem. In the most explicit recent usage, it is an exact criterion for continuous-time random walks (CTRWs) under resetting: beyond a threshold jump number, the probability to jump dominates the probability of reset, so the reset walk remains a meaningful displacement process at large elapsed times (Colantoni et al., 10 Jul 2025). In adjacent parts of the literature, closely related statements concern unique resetting-selected nonequilibrium stationary states (NESSs) (Grange, 2020), symmetry-enforced zero-current stationary states (Sadekar et al., 2020), and universal invariance or ordering properties under restart (Pal et al., 2019, Vidmar, 2019). No work in this corpus derives a universal thermodynamic zeroth-law analogue for stochastic resetting.

1. Conceptual scope and uses of the term

Stochastic resetting is not a single protocol. In the surveyed literature it includes complete resetting to the initial position with arbitrary renewal law, Poissonian resetting to a fixed point, local resetting of constituents in interacting many-body systems, resetting to one of two competing configurations, and finite-time return phases rather than instantaneous teleportation (Colantoni et al., 10 Jul 2025, Grange, 2020, Sadekar et al., 2020, Pal et al., 2019). The precise meaning of any “zero-law” therefore depends on which reset class, state space, and observable are being held fixed.

A second distinction is between kinematic, stationary, and thermodynamic senses of “zero.” In CTRWs, “zero-law” refers to an admissibility condition for long-time displacement (Colantoni et al., 10 Jul 2025). In symmetric exclusion with dichotomous resetting, it refers to exact vanishing of the average current under a symmetry condition, even though the stationary state remains genuinely nonequilibrium (Sadekar et al., 2020). In restart theory, related law-like statements classify which completion-time laws are universally helped, harmed, or left invariant by restart, and identify the exponential law as the unique invariant distribution under broad restart classes (Vidmar, 2019). This suggests that the term is best understood as a family of exact reset-induced selection, cancellation, or invariance principles, not as a single doctrine.

A further source of ambiguity is the relation to equilibrium. Several papers make explicit that vanishing current, flat one-point density, or initial-condition independence of a NESS do not amount to detailed balance or a thermodynamic equilibrium law (Sadekar et al., 2020, Grange, 2020). Thermodynamic treatments of resetting derive first- and second-law balances, resetting entropy production, and integral fluctuation theorems, but they do not introduce an equalizing intensive resetting variable analogous to temperature (Fuchs et al., 2016, Pal et al., 2017).

2. Formal zero-law in CTRWs

The only paper in this set that explicitly introduces the phrase “zero-law of stochastic resetting” studies a general CTRW $X_t$ on $\mathbb{R}$ with i.i.d. waiting times $\tau_i$, i.i.d. jumps $\xi_i$, and complete resetting to the initial position $0$ after i.i.d. reset intervals with density $\varphi(t)$ (Colantoni et al., 10 Jul 2025). The reset process $Y_t$ remains a CTRW with the same jump-size law $\lambda$, but with a modified counting process $Q_t$. At the level of jump counts,

$$p(x;t)=\sum_{n=0}^\infty \rho_n(x)\,\overline{P}_n(t),$$

and the modified counting probabilities satisfy

$\mathbb{R}$0

where $\mathbb{R}$1 is the no-reset survival probability. Resetting thus alters the temporal sector while leaving the spatial jump kernel unchanged (Colantoni et al., 10 Jul 2025).

The zero-law itself is formulated in Laplace space. If $\mathbb{R}$2 and $\mathbb{R}$3 are the Laplace transforms of the waiting-time density and survival function, and $\mathbb{R}$4, $\mathbb{R}$5 those of the resetting law, then for sufficiently large jump number $\mathbb{R}$6,

$\mathbb{R}$7

equivalently

$\mathbb{R}$8

which yields the explicit threshold

$\mathbb{R}$9

The authors interpret this as the condition that, after sufficiently many jumps, the probability to jump to any site is larger than the probability to be reset to the starting site $\tau_i$0 (Colantoni et al., 10 Jul 2025).

The criterion is temporal rather than spatial. It depends only on $\tau_i$1 and $\tau_i$2, not directly on the jump law $\tau_i$3, so it applies equally to Lévy flights and non-Markovian or subdiffusive CTRWs. It is also logically prior to NESS existence criteria: the question is first whether the reset walk remains a meaningful displacement process at all, and only then whether it converges to a stationary distribution (Colantoni et al., 10 Jul 2025). In the same framework, master equations are derived for completely monotone resetting laws, but only under the zero-law condition.

The paper also identifies two asymptotic sectors for power-law waiting and reset laws. For $\tau_i$4 and $\tau_i$5, meaningful displacement holds automatically in the regime $\tau_i$6, which at small $\tau_i$7 means $\tau_i$8; it can also hold in the opposite regime $\tau_i$9 under the alternate inequality written in the paper, which implies $\xi_i$0 (Colantoni et al., 10 Jul 2025).

3. Stationary-state selection, thresholds, and failure modes

A different, narrower “zero-law-like” theme is steady-state selection under resetting. In constant-kernel Smoluchowski aggregation with local Poissonian shattering $\xi_i$1, the mean-field concentrations $\xi_i$2 evolve with conserved total mass $\xi_i$3, and resetting generates a unique NESS determined solely by the dimensionless rate $\xi_i$4 (Grange, 2020). The total cluster density satisfies

$\xi_i$5

with physical stationary solution

$\xi_i$6

and $\xi_i$7 exponentially with relaxation rate $\xi_i$8 (Grange, 2020). The stationary aggregate-size distribution is independent of initial conditions, and for fixed target size $\xi_i$9 the stationary density is maximized at

$0$0

The paper explicitly states, however, that this is not a universal zero-law; it is an exactly solvable example of rate-controlled NESS selection (Grange, 2020).

Resetting can also fail to confine. For one-dimensional diffusion with power-law reset intervals $0$1, there is a sharp threshold at $0$2: for $0$3 no stationary position distribution exists and the width continues to grow as $0$4, whereas for $0$5 a stationary density exists; a second threshold at $0$6 separates existence from divergence of the stationary MSD (Nagar et al., 2015). This gives a concrete asymptotic dichotomy between resetting with effectively zero confining effect and resetting strong enough, on average, to localize.

A more microscopic failure mode appears in the random acceleration process $0$7. Complete resetting of both $0$8 and $0$9 yields a stationary state and finite mean first-passage time, but partial resetting of $\varphi(t)$0 alone does not: the velocity remains unreset, $\varphi(t)$1, the position density scales as $\varphi(t)$2, and the first-passage survival decays as $\varphi(t)$3, so all positive moments of the first-passage time remain divergent (Singh, 2020). By contrast, in underdamped Brownian motion with position resetting only, the velocity sector relaxes to the Maxwellian while the position sector reaches a reset-induced NESS (Gupta, 2018). The comparison shows that whether “partial resetting” is sufficient is model-dependent and hinges on which slow variables remain uncontrolled.

4. Symmetry-induced zero-current states

The most explicit “zero” statement outside the CTRW zero-law is the zero-current state in the half-filled symmetric exclusion process (SEP) with dichotomous resetting to two complementary step configurations $\varphi(t)$4 and $\varphi(t)$5 at rates $\varphi(t)$6 and $\varphi(t)$7 (Sadekar et al., 2020). The total reset rate is $\varphi(t)$8, and the stationary one-point density is

$\varphi(t)$9

so every nonuniform contribution is proportional to $Y_t$0. Hence when

$Y_t$1

the stationary profile is exactly flat (Sadekar et al., 2020).

The average diffusive current across the central bond is

$Y_t$2

and therefore vanishes identically when $Y_t$3. The integrated average current behaves for long times as

$Y_t$4

so symmetric dichotomous resetting gives exact cancellation of the mean drift (Sadekar et al., 2020). This is a genuine zero-current law under a precise reset symmetry.

The same paper shows just as clearly that zero current is not equilibrium. In the zero-current state, current fluctuations differ sharply from equilibrium SEP, the stationary measure is not the uniform measure on fixed-particle-number configurations, spatial correlations are inhomogeneous, temporal correlations decay exponentially rather than algebraically, and the susceptibility to a weak local drive grows as $Y_t$5 rather than $Y_t$6 (Sadekar et al., 2020). The zero-current condition is therefore a first-moment transport statement, not a detailed-balance statement.

5. Universal ordering, invariance, and optimal-restart laws

A broader body of restart theory yields exact universal ordering results that are often the closest mathematical relatives of a “zero-law.” For a completion-time law $Y_t$7 with survival function $Y_t$8, the paper on universal ordering under restart proves that arbitrary stochastic restart and deterministic restart with arbitrary period lead to the same characterization: $Y_t$9 is rendered no bigger under every restart law iff

$\lambda$0

is rendered no smaller iff the inequality is reversed, and is left invariant iff

$\lambda$1

which holds iff $\lambda$2 is exponential (Vidmar, 2019). In mean, the corresponding universal criterion is expressed in terms of the mean residual life $\lambda$3: universal improvement requires $\lambda$4 for all $\lambda$5, universal worsening requires $\lambda$6, and universal invariance again singles out the exponential law (Vidmar, 2019).

Restart class	Universal “no bigger” criterion	Universal invariant law
Arbitrary stochastic or deterministic restart	$\lambda$7	Exponential
Exponential restart at arbitrary rate	$\lambda$8	Exponential

The table also marks a limitation: constant-rate exponential restart is not equivalent to arbitrary stochastic restart, because its universal criterion is a convolution inequality rather than tail supermultiplicativity (Vidmar, 2019).

A different invariance principle arises for resetting with finite return times. For a particle whose reset initiates deterministic return to the origin with space-dependent speed $\lambda$9, the conditional steady-state density during the motion phase satisfies

$Q_t$0

where $Q_t$1 is the stationary density for the corresponding instantaneous-resetting process (Pal et al., 2019). If $Q_t$2 and $Q_t$3 is Laplace,

$Q_t$4

then even the full steady-state density is invariant: $Q_t$5 This is an exact protocol-independence theorem for the steady-state shape of the active phase, and for the Laplace class an even stronger invariance of the full stationary law (Pal et al., 2019).

Optimal-restart theory supplies another law-like statement. For nonlinear diffusion with Poissonian resetting, the paper derives the MFPT, determines the optimal reset rate numerically, and confirms the universal property

$Q_t$6

that is, coefficient of variation equal to unity at the optimum (Chelminiak, 2021). This extends the standard restart universality test beyond linear Brownian diffusion to a density-dependent nonlinear diffusion equation.

6. Limits, misconceptions, and current status

Several negative results delimit the scope of any putative zero-law. First, there is no universal optimality of deterministic restart once the target location is random. For diffusive search with reset-time law $Q_t$7 and random target distance $Q_t$8, the paper on broad target distributions shows that sharp restart is optimal only in the fixed-target or narrow-target regime, while target distributions with exponential or heavier tails favor stochastic resetting over sharp restart (Evans et al., 2024). This rules out a blanket law that deterministic restart is universally best.

Second, thermodynamic resetting theory does not presently contain a zeroth law. Stochastic-thermodynamic analyses derive the first and second law for reset processes, resetting entropy production,

$Q_t$9

and the Landauer-type work bound

$$p(x;t)=\sum_{n=0}^\infty \rho_n(x)\,\overline{P}_n(t),$$0

while fluctuation-theorem work establishes a Hatano–Sasa relation and an auxiliary-dynamics integral fluctuation theorem for systems with resetting (Fuchs et al., 2016, Pal et al., 2017). These are exact nonequilibrium thermodynamic constraints, but neither paper formulates transitivity, equalization under contact, or an intensive resetting variable.

Third, some law-like asymptotic regularizations should not be conflated with a zero-law. In first-passage Brownian functionals, Poisson resetting changes heavy-tailed completion statistics into exponential asymptotic laws, for example

$$p(x;t)=\sum_{n=0}^\infty \rho_n(x)\,\overline{P}_n(t),$$1

and more generally $$p(x;t)=\sum_{n=0}^\infty \rho_n(x)\,\overline{P}_n(t),$$2 for $$p(x;t)=\sum_{n=0}^\infty \rho_n(x)\,\overline{P}_n(t),$$3 (Singh et al., 2022). These are robust reset-induced completion laws, but they are not zeroth-law statements.

The present literature therefore supports a careful synthesis. The expression “zero-law for stochastic resetting” has one formal meaning in CTRWs, where it is an exact admissibility condition for long-time displacement (Colantoni et al., 10 Jul 2025). Beyond that formal usage, the literature contains several exact neighboring principles: zero-current cancellation under reset symmetry, universal invariance of specific stationary sectors, universal ordering under restart, and coefficient-of-variation unity at optimal restart (Sadekar et al., 2020, Pal et al., 2019, Vidmar, 2019, Chelminiak, 2021). What is still missing is a single universal law spanning arbitrary reset protocols, arbitrary observables, and a thermodynamic notion of coexistence.