Position Resetting in Stochastic Processes

• Position resetting is a stochastic protocol that intermittently resets a process’s spatial state, generating non-equilibrium stationary distributions and altered first-passage behaviors.
• It employs a renewal framework where resets to fixed or distributed locations impact diffusion, active dynamics, and search optimizations across various systems.
• Extensions including partial resets, non-instantaneous returns, and resetting coupled with internal states illustrate that resetting efficiency critically depends on the dynamics of unreset degrees of freedom.

Position resetting denotes a class of stochastic protocols in which a process is intermittently returned in position space to a prescribed location, to a distribution of locations, or to a transformed spatial state. In the prototypical formulation introduced for diffusion, a Brownian particle is reset to its initial position at a constant Poissonian rate $r$, thereby generating a nonequilibrium stationary state and qualitatively changing first-passage statistics (Evans et al., 2011). Subsequent work has treated random resetting positions, space-dependent resetting rates, active and persistent dynamics, non-instantaneous return phases, refractory periods, heterogeneous environments, and coupled resetting of position with internal degrees of freedom, establishing position resetting as a broad renewal-based framework rather than a single model (Evans et al., 2019).

1. Basic formulation and renewal structure

In the standard one-dimensional Brownian setting, the probability density $p(x,t|x_0)$ evolves according to

$$\frac{\partial p(x,t|x_0)}{\partial t} = D\frac{\partial^2 p(x,t|x_0)}{\partial x^2} - r\,p(x,t|x_0) + r\,\delta(x-x_0),$$

where $D$ is the diffusion constant and $r$ is the resetting rate (Evans et al., 2011). The loss term $-r p$ removes probability from all positions, while the injection term $r\delta(x-x_0)$ restores probability at the resetting site. In the review formulation, the same process can be written as a renewal decomposition over the time since the last reset,

$$p(x,t|x_0)=e^{-rt}G_0(x,t|x_0)+r\int_0^t d\tau\,e^{-r\tau}G_0(x,\tau|X_r),$$

with $G_0$ the propagator without resetting (Evans et al., 2019).

This renewal structure extends directly to more general protocols. Resetting to a random position $z$ drawn from a distribution $p(x,t|x_0)$0 leads to

$p(x,t|x_0)$1

and position-dependent resetting is represented by a rate field $p(x,t|x_0)$2 rather than a constant (Evans et al., 2011). In active or persistent processes, the renewal construction must specify which variables are reset. The distinction between position-only resetting and resetting of both position and orientation or velocity is central in active Brownian particles, run-and-tumble particles, and randomly accelerated particles, because the positional restart may or may not erase the process’s hidden persistence variables (Kumar et al., 2020).

A useful terminological distinction follows directly from the literature. “Position-only resetting” resets the spatial coordinate while leaving other dynamical variables untouched; “complete resetting” resets position together with orientation, velocity, or other internal states when those variables are part of the Markovian description (Ghosh et al., 9 Jan 2025). This distinction is not cosmetic: it changes stationarity, singular behavior near the resetting point, and first-passage performance.

2. Nonequilibrium stationary states and localization

The most basic stationary consequence of position resetting is the emergence of a localized nonequilibrium stationary state. For diffusion with constant-rate resetting to $p(x,t|x_0)$3, Evans and Majumdar obtained

$p(x,t|x_0)$4

that is, a Laplace distribution rather than the Gaussian associated with ordinary diffusion (Evans et al., 2011). The stationary state is non-Gaussian, has a cusp at the resetting point, and corresponds to a nonequilibrium steady state with probability being continually removed from all $p(x,t|x_0)$5 and reinjected at $p(x,t|x_0)$6 (Evans et al., 2011).

This Laplace structure survives in several generalizations. For resetting to a distribution $p(x,t|x_0)$7, the stationary state becomes

$p(x,t|x_0)$8

so the stationary density is the convolution of the resetting distribution with the exponential kernel set by $p(x,t|x_0)$9 (Evans et al., 2011). In one-dimensional run-and-tumble motion with resetting to a fixed site $$\frac{\partial p(x,t|x_0)}{\partial t} = D\frac{\partial^2 p(x,t|x_0)}{\partial x^2} - r\,p(x,t|x_0) + r\,\delta(x-x_0),$$0, the stationary distribution under symmetric initial velocity conditions is

$$\frac{\partial p(x,t|x_0)}{\partial t} = D\frac{\partial^2 p(x,t|x_0)}{\partial x^2} - r\,p(x,t|x_0) + r\,\delta(x-x_0),$$1

and is independent of the velocity-resetting protocol parameter $$\frac{\partial p(x,t|x_0)}{\partial t} = D\frac{\partial^2 p(x,t|x_0)}{\partial x^2} - r\,p(x,t|x_0) + r\,\delta(x-x_0),$$2 (Evans et al., 2018). This is a direct active-matter analogue of the diffusive Laplace steady state.

In two-dimensional active Brownian motion, the stationary density depends strongly on whether orientation is reset. For protocol II of Kumar, Sadekar, and Basu—position reset without orientation reset—the position distribution reaches a stationary state and remains isotropic; in the large-$$\frac{\partial p(x,t|x_0)}{\partial t} = D\frac{\partial^2 p(x,t|x_0)}{\partial x^2} - r\,p(x,t|x_0) + r\,\delta(x-x_0),$$3 regime,

$$\frac{\partial p(x,t|x_0)}{\partial t} = D\frac{\partial^2 p(x,t|x_0)}{\partial x^2} - r\,p(x,t|x_0) + r\,\delta(x-x_0),$$4

with a logarithmic divergence near the origin (Kumar et al., 2020). For anisotropic active Brownian particles in two dimensions, position-only resetting again produces a stationary state, but the long-time spatial statistics become isotropic, with

$$\frac{\partial p(x,t|x_0)}{\partial t} = D\frac{\partial^2 p(x,t|x_0)}{\partial x^2} - r\,p(x,t|x_0) + r\,\delta(x-x_0),$$5

so the stationary MSD is independent of the anisotropy $$\frac{\partial p(x,t|x_0)}{\partial t} = D\frac{\partial^2 p(x,t|x_0)}{\partial x^2} - r\,p(x,t|x_0) + r\,\delta(x-x_0),$$6 (Ghosh et al., 9 Jan 2025).

These examples show that position resetting acts as a localization mechanism, but the form of localization depends on whether additional dynamical variables continue to evolve between resets. A plausible implication is that stationary non-Gaussianity is generic, whereas the singular structure at the resetting point is protocol-specific.

3. First-passage theory and optimal resetting

The canonical first-passage result is the mean time to hit an absorbing target at the origin. For a diffusive particle starting at $$\frac{\partial p(x,t|x_0)}{\partial t} = D\frac{\partial^2 p(x,t|x_0)}{\partial x^2} - r\,p(x,t|x_0) + r\,\delta(x-x_0),$$7,

$$\frac{\partial p(x,t|x_0)}{\partial t} = D\frac{\partial^2 p(x,t|x_0)}{\partial x^2} - r\,p(x,t|x_0) + r\,\delta(x-x_0),$$8

which diverges at $$\frac{\partial p(x,t|x_0)}{\partial t} = D\frac{\partial^2 p(x,t|x_0)}{\partial x^2} - r\,p(x,t|x_0) + r\,\delta(x-x_0),$$9 but is finite for every $D$0 (Evans et al., 2011). Minimization with respect to $D$1 gives the optimality condition

$D$2

with

$D$3

Thus, resetting creates a finite and optimizable search time in a problem where pure diffusion has infinite mean search time (Evans et al., 2011).

The same problem acquires a different asymptotic structure for many searchers. If $D$4 independent searchers reset to their own initial positions and $D$5 is the density, then the target survival probability satisfies

$D$6

At late times, the average and typical survival probabilities behave differently: $D$7 with $D$8 (Evans et al., 2011). Position resetting therefore separates annealed and quenched asymptotics: the average decays as a power law, while the typical value decays exponentially.

A recurring misconception is that positional restart always improves search. The literature is more specific. In one-dimensional Brownian search with resetting to random positions, an optimal reset rate exists for distributed resetting to a finite interval if the target lies outside the interval, but no optimal reset rate exists when the target belongs to the resetting interval or when the interval is infinite; in those cases there is instead an optimal interval width or optimal characteristic scale (Mendez et al., 2024). In randomly accelerated motion, partial resetting of position alone does not render the mean first passage time finite, whereas complete resetting of both position and velocity does (Singh, 2020). Likewise, for a run-and-tumble particle in one dimension, the mean time to absorption is always less for velocity randomization than for position-only resetting (Evans et al., 2018). Position resetting is therefore beneficial only relative to the structure of the underlying state space and target geometry.

4. Distributed resetting positions and optimization landscapes

Resetting need not be to a single site. For a Brownian searcher reset at rate $D$9 to a random position $r$0 drawn from a density $r$1, the averaged mean first-passage time is

$r$2

(Mendez et al., 2024). For a symmetric uniform distribution on $r$3 with $r$4, distributed resetting is always more efficient than resetting to a point, and an optimal reset rate exists (Mendez et al., 2024). The same work shows that the averaged first-passage density at small $r$5 depends only on the mean first-passage time,

$r$6

which leads to an “Equivalent Resetting Point”

$r$7

At the optimal reset rate, when it exists, the coefficient of variation satisfies $r$8 (Mendez et al., 2024).

Evans and Majumdar also considered optimal resetting for a distributed target $r$9. Averaging over reset position and target position gives

$-r p$0

and calculus of variations yields the ideal unconstrained stationary density

$-r p$1

The resetting distribution must then satisfy

$-r p$2

subject to the physicality constraint $-r p$3 (Evans et al., 2011). For the exponentially decaying target distribution $-r p$4, a transition occurs at $-r p$5: below that threshold the optimal $-r p$6 is a mixture of an exponential and a delta at the origin, while above it the optimum becomes $-r p$7 (Evans et al., 2011).

A more recent development is the discovery of discontinuous transitions in optimal resetting rates for multi-site or compactly supported resetting distributions. For two resetting points $-r p$8, with reset rates $-r p$9 and $r\delta(x-x_0)$0,

$r\delta(x-x_0)$1

and the optimal resetting rate can jump discontinuously as $r\delta(x-x_0)$2 or $r\delta(x-x_0)$3 is varied (Julián-Salgado et al., 2023). The critical point exists only for

$r\delta(x-x_0)$4

while, for the averaged initial-position setting, a discontinuity exists for $r\delta(x-x_0)$5 (Julián-Salgado et al., 2023). Closely related behavior occurs for compactly supported distributed resetting that does not include the target: the mean first-passage time can develop two local minima in $r\delta(x-x_0)$6, the optimal rate can jump discontinuously, and the last resetting position before absorption is distributed as

$r\delta(x-x_0)$7

thereby distinguishing strategies dominated by likely but distant reset points from strategies dominated by less likely but closer ones (Julián-Salgado et al., 19 Jul 2025).

5. Active, persistent, and higher-order dynamics

Position resetting has been generalized extensively to active matter and persistent random motion. For active Brownian particles in two dimensions, three protocols have been analyzed: complete resetting of position and orientation, position-only resetting, and orientation-only resetting (Kumar et al., 2020). Position resets are sufficient for a stationary position distribution in the first two cases, whereas orientation-only resetting does not produce a stationary position state (Kumar et al., 2020). In the rapid-resetting regime, complete resetting yields a strongly anisotropic stationary state, while position-only resetting yields an isotropic $r\delta(x-x_0)$8-profile with logarithmic divergence at the origin (Kumar et al., 2020).

In a harmonic trap, stochastic position and orientation resetting permit exact moment calculations for a two-dimensional active Brownian particle. The steady-state mean parallel displacement is

$r\delta(x-x_0)$9

which peaks at $$p(x,t|x_0)=e^{-rt}G_0(x,t|x_0)+r\int_0^t d\tau\,e^{-r\tau}G_0(x,\tau|X_r),$$0, and the steady-state MSD is

$$p(x,t|x_0)=e^{-rt}G_0(x,t|x_0)+r\int_0^t d\tau\,e^{-r\tau}G_0(x,\tau|X_r),$$1

The excess kurtosis

$$p(x,t|x_0)=e^{-rt}G_0(x,t|x_0)+r\int_0^t d\tau\,e^{-r\tau}G_0(x,\tau|X_r),$$2

separates a Gaussian crossover regime from an activity-dominated regime with $$p(x,t|x_0)=e^{-rt}G_0(x,t|x_0)+r\int_0^t d\tau\,e^{-r\tau}G_0(x,\tau|X_r),$$3 and a resetting-dominated regime with $$p(x,t|x_0)=e^{-rt}G_0(x,t|x_0)+r\int_0^t d\tau\,e^{-r\tau}G_0(x,\tau|X_r),$$4 (Shee, 2024).

For chiral active Brownian particles with stochastic position-orientation resetting, the steady-state MSD is

$$p(x,t|x_0)=e^{-rt}G_0(x,t|x_0)+r\int_0^t d\tau\,e^{-r\tau}G_0(x,\tau|X_r),$$5

and the line $$p(x,t|x_0)=e^{-rt}G_0(x,t|x_0)+r\int_0^t d\tau\,e^{-r\tau}G_0(x,\tau|X_r),$$6 marks the transition from oscillatory to monotonic orientation autocorrelation (Shee, 17 Aug 2025). The same study identifies three regimes—activity-dominated chiral, resetting-dominated with chirality, and resetting-dominated without chirality—using the autocorrelation and excess kurtosis (Shee, 17 Aug 2025).

Run-and-tumble particles provide a complementary persistent model. In two dimensions, resetting both position and orientation randomization produces a stationary radial distribution that approaches a constant value as $$p(x,t|x_0)=e^{-rt}G_0(x,t|x_0)+r\int_0^t d\tau\,e^{-r\tau}G_0(x,\tau|X_r),$$7, while the stationary $$p(x,t|x_0)=e^{-rt}G_0(x,t|x_0)+r\int_0^t d\tau\,e^{-r\tau}G_0(x,\tau|X_r),$$8-marginal diverges logarithmically as $$p(x,t|x_0)=e^{-rt}G_0(x,t|x_0)+r\int_0^t d\tau\,e^{-r\tau}G_0(x,\tau|X_r),$$9; both marginals decay exponentially far from the origin (Santra et al., 2020). The relaxation to stationarity occurs through a front at

$G_0$0

and the mean first-passage time can be minimized by tuning the reset rate when resetting occurs away from the absorbing boundary (Santra et al., 2020).

For higher-order stochastic motion, the role of position resetting becomes more delicate. In the random acceleration process, complete resetting $G_0$1 produces a stationary joint state and finite mean first-passage time, but partial resetting of position alone leaves the system transient and does not render the mean first-passage time finite, because the velocity fluctuations continue to grow as $G_0$2 (Singh, 2020). This is a precise example of a broader principle already visible in active systems: position resetting can fail if the unreset degrees of freedom retain enough memory to dominate long excursions.

6. Non-instantaneous return, refractory periods, and generalized position resetting

A major generalization replaces instantaneous position resetting by a two-phase reset-return process. In the formulation of Bodrova and Sokolov, a displacement phase is terminated by a resetting event, after which the particle returns to the origin according to a specified equation of motion; one run has duration

$G_0$3

The stationary density is

$G_0$4

where $G_0$5 and $G_0$6 are the rescaled densities in the displacement and return phases (Bodrova et al., 2019). For Brownian motion with exponential resetting and return at constant speed or constant acceleration, the stationary PDF is invariant under the return speed or acceleration, but the mean hitting time still depends explicitly on the return dynamics (Bodrova et al., 2019).

A related finite-time-reset model uses a linear confining potential $G_0$7 during the return phase. Here the probability density is decomposed into diffusion and return sectors, the steady state is a mixture of exponentials, and relaxation occurs by “cone spreading” with travelling fronts separating an inner core that has reached steady state from an outer region that has not (Gupta et al., 2020). The corresponding large-deviation function is non-analytic at the front velocity, yielding a dynamical transition analogous to the instantaneous-resetting case (Gupta et al., 2020).

Refractory periods add another layer. If a process is reset to the origin and then remains quiescent for a random refractory time $G_0$8 drawn from $G_0$9, the stationary density is

$z$0

so the stationary law acquires a delta peak at the resetting position whose weight is controlled by $z$1 (Evans et al., 2018). For power-law refractory periods, the relaxation to stationarity becomes algebraically slow (Evans et al., 2018).

Generalized position resetting may also involve resetting other functionals together with position. In diffusion with position and occupation-time resetting, both the position $z$2 and the internal state $z$3 are reset to their initial values at rate $z$4, which is mathematically equivalent to resetting the occupation time $z$5. The survival probability obeys the renewal relation

$z$6

and the MFPT is

$z$7

(Bressloff, 2022). This framework makes explicit that positional restart may need to be accompanied by reset of accumulated internal variables if threshold absorption depends on the process history.

Two further variants show how far the notion of position resetting can be extended while retaining exact control. In bounded heterogeneous environments with space-dependent diffusivity $z$8, the MFPT with resetting satisfies a backward equation whose exact closed-form solution is available in the Stratonovich prescription after the transformation

$z$9

and the efficiency of resetting depends strongly on whether $p(x,t|x_0)$00 decreases or increases away from the target (Jr et al., 2024). In “resetting by rescaling,” the update rule $p(x,t|x_0)$01 replaces reset to a fixed site; for $p(x,t|x_0)$02 a stationary state exists with a Gaussian peak near $p(x,t|x_0)$03 and exponential tails, and although the MFPT has an optimal resetting rate for all $p(x,t|x_0)$04, only negative rescaling improves search relative to standard resetting to the origin (Biroli et al., 2024).

These extensions clarify that “position resetting” is best understood as a family of spatial restart mechanisms. The common ingredients are renewal structure, nonequilibrium localization, and strong sensitivity of first-passage behavior to what is reset, where reset positions are allowed, and whether reset is instantaneous or dynamically resolved.