--- title: Position Resetting in Stochastic Processes url: https://www.emergentmind.com/topics/position-resetting type: topic --- # Position Resetting in Stochastic Processes 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 [1102.2704]. 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 [1910.07993]. ## 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 [1102.2704]. 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 [1910.07993]. This renewal structure extends directly to more general protocols. Resetting to a random position \(z\) drawn from a distribution \({\cal P}(z)\) leads to \[ \frac{\partial p(x,t)}{\partial t} = D\frac{\partial^2 p(x,t)}{\partial x^2} -rp(x,t)+r{\cal P}(x), \] and position-dependent resetting is represented by a rate field \(r(x)\) rather than a constant [1107.4225]. 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 [2008.03294]. 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 [2501.05149]. 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 \(x_0\), Evans and Majumdar obtained \[ p_{\rm st}(x|x_0)=\frac{\alpha_0}{2}\exp\!\left[-\alpha_0|x-x_0|\right], \qquad \alpha_0=\sqrt{\frac{r}{D}}, \] that is, a Laplace distribution rather than the Gaussian associated with ordinary diffusion [1102.2704]. 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 \(x\) and reinjected at \(x_0\) [1102.2704]. This Laplace structure survives in several generalizations. For resetting to a distribution \({\cal P}(z)\), the stationary state becomes \[ p^*(x)=\frac{\alpha_0}{2}\int dz\,{\cal P}(z)\exp(-\alpha_0|x-z|), \] so the stationary density is the convolution of the resetting distribution with the exponential kernel set by \(\alpha_0^{-1}\) [1107.4225]. In one-dimensional run-and-tumble motion with resetting to a fixed site \(X_r\), the stationary distribution under symmetric initial velocity conditions is \[ P_r^{st}(x)=\frac{\lambda_r}{2}e^{-\lambda_r|x-X_r|}, \qquad \lambda_r=\sqrt{\frac{r(r+2\gamma)}{v_0^2}}, \] and is independent of the velocity-resetting protocol parameter \(\eta\) [1808.06450]. 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-\(r\) regime, \[ P_{\rm st}(x)=\frac{r}{\pi v_0}K_0\!\left(\frac{r|x|}{v_0}\right), \] with a logarithmic divergence near the origin [2008.03294]. 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 \[ (\sigma_x^2)_{pr}=(\sigma_y^2)_{pr} = \frac{v_0^2}{r(r+D_\theta)}+\frac{2\bar D}{r}, \] so the stationary MSD is independent of the anisotropy \(D_\parallel-D_\perp\) [2501.05149]. 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 \(x_0>0\), \[ T(x_0)=\frac{1}{r}\left(\exp(\alpha_0 x_0)-1\right), \] which diverges at \(r=0\) but is finite for every \(r>0\) [1102.2704]. Minimization with respect to \(r\) gives the optimality condition \[ \frac{z^*}{2}=1-e^{-z^*}, \qquad z^*=1.59362\ldots, \] with \[ r^*=\frac{(z^*)^2D}{x_0^2}, \qquad z=\alpha_0x_0=x_0\sqrt{\frac{r}{D}}. \] Thus, resetting creates a finite and optimizable search time in a problem where pure diffusion has infinite mean search time [1102.2704]. The same problem acquires a different asymptotic structure for many searchers. If \(N\) independent searchers reset to their own initial positions and \(\rho=N/L\) is the density, then the target survival probability satisfies \[ P_s(t)=\prod_{i=1}^N Q(x_i,t). \] At late times, the average and typical survival probabilities behave differently: \[ P_s^{\rm av}(t)\sim t^{-2\rho\sqrt{D/r}}, \qquad P_s^{\rm typ}(t)\sim \exp\!\left[-K\rho\sqrt{Dr}\,t\right], \] with \(K=8(1-\ln 2)\) [1102.2704]. 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 [2401.01125]. 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 [2007.05576]. 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 [1808.06450]. 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 \(r\) to a random position \(x_0\) drawn from a density \(f(x_0)\), the averaged mean first-passage time is \[ \langle T\rangle_{x_0}=\frac{1}{r}\left[\frac{1}{I(r,x_T)}-1\right], \qquad I(r,x_T)=\int_{-\infty}^{\infty}e^{-|x_0-x_T|\sqrt{r/D}}f(x_0)\,dx_0 \] [2401.01125]. For a symmetric uniform distribution on \([-L,L]\) with \(x_T>L\), distributed resetting is always more efficient than resetting to a point, and an optimal reset rate exists [2401.01125]. The same work shows that the averaged first-passage density at small \(s\) depends only on the mean first-passage time, \[ \langle p_f(s)\rangle_{x_0}\simeq \frac{1}{1+s\langle T\rangle_{x_0}}, \] which leads to an “Equivalent Resetting Point” \[ |x_e-x_T|=-\sqrt{\frac{D}{r}\ln I(r,x_T)}. \] At the optimal reset rate, when it exists, the coefficient of variation satisfies \(\mathrm{CV}=1\) [2401.01125]. Evans and Majumdar also considered optimal resetting for a distributed target \(P_T(x)\). Averaging over reset position and target position gives \[ \overline{T} = \frac{1}{r} \left[ \frac{\alpha_0}{2}\int dx_T\,\frac{P_T(x_T)}{p^*(x_T)}-1 \right], \] and calculus of variations yields the ideal unconstrained stationary density \[ p^*_{\mathrm{opt}}(x)= \frac{P_T^{1/2}(x)}{\int dz\,P_T^{1/2}(z)}. \] The resetting distribution must then satisfy \[ {\cal P}(x)=p^*(x)-\frac{1}{\alpha_0^2}\frac{\partial^2 p^*(x)}{\partial x^2}, \] subject to the physicality constraint \({\cal P}(x)\ge 0\) [1107.4225]. For the exponentially decaying target distribution \(P_T(x)=\frac{\beta}{2}e^{-\beta|x|}\), a transition occurs at \(\beta=2\alpha_0\): below that threshold the optimal \({\cal P}(z)\) is a mixture of an exponential and a delta at the origin, while above it the optimum becomes \(\delta(x)\) [1107.4225]. 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 \(x_1m_c=6.6008\ldots\) [2311.11897]. 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 \(r\), the optimal rate can jump discontinuously, and the last resetting position before absorption is distributed as \[ \rho_l(z)=\mathcal C\,\rho(z)e^{-\alpha_0|z|}, \] thereby distinguishing strategies dominated by likely but distant reset points from strategies dominated by less likely but closer ones [2507.14483]. ## 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 [2008.03294]. 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 [2008.03294]. In the rapid-resetting regime, complete resetting yields a strongly anisotropic stationary state, while position-only resetting yields an isotropic \(K_0\)-profile with logarithmic divergence at the origin [2008.03294]. 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 \[ \langle r_\parallel\rangle_r^{st} = \frac{r\,Pe}{(r+\beta)(1+r)}, \] which peaks at \(r_{\max}=\sqrt{\beta}\), and the steady-state MSD is \[ \langle r^2\rangle_r^{st} = \frac{4}{2\beta+r} + \frac{2Pe^2}{(1+\beta+r)(2\beta+r)}. \] The excess kurtosis \[ \mathcal K_r=\frac{\langle r^4\rangle_r}{2\langle r^2\rangle_r^2}-1 \] separates a Gaussian crossover regime from an activity-dominated regime with \(\mathcal K_r<0\) and a resetting-dominated regime with \(\mathcal K_r>0\) [2409.06920]. For chiral active Brownian particles with stochastic position-orientation resetting, the steady-state MSD is \[ \left\langle \mathbf r^2\right\rangle_r^{\rm st} = \frac{4D}{r} + \frac{2(r+D_r)v_0^2}{r\big[(r+D_r)^2+\Omega_0^2\big]}, \] and the line \(r+D_r=\Omega_0\) marks the transition from oscillatory to monotonic orientation autocorrelation [2508.12223]. The same study identifies three regimes—activity-dominated chiral, resetting-dominated with chirality, and resetting-dominated without chirality—using the autocorrelation and excess kurtosis [2508.12223]. 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 \(r\to 0\), while the stationary \(x\)-marginal diverges logarithmically as \(x\to 0\); both marginals decay exponentially far from the origin [2009.09891]. The relaxation to stationarity occurs through a front at \[ r_0(t)=x_0(t)= \frac{\sqrt{\alpha^2+2\alpha\gamma}}{\alpha+\gamma}\,v_0 t, \] and the mean first-passage time can be minimized by tuning the reset rate when resetting occurs away from the absorbing boundary [2009.09891]. For higher-order stochastic motion, the role of position resetting becomes more delicate. In the random acceleration process, complete resetting \((x,v)\to(x_0,v_0)\) 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 \(\sim \sqrt{t}\) [2007.05576]. 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 \[ t_{\rm run}=t_{\rm res}+t_{\rm ret}. \] The stationary density is \[ P(x)=\frac{\rho_1(x)+\rho_2(x)} {\langle t_{\rm res}\rangle+\langle t_{\rm ret}\rangle}, \] where \(\rho_1\) and \(\rho_2\) are the rescaled densities in the displacement and return phases [1907.12326]. 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 [1907.12326]. A related finite-time-reset model uses a linear confining potential \(V(x)=\lambda |x|\) 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 [2012.12878]. The corresponding large-deviation function is non-analytic at the front velocity, yielding a dynamical transition analogous to the instantaneous-resetting case [2012.12878]. Refractory periods add another layer. If a process is reset to the origin and then remains quiescent for a random refractory time \(\tau\) drawn from \(W(\tau)\), the stationary density is \[ P^{\rm st}(x) = \frac{r}{1+r\langle\tau\rangle} \left[ \tilde G_0(x,r)+\delta(x)\langle\tau\rangle \right], \] so the stationary law acquires a delta peak at the resetting position whose weight is controlled by \(r\langle\tau\rangle\) [1809.01551]. For power-law refractory periods, the relaxation to stationarity becomes algebraically slow [1809.01551]. Generalized position resetting may also involve resetting other functionals together with position. In diffusion with position and occupation-time resetting, both the position \(\mathbf X_t\) and the internal state \(\mathcal U_t\) are reset to their initial values at rate \(r\), which is mathematically equivalent to resetting the occupation time \(A_t\to 0\). The survival probability obeys the renewal relation \[ S_r(x_0,s)=\frac{S(x_0,r+s)}{1-rS(x_0,r+s)}, \] and the MFPT is \[ T_r(x_0)=\frac{S(x_0,r)}{1-rS(x_0,r)} \] [2205.13989]. 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 \(D(x)\), the MFPT with resetting satisfies a backward equation whose exact closed-form solution is available in the Stratonovich prescription after the transformation \[ y(x)=\int_0^x \frac{dx'}{\sqrt{D(x')}}, \] and the efficiency of resetting depends strongly on whether \(D(x)\) decreases or increases away from the target [2408.04726]. In “resetting by rescaling,” the update rule \(x\to a x\) replaces reset to a fixed site; for \(|a|<1\) a stationary state exists with a Gaussian peak near \(x=0\) and exponential tails, and although the MFPT has an optimal resetting rate for all \(-1