Transition path theory for diffusive search with stochastic resetting

Many chemical reactions can be formulated in terms of particle diffusion in a complex energy landscape. Transition path theory (TPT) is a theoretical framework for describing the direct (reaction) pathways from reactant to product states within this energy landscape, and calculating the effective reaction rate. It is now the standard method for analyzing rare events between long lived states. In this paper, we consider a completely different application of TPT, namely, a dual-aspect diffusive search process in which a particle alternates between collecting cargo from a source domain A and then delivering it to a target domain B. The rate of resource accumulation at the target, k AB , is determined by the statistics of direct (reactive or transport) paths from A to B. Rather than considering diffusion in a complex energy landscape, we focus on pure diffusion with stochastic resetting. Resetting introduces two non-trivial problems in the application of TPT. First, the process is not time-reversal invariant, which is reflected by the fact that there exists a unique non-equilibrium stationary state (NESS). Second, calculating k AB involves determining the total probability flux of direct transport paths across a dividing surface S between A and B. This requires taking into account discontinuous jumps across S due to resetting. We derive a general expression for k AB and show that it is independent of the choice of dividing surface. Finally, using the example of diffusion in a finite interval, we show that there exists an optimal resetting rate at which k AB is maximized. We explore how this feature depends on model parameters.


Introduction
A classical problem in statistical physics is the diffusive search for some target U in a bounded domain Ω ⊂ R d , see Fig. 1(a) [21,4,6].If the boundary ∂Ω is totally reflecting then the probability of eventually finding the target is unity.One typically formulates the search process as a first passage time (FPT) problem in which the target surface ∂U is taken to be totally absorbing.The FPT is defined according to T (x 0 ) = inf{t > 0, X(t) ∈ ∂U|X(0) = x 0 }, where X(t) is the position of the diffusing particle or searcher at time t and x 0 is its initial position.The mean FPT (MFPT) can be determined by solving a backward Kolmogorov equation or by calculating the probability flux into the target, whose Laplace transform is the generator of the FPT density.Higher-order moments can be calculated in a similar fashion.Various extensions include diffusive search within some energy landscape, diffusive search with stochastic resetting (as reviewed in Ref. [13]), and modifications in the absorption process itself.The last extension could involve taking the target surface ∂U to be partially absorbing [14].Alternatively the whole domain U could be partially absorbing, which means that the searcher freely enters and exits the target domain and can only be absorbed within the target interior [8,22].
There are two complementary interpretations of the search-and-capture process shown in Fig. 1(a): [I] the diffusing particle transports and delivers resources to the target (eg.intracellular vesicular transport) or [II] the particle searches for a target in order to extract resources from the target (eg.animal foraging).In this paper we consider a more complex diffusive search problem that is an alternating sequence of [I] and [II], see Fig. 1(b).The main idea is to assume that resources are initially located within a source domain A. This means that a particle first has to find the domain A in order to be supplied with a packet of resources (cargo).The resulting particle-cargo complex then searches for a target domain B in order to deliver its cargo and return to the bare particle state; if the complex returns to A before reaching B it does not load any additional cargo.In addition, we assume that the loading and unloading The time to find the target is identified with the FPT to be absorbed at the target surface ∂U .After finding the target, the particle either loads or unloads a packet of resources.(b) Dual-aspect diffusive search process that alternates between the search for a resource domain A where a particle collects cargo and the subsequent search for a target domain B where the cargo is delivered.The rate of resource accumulation at the target is determined by the statistics of direct (reactive or transport) paths from A to B. of cargo does not disrupt particle diffusion.(A more general model would allow for the particle to temporarily stop searching during the loading and unloading of cargo, which would introduce refractoriness into the model.)Finally, once the particle has delivered its cargo it continues searching for the source domain in order to load more cargo, and the process repeats (assuming that the amount of resources within A and the capacity of the target B are both unbounded) ‡.One quantity of interest for the dual-aspect search process outlined in Fig. 1(b) is the mean rate at which the target accumulates resources.This can be calculated using the mathematical framework of transition path theory (TPT) [15,29,18,30].The latter was originally developed within the context of analyzing chemical reaction rates.In the latter case, the path traced out by a diffusing particle in Fig. 1(b) corresponds to the trajectory of a chemical system in some energy landscape, with A representing reactant states and B representing product states.One way to approximate the reaction rate is in terms of the mean crossing frequency of transitions from A to B, which is proportional to the total flux of reactive trajectories across any dividing surface S separating the reactant states A from the product states B. However, this clearly overestimates the reaction rate, since reactive trajectories can recross the surface S many times during a single reaction, including multiple returns to the reactant states A prior to reaching B. TPT deals with this overcounting by characterizing the statistical properties of the ensemble of reactive trajectories under the constraint that each reactive trajectory is a direct path from A to B.
The mapping of the dual-aspect diffusive search problem to TPT is shown in Fig. ‡ This is of course an idealization in which the searcher is unable to memorize the locations of the source and target once they have been found for the first time.One possible scenario where a lack of memory might be advantageous is if there are multiple sources and multiple targets within the search domain, particularly given that a memory device costs resources.
2. Let X(t), −∞ < t < ∞, denote the position of the searcher at time t.First, we distinguish between sections of the trajectory along which the particle is carrying cargo (red curves) from sections along which the particle is free (black curves).Thus any trajectory exiting A is red, whereas any trajectory exiting B is black.We further subdivide each red section into the part that forms a direct connection from A to B (solid red curves), which we will call a transport trajectory, and the remainder that returns to A without reaching B (dashed red curves).Let N T denote the number of transport trajectories observed during the time interval [−T, T ], then the mean frequency at which these trajectories are observed within the given trajectory is Assuming that the stochastic process is ergodic, we can equate k AB with an ensemble average that is independent of the particular trajectory and can thus be calculated using TPT.Within the context of stochastic transport processes, k AB is equivalent to the mean frequency at which units of cargo are delivered to B.
The mathematical analysis of TPT is typically developed within the context of stochastic differential equations (SDEs) as exemplified by a multi-dimensional Langevin equation [29,18,30].One of the novel aspects of stochastic transport processes is that the searcher dynamics may be described by a more general stochastic process [7].Examples include active motor transport [19], active Brownian and runand-tumble particles [23], facilitated diffusion [5,16], and Lèvy flights [26].In this paper we construct TPT for yet another example, namely, diffusion with stochastic resetting.That is, the position of the particle resets to a fixed location x r ∈ Ω\(A∪B) at a constant rate r [10,11,12].It is well known that such a process is not time-reversal invariant, which is reflected by the fact that there exists a unique non-equilibrium stationary state (NESS).Such a state exists even if the dynamics is unbounded, that is, Ω = R d .The application of TPT is thus non-trivial.
The structure of the paper is as follows.In section 2, we review TPT for an overdamped Brownian particle in R d [29,18,30].The main analysis is developed in section 3, where we extend TPT to include the effects of stochastic resetting.First, we construct the reverse-time diffusion process with resetting by extending the formalism of Ref. [1].(This is necessary in order to determine the so-called backward committor function.)We then calculate the accumulation rate k AB by generalizing the derivations presented in the appendices of Ref. [18].In particular, we equate k AB with the time-averaged probability flux of transport trajectories across a dividing surface between the source and target domains.There are two distinct contributions to k AB corresponding, respectively, to paths that cross S smoothly and those that jump across S due to stochastic resetting.We also prove that k AB is independent of the choice of dividing surface S, and that it is a positive quantity.Finally, in section 4 we illustrate the theory by considering pure diffusion with resetting in a finite interval.We show that there exists an optimal resetting rate at which k AB is maximized, and explore how this feature depends on model parameters.

TPT for overdamped Brownian motion
Let X(t) ∈ R d , 0 ≤ t ≤ T , denote the position of an overdamped Brownian particle evolving according to the SDE where D is the diffusivity and W j (t), j = 1, . . ., d, are independent Wiener processes.Let with the initial condition p 0 (x, 0) = p 0 (x).(The subscript on p 0 indicates that there is no stochastic resetting.)The probability density p 0 (x, t) evolves according to the forward Fokker-Planck (FP) equation We will assume that for the given drift functions f j (x), there exists a stationary measure ρ 0 (x)dx such that lim t→∞ p 0 (x, t) = ρ 0 (x). (2.3) In particular, suppose that we have a conservative force for which , where U (x) is a potential energy function and γ is a friction coefficient satisfying the Einstein relation Dγ = k B T .The stationary density is then given by the Boltzmann distribution ρ 0 (x) = Z −1 e −U (x)/k B T with Z = ´Rd e −U (y)/k B T dy < ∞.
In addition, the diffusion process is time reversible and all equilibrium probability currents vanish.

Ensemble of transport trajectories
Let X(t), −∞ < t < ∞, be an infinitely long sample trajectory of the SDE (2.1), which is ergodic with respect to the equilibrium probability density ρ 0 (x).That is, given any suitable observable Φ(x), we have the equivalence of time and ensemble averages:

.4)
Suppose that A ⊂ R d and B ⊂ R d are two bounded regions in the phase space R d that specify the source and target domains, respectively, see Fig. 2. Any trajectory X(t), −∞ < t < ∞, can be partitioned into pieces that are either transport trajectories or their complement.Each transport trajectory connects ∂A to ∂B.That is, it starts at a point on the boundary ∂A and ends at a point on the boundary ∂B without ever returning to A. It follows from ergodicity that the set of all transport trajectories forms an ensemble Γ whose statistical properties are independent of the particular trajectory X(t) used to generate the ensemble.Suppose that X(t) / ∈ A ∪ B at time t and set (2.7) One of the major objects of interest in transition path theory is the probability density that a trajectory passing through x / ∈ A ∪ B at time t is a transport trajectory at time t.For a given trajectory X(t), let R denote the set of times for which X(t) is on a transport trajectory.The probability density of transport trajectories ρ where It can be proven that [29,18,30] where q 0 (x) is the probability that the transport trajectory reaches first B before A, and q 0 (x) is the probability that the transport trajectory came from A rather than B.

Committor functions
The probabilities q 0 (x) and q 0 (x) are known as the forward and backward committor functions.The former is equivalent to the splitting probability that a trajectory starting at x reaches B before A and thus satisfies the backward Kolmogorov equation Determining the corresponding Kolmogorov equation for the backward committor function is more involved.Intuitively speaking, we can identify q 0 (x) as the splitting probability of the corresponding reverse-time diffusion process that starts at x and reaches A before B. Hence, the nontrivial step is determining the evolution equation for the reverse-time diffusion process.Let X(t) = X(T −t).In the absence of resetting and under mild conditions on the drift vector f (x) and initial density p 0 (x), it can be proven that the reverse-time process X(t) satisfies an SDE of the form [1] (see also section 3.2) where (2.12) In particular, taking p 0 (x) = ρ(x), we have with

.14)
The corresponding forward FP equation for the reverse-time process is where It now follows that q(x) satisfies the backward Kolmogorov equation In the particular case of an overdamped Brownian particle subject to a conservative force, we have a reversible diffusion process for which q 0 (x) = 1 − q 0 (x).This follows from substituting for ρ 0 (x) in (2.14) using the Boltzmann distribution: (2.17) However, if the force is non-conservative force or stochastic resetting is included (see section 3.1), then time reversibility no longer holds.Finally, note that although the boundary value problems for the committor functions are defined in R d \A ∪ B, we extend their domains of definition by taking q 0 (x) = 0, q0 (x) = 1 for all x ∈ A, q 0 (x) = 1, q0 (x) = 0 for all x ∈ B. (2.18)

Probability current and transition rate
Another quantity of interest is the probability current J (2.19) The vector n(x) denotes the unit normal to S pointing towards B and dσ(x) is the surface element on S. It can be proven that [18] (see section 3.2) where J (0) (x) is the equilibrium probability current with components Moreover, the reaction rate can be rewritten as the volume integral In the particular case of a conservative force, we have J (0) j (x) = 0 for all j = 1, . . ., d and q 0 (x) = 1 − q 0 (x), which means that (2.23)

Overdamped Brownian motion with stochastic resetting
Now suppose that the Brownian particle resets to a fixed position x r at a rate r, see Fig. 3.When resetting is included, the SDE is modified according to The probability density p(x, t) evolves according to the modified forward FP equation [10,11,12] ∂p(x, t) ∂t with p(x, 0) = p(x).Similarly, we define the propagator p(x, t|y, t 0 ) as the solution to equation (3.2) under the initial condition p(x, t 0 |y, t 0 ) = δ(x − y).The propagator also satisfies the backward FP equation (Under time translation invariance, we have p(x, t|y, t 0 ) = p(x, t − t 0 |y, 0).) In the absence of stochastic resetting, ergodicity with respect to a stationary density ρ 0 (x) depends on the properties of the force vector f (x).If the external force field is zero (flat energy landscape), then Brownian motion in an unbounded domain is non-ergodic, reflecting the fact that the stationary density is zero pointwise.One of the important consequences of stochastic resetting is the existence of a nontrivial stationary density ρ(x) = lim t→∞ p(x, t|x 0 , 0) for unbounded Brownian motion [10,11,12].In particular, ρ(x) represents an NESS because there exist nonzero probability fluxes.That is the point x r acts as a probability source, whereas all positions x ̸ = x r are potential probability sinks.An immediate issue is whether or not the resulting stochastic process is ergodic with respect to the NESS.The ergodicity of diffusion processes with stochastic resetting has recently been explored in a number of studies [24,27,25,28,2].Although it has not been proven rigorously, normal diffusion processes with Poissonian resetting appear to be ergodic, and we will assume this in the following.
Taking the large-t limit of equation (3.2) shows that ρ(x) (if it exists for a given force field f (x)) satisfies the stationary equation It immediately follows that the current J(x) is not divergence-free.An alternative way to determine the NESS is to note that the propagator satisfies the last renewal equation [13] p(x, t|x 0 , 0) = e −rt p 0 (x, t|x 0 , 0) + r ˆt 0 p 0 (x, τ |x r , 0)e −rτ dτ, where p 0 is the corresponding propagator without resetting.The stationary state ρ(x) is obtained by taking the limit t → ∞ in equation (3.5): That is, ρ(x) is determined by the r-Laplace transform of the propagator p 0 (assuming it exists).In addition, the backward FP equation (3.3) implies that the forward committor function q(x) satisfies the boundary value problem L r q(x) = 0, q(x) = 0 for x ∈ ∂A, q(x) = 1 for x ∈ ∂B, with In order to determine the corresponding Kolmogorov equation for the backward committor function, we need to derive the FP equation for the reverse-time diffusion process with resetting.

Reverse-time diffusion process with resetting and the backward committor function
We obtain the reverse-time Markov process by extending the approach presented in Ref. [1].Consider the joint probability density ρ(x, t, y, t 0 ) = p(x, t|y, t 0 )p(y, t 0 ), t 0 < t, where p(y, t 0 ) evolves according to equation (3.2) with x → y, t → t 0 and p(y, 0) = p 0 (y).Differentiating both sides with respect to t 0 we have The last line represents jumps x r → y with transition rate We can reinterpret equation (3.10) as the forward FP equation for a time reversed process by setting p(x, t|X, 0) = ρ(X, T, x, T − t)/p(X, T ) so that with f j (x, t) given by equation (2.12).This describes a Markov diffusion process of the following form: if X(t) ̸ = x r then with probability p(x, T − t)rdt p(x r , T − t) . ( Finally, taking p(x, 0) = ρ(x) we have with f j (x) given by equation (2.14), whereas if X(t) = x r then x j − x r,j with probability ρ(x)rdt ρ(x r ) . (3.14b) The forward FP equation becomes Similarly, the backward FP equation for the reverse-time process with stochastic resetting is Finally, the backward committor functions is obtained from the equation L r q(x) = 0, q(x) = 1 for x ∈ ∂A, q(x) = 0 for x ∈ ∂B, (3.17) where ˆRd ρ(x)q(x)dx − q(x r ) . (3.18)

Calculation of the transition rate k AB
In order to calculate the target accumulation rate k AB , we need to determine the generalizations of equations (2.19) and (2.22) in the presence of resetting.We proceed by extending the derivations presented in the appendices of Ref. [18].A crucial assumption in these derivations is that the underlying stochastic process is ergodic with respect to the stationary density along the lines of equation (2.4).It is clear from Fig. 3 that there are two distinct ways in which the particle can cross the surface S: either continuously, as at the points a, b, c, a ′ , b ′ , or as a jump via stochastic resetting.These two cases will emerge from the analysis.The starting point is an equation equating k AB with the time-avergae of the total probability flux of transport trajectories across S [18]: Assuming the stochastic process is ergodic, we first take the limit T → ∞ to obtain The committor functions q(y) and q(x(s)) ensure that we only sum over transport trajectories.In addition E y denotes expectation conditional on x(0) = y.In the case of a smooth function ϕ(x), we have lim In order to evaluate the expectations in equation (3.20) using equation (3.22), we need to regularize the discontinuous indicator functions χ Ω S and χ Ω c S .Following Ref. [18], we introduce a differentiable interfacial function h δ with h δ (y) = 1 for y ∈ Ω s and dist(y, S) > δ 0 for y ∈ Ω c s and dist(y, S) > δ , (3.23) where dist(y, S) = min z∈S |y − z|, and which interpolates smoothly between 0 and 1 within the boundary layer of width 2δ.It follows that equation (3.20) is the limit as δ → 0 of Using the result we can now apply equation (3.22) to yield since L r q(y) = 0.If we now substitute the explicit form for L r , see equation (3.3), we find that Finally, integrating by parts the term involving the second order derivative of h δ gives ∂h δ (y) ∂y j q(y)q(y)J j (y) + Dρ(y)q(y) ∂q(y) ∂y j − Dρ(y)q(y) ∂q(y) ∂y j dy where J j is the current defined in equation (3.4).
Recall that for any sutiably defined vector field v(y) = (v 1 (y), . . .v d (y)) ⊤ , we have We have used integration by parts, the definition of h δ (y), and the divergence theorem.Hence, taking the limit δ → 0 in equation (3.28) yields where AB,j without resetting, in which the triplet (ρ 0 , q 0 , q 0 ) is replaced by (ρ, q, q).Moreover, the NESS ρ(x) is given by equation (3.6), while the committor functions q(x) and q(x) satisfy equations (3.7) and (3.17), respectively.It follows that J AB (x) is an implicit function of the resetting rate r.The first term on the right-hand side of equation (3.29) represents the contribution from smooth paths crossing S, see Fig. 3.The two additional terms on the right-hand side of equation (3.29) take into account paths that jump across S due to resetting.In particular, the contribution to k AB due to resetting can be decomposed according to rq(x r ) ˆΩS ρ(y)q(y)dy for x r ∈ Ω c S and − rq(x r ) ˆΩc S ρ(y)q(y)dy for x r ∈ Ω S .
We now need to check that k AB is independent of the dividing surface S. In the absence of stochastic resetting, the current J (0) AB is divergence-free and the result follows immediately from equation (2.19) and (2.20).Let us determine the divergence of the corresponding current J AB in the presence of resetting: Using equations (3.2), (3.7) and (3.17), this reduces to Consider two concentric domains Ω S and Ω S ′ , Ω ′ S ⊂ Ω S with corresponding dividing surfaces S and S ′ .Denote the corresponding accumulation rates by k AB (S) and k AB (S ′ ).In order to compare the two rates, it is necessary to specify the location of x r with respect to the two domains, see Fig. 5. First, suppose that x r ∈ Ω c S (case (i)).Equation (3.29) The last line follows from equation (3.31) and the assumption that x r / ∈ Ω S − Ω S ′ , which implies that only the first term on the right-hand side of (3.31) contributes.On the other hand, if x r ∈ Ω S − Ω S ′ (case (ii)), then The additional integral over R d is cancelled by the term involving the Dirac delta function in equation (3.31).Finally, it can be checked that k AB (S) = k AB (S ′ ) when x ∈ Ω S ′ (case (iii)).We conclude that although the probability current J AB is not divergence-free in the presence of stochastic resetting, the resulting rate k AB is independent of the dividing surface S.
Having obtained the resetting-dependent version of equation (2.19), we now turn to the analog of equation (2.22).Following Ref. [18], we introduce the forward isocommittor surface M(ξ) = {x ∈ R d : q(x) = ξ} with ξ ∈ [0, 1] and define the integral Note that M(0) = ∂A, where ∂A is the surface of the source domain.Since q(x) = 0 and q(x) = 1 on ∂A, it follows from equation (3.30) that and hence We have also used equation (3.29) with Ω S = A. In addition, rewriting A(ξ) as we find We have performed an integration by parts and used equation (3.7).Using equation (3.4), we can combine the two terms in square brackets to give where Ω(ξ) is the domain containing A whose surface is M(ξ).Hence, In order to integrate equation (3.38) with respect to ξ, we make use of the following identities.First, for any integrable function ϕ(x), we have ˆΩ(ξ) ϕ(x)dx = ˆξ 0 dξ ′′ ˆRd ϕ(x)δ(q(x) − ξ ′′ )dx + ˆA ϕ(x)dx.(3.39) Second, integrating both sides with respect to ξ implies that The last line follows from another application of equation (3.39).Now integrating both sides of equation (3.38) with respect to ξ using equations (3.39) and (3.40) gives We have also used equation (3.36) and q(x) = 0 for all x ∈ A. Finally, integrating both sides of equation (3.41) with respect to ξ and using equation (3.37) leads to the result We evaluate the double integral on the second line using the identity (3.39): where Combining our various results, we have Rearranging various terms, we have This is supplemented by the reflecting boundary conditions d p 0 (x, s|x 0 , 0) We can identify p 0 (x, s|x 0 ) as a Green's function of the modified Helmholtz equation on [−a, L + b].Imposing continuity of p(x, s|x 0 ) across x 0 and matching the discontinuity in the first derivative yields the solution and The 1D version of equation (3.46) for k AB is The term in curly brackets can be written as Using the fact that 0 < x r < L, we have Using the small-r behavior of ρ(x) and q(x), we find from equation (4.15) that lim (4.18) Moreover, in the limit r → ∞, we have k AB → 0 since the particle resets so frequently that it never has a chance to collect resources and deliver them to the target.As with previous studies of search processes with resetting in the interval [9,20], we wish to determine whether or not there exists an optimal resetting rate at which k AB is maximized.This is explored in Fig. 8, where we plot k AB as a function of r for a = b = 0.5, L = 1 and various values of x r .(Our results are obtained by numerically integrating the various terms in equation (3.46.)As expected, k AB → 0.5 as r → 0. We find that for 0 < x r < 0.5, there exists an optimal resetting rate, which is an increasing function of x r .(Since the source and target domains have the same length, it follows that k AB is invariant under the mapping x r → 1 − x r .The latter no longer holds when a ̸ = b.This is illustrated in Fig. 9 for the case a > b, which shows that if x r < 0.5 then the accumulation rate for fixed r increases under the mapping x r → 1 − x r .Such a result makes sense, since taking the reset position to be closer to the target means that the particle is less likely to waste time diffusing within the source domain.In Fig. 10, we plot k AB against r for various lengths L and fixed x r , a, b.As expected, the accumulation rate is a a decreasing function of L. We also find that the optimal reset rate decreases with increasing L. Finally, note that in the limit a, b → ∞ and r = 0, the analysis breaks down since diffusion in an unbounded domain is non-ergodic.x r = 0.5 x r = 0.4 x r = 0.3 x r = 0.2 x r = 0.1

Discussion
In this paper we developed a novel application of TPT to a dual-aspect search process, in which a diffusing particle first has to find a source domain A in order to collect cargo, and then has to find a distinct target domain B where the cargo is delivered, see Fig. 1(b).For simplicity, we assumed that (i) cargo loading and unloading do not interrupt the ongoing search process, and (ii) the particle does not maintain a memory of the locations of A and B. The mapping of the dual-aspect search process to TPT is based on the identification of the rate k AB at which the target accumulates resources with the time-averaged probability flux across a dividing surface between A and B. The calculation of k AB then assumes that the underlying search process is ergodic.In this paper, we focused on the particular example of diffusion with stochastic resetting in the absence of an external potential.The calculation of k AB required taking into account of the fact that the stochastic process is not time-reversal invariant, and that transition paths can jump discontinuously across S via resetting.In the case of diffusion in the interval, we established that the accumulation rate k AB is a nontrivial function of the reset rate r.In particular, there exists an optimal reset rate at which k AB is maximized.There are a variety of possible extensions of the current work.First, one could consider higher-dimensional examples of the basic model developed in this paper.Second, one could explore what happens if assumption (i) or (ii) is relaxed.Third, there are a wide range of other stochastic processes that could be incorporated into the dual-aspect search process, most notably, active Brownian motion, run-and-tumble x r = 0.3 x r = 0.25 x r = 0.1 reset rate r  dynamics, and Lèvy flights.A third non-trivial extension is to consider multiple

Figure 1 .
Figure1.(a) Classical search problem in which a diffusing particle searches for a target U in a bounded domain Ω ⊂ R d with a totally reflecting exterior boundary ∂Ω.The time to find the target is identified with the FPT to be absorbed at the target surface ∂U .After finding the target, the particle either loads or unloads a packet of resources.(b) Dual-aspect diffusive search process that alternates between the search for a resource domain A where a particle collects cargo and the subsequent search for a target domain B where the cargo is delivered.The rate of resource accumulation at the target is determined by the statistics of direct (reactive or transport) paths from A to B.

Figure 2 .
Figure 2. Mapping of the sequential transport process to TPT.Direct paths or transport trajectories are shown as solid red curves.The effective rate k AB at which cargo is delivered from the source A to the target B is determined by the number of transition trajectories N T observed over a time interval [−T, T ] according to k AB = lim T →∞ N T /2T .(See text for further details.) AB (x) of transport trajectories crossing a dividing surface S ⊂ Ω AB , with A on one side and B on the other side of the surface.(The superscript (0) again indicates that there is no resetting.)Integrating J (0) AB (x) over S yields the total probability flux of transport trajectories across this surface, which determines the target accumulation rate k (0) AB according to k (0) AB = ˆS J (0) AB (x) • n(x)dσ(x).

Figure 3 .
Figure 3. Transport process with stochastic resetting.The dividing surface S ∈ Ω AB is the boundary of a region Ω S with A ∩ Ω S = A and B ∩ Ω S = ∅.The sample trajectory smoothly crosses S from left to right at the points a, b, c and from right to left at the points a ′ , b ′ .The trajectory can also jump across S by resetting from a point x ∈ Ωs if xr ∈ Ω c S ≡ R d \Ω S (or from a point x ∈ Ω c s if xr ∈ Ω S -not shown).

Figure 4 .
Figure 4. Two concentric domains Ω S and Ω S ′ , Ω S ′ ⊂ Ω S , with corresponding dividing surfaces S and S ′ .Also shown are three different positions of the reset point xr relative to these surfaces.

4 .
.46) It can be checked that all terms on the right-hand side of equation (3.46) are positive.Equation (3.46) is the resetting analog of equation (2.22).Diffusion with resetting in the interval As an illustrative example of the analysis developed in section 3, consider pure diffusion in the finite interval Ω = [−a, L+b], a, b > 0, with reflecting boundaries at x = −a and x = L + b.Take the source and target domains to be A = [−a, 0] and B = [L, L + b],respectively, see Fig.5(a).We assume that the particle resets to a point x r ∈ (0, L) at a rate r, see Fig.5(b).As in Fig.1(b), the particle carries cargo along sections of the trajectory that link A to B.In order to determine the NESS ρ(x), we consider the Laplace transformed diffusion equation in [−a, L + b] without resetting:

Figure 8 .
Figure 8. Plot of target accumulation rate k AB as a function of the reset rate r for various values of the reset point xr.Other parameters are L = 1, a = b = 0.5 and D = 1.There exists an optimal resetting rate for all xr ̸ = 0.5.(The curves are invariant under the mapping xr → 1 − xr.The maximum when xr = 0.4, 0.6 occurs at r ≈ 140 -not shown.)

Figure 9 .
Figure 9. Plot of target accumulation rate k AB as a function of the reset rate r for various values of the reset point xr.Other parameters are L = 1, a = 0.5, b = 0, and D = 1.Dashed curves correspond to the case 1 − xr.

Figure 10 .
Figure 10.Plot of target accumulation rate k AB as a function of the reset rate r for various values of the domain size L for xr = 0.1.Other parameters are L = 1, a = b = 0.5 and D = 1.