Now, it is instructive to ask: given that the position of the particle at time t has many different values, what is the full distribution of the position at time t? In other words, what is the form of P(x, t), defined such that P(x, t)dx is the probability to find the particle between positions x and x + dx at time t, given that the particle started from position x ₀ at time t = 0: P(x, 0) = δ(x − x ₀). The probability density is of course normalized as ∫dx P(x, t) = 1 ∀ t. The question just posed may be answered by writing down and solving the so-called Fokker-Planck equation for the time evolution of P(x, t). The equation reads ∂ P x , t ∂ t = D ∂ 2 P x , t ∂ x 2 , (5) and needs to be solved subject to the initial condition P(x, 0) = δ(x − x ₀).

The Fokker-Planck equation may be derived by noting down the ways in which the probability density P(x, t) changes in a small time 0 < Δt ≪ 1, and finally taking the limit Δt → 0. One has P x , t + Δ t = P x , t + ∫ d Δ x P x − Δ x , t ϕ Δ t Δ x − ∫ d Δ x P x , t ϕ Δ t Δ x , (6) where ϕ _Δt (Δx) is the probability density for x to change by an amount Δx in time Δt, and is taken to be independent of x. The probability density satisfies the normalization ∫d(Δx) ϕ _Δt (Δx) = 1. Note that Eq. 1 implies that this jump probability is independent of the value of x from which the jump is taking place. In Eq. 6, the second term on the right hand side (rhs) denotes gain in probability due to a change taking place from x − Δx to x, while the third term on the rhs denotes loss in probability due to a change taking place from x. Next, noting that the dynamics (1) implies that for small Δt, the change Δx that x undergoes is also small, and assuming that P(x, t) is a slowly varying function of x, we may Taylor expand the left hand side (lhs) of Eq. 6 in powers of Δt and the second term on the right hand side in powers of Δx. Executing such an expansion, and noting that Eq. 1 implies that ⟨Δx⟩≡∫d(Δx) Δx ϕ _Δt (Δx) = 0 and ⟨(Δx)²⟩≡∫d(Δx) (Δx)² ϕ _Δt (Δx) = 2DΔt, one obtains Eq. 5.

The solution to Eq. 5 is readily obtained by expressing P(x, t) in terms of its Fourier component P ̃ ( k , t ) = ∫ − ∞ ∞ d x P ( x , t ) exp ( − i k x ) that satisfies a linear differential equation given by ∂ P ̃ ( k , t ) / ∂ t = − D k 2 P ̃ ( k , t ) subject to the initial condition P ̃ ( k , 0 ) = exp ( − i k x 0 ) ; this equation has the solution P ̃ ( k , t ) = exp ( − D k 2 t − i k x 0 ) . On performing inverse Fourier transformation according to the prescription P ( x , t ) = 1 / ( 2 π ) ∫ − ∞ ∞ d k P ̃ ( k , t ) exp ( i k x ) , one finally obtains P x , t = 1 4 π D t e − x − x 0 2 / 4 D t , (7) which is a Gaussian distribution centred at x ₀. Eq. 7 implies that the probability of finding the particle at a given distance from its initial location grows with time. Consequently, as time goes by, one has an increased probability of finding the particle far off from its initial location, although at any given time, the most probable location of the particle is at its initial location. These features are evident from the plot of P(x, t) at different times depicted in Figure 1.

The mean-squared displacement (MSD) of the particle about its initial location is obtained straightforwardly from Eq. 7 as ⟨ x − x 0 2 ⟩ t = ∫ − ∞ ∞ d x x − x 0 2 P x , t = 2 D t , (8) matching with the result (4), and which implies that the MSD grows forever as a function of time. Such a behavior in which the MSD grows linearly in time is referred to as normal diffusive behavior.

The aspect of the probability of finding the particle far off from x ₀ increasing with time may be traced back to the fact that since the particle is diffusing in free space, it is no wonder that at longer times, it would have spread to a larger region of the available space than at shorter times. Let us then ask this question: Is there a way to stop this spread? Well, one possible way could be to have the particle diffuse not in free space but in a bounded domain of finite extent, say, between points x = − L and x = + L on the x-axis. In this case, the probability of finding the particle at a given location would keep increasing with time, until it can increase no further, that is, until the system attains a stationary state. In this state, attained in the limit t → ∞, it may be argued that this probability would be the same at all locations inside the bounded domain. In other words, the particle would be equally likely to be found anywhere within the region x ∈ [ − L, L].

Suppose we ask: what is the probability for the next reset to happen after a certain time t, in the interval [t, t + dt]? Let us discretize time in equal steps of length 0 < Δt ≪ 1; we will eventually consider the limit Δt → 0. The number of such steps during time duration t is obviously given by N = t / Δ t . During each step of length Δt, the probability of reset equals rΔt, while the same for no reset equals 1 − rΔt. Consequently, the probability that the next reset happens after time t, in the interval [t, t + Δt], is given by ( 1 − r Δ t ) ︸ N r Δ t = ( 1 − r Δ t ) t / Δ t r Δ t . In the limit Δt → 0, the latter quantity equals exp( − rt) rdt. We thus conclude that the probability for no reset to occur during a given time duration t equals exp( − rt), while the probability for the next reset to occur after time t, that is, in the interval [t, t + dt], equals exp( − rt) rdt. For latter purpose, let us note that at a given time instant t, the probability that the last reset happened in the interval [t − τ − dτ, t − τ], with τ ∈ [0, t], equals exp( − rτ) rdτ.

Note that as mentioned earlier, the dynamics (1) represents infinitesimal change in x in infinitesimal time interval dt, while incorporating resetting into the dynamics has in the resulting dynamics reset events leading to any amount of change in x in infinitesimal interval dt. Indeed, the reset destination is always the same, namely, x ₀, irrespective of the location the particle resets from, so that the particle may execute a very long jump during a reset event.

It should be apparent from the aforementioned dynamical rules that introducing resetting into the dynamics would not let the particle go at any time, long or short, very far from the initial location. Is it actually the case? To answer this, we need to study just as in the case of free diffusion discussed above the time evolution of the probability density P _r(x, t) in presence of resetting, which may be derived by writing down the ways in which P _r(x, t) changes in a small time 0 < Δt ≪ 1 and then taking the limit Δt → 0. One has P r x , t + Δ t = P r x , t + 1 − r Δ t [ ∫ d Δ x P r x − Δ x , t ϕ Δ t Δ x − ∫ d Δ x P r x , t ϕ Δ t Δ x ] − r Δ t P r x , t + r Δ t δ x − x 0 . ( 10 )

Here, the second and the third term arise from diffusion, while the fourth and the fifth term are owing to resetting events. While the former two terms are as in the free diffusion case, Eq. 6, the latter two terms may be understood as follows. For r ≠ 0, there is a loss in probability at all locations other than x ₀ due to resetting events, which is represented by the fourth term on the rhs of Eq. 10, while the corresponding gain in probability at x = x ₀, given by rΔtδ(x − x ₀)∫dx P _r(x, t) = rΔtδ(x − x ₀), yields the last term on the rhs. Here we have used the normalization condition ∫dx P _r(x, t) = 1 ∀ t. Implementing appropriate Taylor expansions on both sides of Eq. 10, then taking the limit Δt → 0, the probability P _r(x, t) may be shown to be satisfying the Fokker-Planck equation [2] ∂ P r x , t ∂ t = D ∂ 2 P r x , t ∂ x 2 − r P r x , t + r δ x − x 0 . (11)

The initial condition is P _r(x, 0) = δ(x − x ₀). Eq. 11 may be solved for P _r(x, t); here, we report only the solution, referring the reader to Ref. [3] for details of obtaining it: P r x , t = e − r t − x − x 0 2 4 D t 4 π D t + 1 4 r D e − r D | x − x 0 | E r f c | x − x 0 | 2 D t − r t − 1 4 r D e r D | x − x 0 | E r f c | x − x 0 | 2 D t + r t , (12) where E r f c ( x ) ≡ 2 / π ∫ x ∞ d y e − y 2 is the complementary error function. The MSD of the particle may be obtained from Eq. 12 as ⟨ x − x 0 2 ⟩ t = 2 D r 1 − e − r t . (13)

Eq. 12 implies that in the limit t → ∞, one has a stationary-state form [2]: P r, ss x = 1 2 r D e − | x − x 0 | r / D . (14)

Correspondingly, the MSD relaxes at long times to the stationary-state value ⟨ x − x 0 2 ⟩ r, ss = 2 D r . (15)

Figure 3 shows the probability distribution P _r ( x, t) in presence of resetting and at four different times, as given by Eq. 12, showing in particular convergence in time to the stationary state (14).

The stationary-state distribution is an exponential centred at the resetting location; since the particle keeps resetting to x ₀ in time, it is no wonder that the most likely position of the particle is x ₀. More importantly, an exponential profile implies that there is a characteristic length scale within which the particle is to be found with significant probability and beyond which one has an exponentially-small probability of finding the particle. In conclusion, we have been able to achieve our goal: by modifying not the boundary conditions but rather the dynamics of free diffusion, one induces a stationary state in the system. While we have discussed in the foregoing the case of resetting at exponentially-distributed random times, the case of resetting at power-law-distributed random times has also been considered in the literature, see Ref. [4]. We remark in passing that resetting creates a source of probability at x ₀ and a sink at all other locations x ≠ x ₀, so that the condition of detailed balance, which characterizes an equilibrium stationary state [5], is manifestly violated in the stationary state induced by resetting. Consequently, the stationary state (14) is a generic nonequilibrium stationary state [2].

In order to discuss the concept and utility of first-passage times, let us come back to free diffusion. Referring to Figure 4, suppose we ask: when does a trajectory of free diffusion starting at x ₀ > 0 at time t = 0 cross the origin for the first time? Clearly the time t _f that it happens, aptly termed the first-passage time (FPT), is a random variable that varies from trajectory to trajectory, and one may ask: what is the form of its distribution P ( x 0 , t f ) ? This distribution would explicitly depend on x ₀, a fact encoded in the defining symbol for the distribution. In order to answer this question, noting that t _f is by definition a positive quantity, it proves convenient to consider the Laplace transform of P ( x 0 , t f ) : P ̃ x 0 , s ≡ ∫ 0 ∞ d t f e − s t f P x 0 , t f = ⟨ e − s t f ⟩ , (16) where the angular brackets denote averaging with respect to the distribution P ( x 0 , t f ) , s is the Laplace variable, and tilde denotes the function obtained on performing Laplace transform L of a function f(t), that is, L [ f ( t ) ] = f ̃ ( s ) . Now, let us understand that a given value of t _f corresponds to all trajectories { x ( τ ) } 0 ≤ τ ≤ t f between times 0 and t _f satisfying x(0) = x ₀ and x(0 < τ < t _f ) > 0. Consequently, averaging with respect to t _f is tantamount to averaging with respect to all trajectories that start at x ₀ and propagate for time t _f without ever crossing the origin, and such an averaging is what the angular brackets in Eq. 16 would also stand for.

In order to proceed, we now derive a differential equation for P ̃ ( x 0 , s ) . To this end, following Ref. [6], we split the time interval [0, t _f ] into two parts. During the first interval [0, Δτ], the aforementioned set of trajectories start from x ₀ and propagate up to x ₀ + Δx, where Δx is of course different for different trajectories. In the second interval [Δτ, t _f ], these trajectories start at x ₀ + Δx and reach 0 at t _f . We take Δτ to be a small time interval with 0 < Δτ ≪ 1, and eventually, we would take the limit Δτ → 0. We get, to leading order in Δτ, that P ̃ x 0 , s = e − s Δ τ P ̃ x 0 + Δ x , s Δ x ≈ 1 − s Δ τ P ̃ x 0 + Δ x , s Δ x , (17) where the average denoted by the angular brackets is to be performed over all realizations of Δx. Now, Δτ being small implies that so is Δx, so that the quantity P ̃ ( x 0 + Δ x , s ) on the rhs of the above equation may be expanded in a Taylor series in Δx. One gets P ̃ x 0 , s ≈ 1 − s Δ τ P ̃ x 0 , s + P ̃ ′ x 0 , s Δ x + P ̃ ″ x 0 , s Δ x 2 2 + ⋯ Δ x . (18)

Here, the prime denotes derivative with respect to x ₀. Now, Eq. 1 gives Δx = η(0)Δτ, so that averaging with respect to Δx is equivalent to averaging with respect to different realizations of η(0). Using ⟨η(0)⟩ = 0 and ⟨η ²(0)⟩ = 2D/Δτ (see Appendix) so that ⟨Δx⟩_Δx = 0 and ⟨ ( Δ x ) 2 ⟩ Δ x = 2 D Δ τ , we get from Eq. 18 on taking the limit Δτ → 0 that D P ̃ ″ x 0 , s − s P ̃ x 0 , s = 0 . (19)

The above equation, referred to as the backward Fokker-Planck equation, is to be solved subject to two boundary conditions: 1) P ̃ ( x 0 → 0 , s ) = 1 , since as x ₀ → 0, we have t _f → 0, and 2) P ̃ ( x 0 → ∞ , s ) = 0 , since as x ₀ → ∞, we have t _f → ∞. The solution of Eq. 19 subject to these boundary conditions is P ̃ x 0 , s = e − s / D x 0 . (20)

On performing inverse Laplace transformation, one readily gets [6]. P x 0 , t f = x 0 4 π D e − x 0 2 / 4 D t f t f 3 / 2 . (21)

(For x ₀ < 0, the corresponding FPT distribution is obtained from the above equation with the replacement x ₀ → |x ₀|.) We thus find that the FPT distribution has a tail ∼ t f − 3 / 2 , and moreover, that the average FPT (the mean first-passage time (MFPT)) given by ⟨ t f ⟩ = ∫ 0 ∞ d t f t f P ( x 0 , t f ) is infinite ! The latter result implies that the dynamics (1) allows for trajectories which starting at x ₀ start to move away from the origin and hence take a very long time to hit the origin for the first time (e.g., the blue trajectory in Figure 4). Does repeated resetting to x ₀ kill these trajectories, given that it allows the particle to remain effectively in a finite region around x ₀? Let us then investigate the distribution of the FPT in presence of resetting.

In order to proceed, define P r ( x 0 , t f ) as the FPT distribution in presence of resetting, and S _r(x ₀, t) as the so-called survival probability, that is, the probability that a trajectory starting at x ₀ at time t = 0 has not crossed the origin up to time t. Note that resetting is taking place not to the origin but to the initial location x ₀ of the particle. Obviously then one has S r ( x 0 , t + Δ t ) − S r ( x 0 , t ) = P r ( x 0 , t ) Δ t for small Δt, which in the limit Δt → 0 gives P r ( x 0 , t ) = − ∂ S r ( x 0 , t ) / ∂ t . In terms of S _r(x ₀, t), the mean FPT is given as ⟨ t f ⟩ r ≡ ∫ 0 ∞ d t f t f P r ( x 0 , t f ) = − ∫ 0 ∞ d t f t f ∂ S r ( x 0 , t f ) / ∂ t f = − [ t f S r ( x 0 , t f ) ] 0 ∞ + ∫ 0 ∞ d t f S r ( x 0 , t f ) = ∫ 0 ∞ d t f S r ( x 0 , t f ) = S ̃ r ( x 0 , s = 0 ) , using S _r(x ₀, ∞) = 0. A trajectory starting from x ₀ that reaches the origin for the first time at time t may have had no reset at all since the time instant t = 0 or may have had its last reset at an earlier time instant in the interval [t − τ − dτ, t − τ], with τ ∈ [0, t], and has not passed through the origin before that. Consequently, one has P r x 0 , t = − ∂ S r x 0 , t ∂ x 0 = ∫ 0 t d τ S r x 0 , t − τ r e − r τ P x 0 , τ + e − r t P x 0 , t . (22)

Indeed, trajectories that did not have any reset during the time interval [0, t] (the probability of which, according to our earlier discussions, is exp( − rt)) give rise to the second term on the rhs of the second equality above. On the other hand, trajectories that at time t have had last reset in the interval [t − τ − dτ, t − τ] (the probability of which is exp( − rτ) rdτ), with τ ∈ [0, t], and have not hit the origin before that give rise to the first term on the rhs of the second equality.

Using Eq. 21 in Eq. 22, taking Laplace transform of both sides, and using S _r(x ₀, t = 0) = 1, we finally get S ̃ x 0 , s = 1 − A r A + s , (23) where the quantity A is defined as A ≡ A s , r = L x 0 e − r t − x 0 2 / 4 D t 4 π D t 3 = e − x 0 2 / D s + r . (24)

Here, we have used L [ t − 3 / 2 ⁡ exp ( − a / ( 4 t ) ) ] = 2 π / a exp ( − a s ) for Re(a) > 0. Consequently, we obtain the MFPT in presence of resetting as [2]. ⟨ t f ⟩ r = e x 0 r / D − 1 r . (25)

We see from Eq. 25 that for fixed x ₀ and D, the MFPT in presence of resetting is finite for finite r, so that resetting has a drastic consequence on rendering the MFPT of free diffusion finite. Moreover, ⟨ t f ⟩ r diverges in two extreme limits, namely, as r → 0 and as r → ∞, so it has to be that there exists a value r ^⋆ of r at which ⟨ t f ⟩ r as a function of r has a minimum. Note that the limit r → 0 corresponds to free diffusion, so that the fact of a diverging ⟨ t f ⟩ r as r → 0 is consistent with what we observed above regarding the MFPT for free diffusion. The divergence of MFPT as r → ∞ may be understood on the basis of the fact that in this limit, the particle resets so often that it does not effectively get a chance to move away from x ₀ and hit the origin. The quantity r ^⋆ satisfying d ⟨ t f ⟩ r / d r | r = r ⋆ = 0 yields the following transcendental equation [2]. z ⋆ 2 = 1 − e − z ⋆ ; r ⋆ = z ⋆ 2 D / x 0 2 , (26) which has a unique non-zero solution z ^⋆ = 1.593 62 …. It then follows that there is an optimal resetting rate r ^⋆, such that the MFPT in presence of resetting attains its minimum as a function of r. The variation of ⟨ t f ⟩ r with r, as given by Eq. 25, is shown in Figure 5, which clearly shows the existence of a minimum.

What has then a minimum MFPT got to do with search processes? Imagine having misplaced one of your favourite or essential belongings. It could be your car key or your mobile phone. After getting into your office, you usually keep the key on your office table. Having misplaced it, a typical tendency to locate it is to begin searching in random directions for the key (well, you do not know beforehand which particular direction to search for, so random directions would be the best bet!) while starting from your office. Having searched unsuccessfully for sometime, you realize that perhaps you did not search well enough in the region around the table (after all, you usually keep the key on the table itself), so you return to the table, and re-initiate your search. You keep repeating the process until you locate the key, and of course the first time you locate it, you stop searching any further. Figure 6 models this search strategy in one dimension: the brown filled circle denotes your misplaced belonging, while searching in random directions for random times followed by re-initiation of the search process from your starting position may be modelled as free diffusion interspersed in time with stochastic resets to the initial location (the two space-time trajectories in the plot), namely, what has been our model of study in this work. Had there been no resets, that is, if you had continued searching in random directions without ever coming back to your initial location, we would arrive at the following conclusion based on our analysis of free diffusion pursued above. If you had repeated the process over and over again, then, in some realizations of the process, you would have located the brown circle in a finite time, while there would be happily a large number of other realizations in which you fail to locate it in finite time, so that the MFPT through the brown circle would diverge ! On the other hand, introducing stochastic resetting into the dynamics would make the MFPT finite, as our theoretical analysis of diffusion with stochastic resetting has revealed. Not just that, there exists an optimal value for the rate r at which stochastic resets are executed such that the corresponding MFPT attains its minimum value. Thus, stochastic resetting offers a very efficient way to locate your misplaced belonging. Of course, for resetting to be efficient, the quantity x ₀, the initial location of the searcher to which it repeatedly resets, has to be small enough. In other words, the location x ₀ has to be close enough to the target for resetting to do a good job. Indeed, as is evident from Eq. 25, if x ₀ is large, the MFPT would be very large, and then resetting would not help to locate the target in a finite time.

Until now, we have considered standard Brownian motion (1) in discussing the effects of stochastic resetting. We now discuss briefly the corresponding effects on the so-called scaled Brownian motion (SBM) [16, 17], which in appropriate limits reduces to the standard Brownian motion. SBM is used in such contexts as fluorescence recovery following photobleaching [18], anomalous diffusion in biophysics [19, 20], granular gas of viscoelastic particles [21], etc. In contrast to the standard Brownian motion (1) that involves a time-independent diffusion coefficient, the SBM involves a diffusion coefficient that is time dependent and in fact which scales as a power-law in time. Thus, the corresponding dynamics is given by [16]. d x d t = η t , (48) with η(t) a Gaussian, white noise with properties 〈 η t 〉 = 0 , 〈 η t η t ′ 〉 = 2 D t δ t − t ′ ; (49) here, we have D t = α K α t α − 1 ; α > 0 , (50) with K _α a constant. Note that setting α = 1 and K ₁ = D reduces the SBM to the standard BM (1). In the case of the SBM, the position probability distribution may be shown to have the form [16]. P x , t = 1 4 π K α t α e − x − x 0 2 / 4 K α t α , (51) which as expected reduces for α = 1 to the result (7). The MSD for the SBM reads [16]. ⟨ x − x 0 2 ⟩ = 2 K α t α . (52)

The behavior of the MSD as a function of time depends explicitly on the value of α. Obviously, α = 1, the case of the standard BM, results in diffusive behavior. For 0 < α < 1, one has a subdiffusive behavior, while the behavior is superdiffusive for α > 1. For α = 2, one has ballistic growth of the MSD in time, and α > 2 results in superballistic behavior. In the limit α → 0, one has logarithmic time dependence of the MSD.

In discussing resetting of the SBM, two separate situations were considered: one in which only the position of the particle resets to its initial value (referred to as nonrenewal resetting [16]), and other in which both the position and the diffusion coefficient reset to their respective initial values (referred to as renewal resetting [17]). Let us discuss the effects of resetting in the two cases. In the case of nonrenewal resetting at exponentially-distributed random times (the set-up of Eq. 9), it was shown that in the long-time limit (more precisely, for times t satisfying t α + 1 ≫ ( x − x 0 ) 2 / ( K α r ) ), the position probability distribution in presence of resetting is given by [16]. P r x , t ≃ 1 2 r α K α t 1 − α / 2 ⁡ exp − r α K α | x − x 0 | t 1 − α / 2 . (53)

The above equation implies a probability distribution that is non-Gaussian and also time-dependent (unless α = 1), with a cusp at the location of resetting x = x ₀; For α = 1, the result yields the stationary-state (14) for the standard BM. The MSD in presence of resetting is given in the limit of long times by [16]. ⟨ x − x 0 2 ⟩ t ≃ 2 α K α r t α − 1 . (54)

We thus see that with respect to the result for the SBM in absence of resetting, Eq. 52, the MSD in presence of resetting has a time dependence characterized by an exponent that is smaller by unity. For α = 1, Eq. 54 yields the stationary-state result (15) for the standard BM. For α = 2, when the SBM in absence of resetting shows ballistic motion, the MSD in presence of resetting exhibits normal diffusive behavior. Superdiffusive SBM corresponding to α in the range 1 < α < 2 exhibits a subdiffusive behavior in presence of resetting. For 0 < α < 1, when the SBM in absence of resetting shows a subdiffusive behavior, Eq. 54 shows that the MSD in presence of resetting decays to zero as a power law, implying thereby that the particle at long times remains in the close vicinity of the resetting location. We thus see that in the case of nonrenewal resetting of the SBM at exponentially-distributed random times, one ends up with a rather rich variety of behavior of the position probability distribution and the MSD depending on the value of α. In the foregoing, we discussed the case of resetting at exponentially-distributed times; for discussion on the effects of resetting at power-law-distributed times, the reader is referred to Ref. [16].

We now turn to the case of renewal resetting of the SBM. In this case, if there is a resetting at time instant t _r , the particle resets its position to x ₀, and also the diffusion constant at a later time t > t _r becomes D ( t ) = α K α ( t − t r ) α − 1 (in the nonrenewal case discussed above, the diffusion coefficient is by contrast given by D(t) = αK _α t ^α−1). The case has been studied in detail for both exponential and power-law resetting in Ref. [17]. Here, in the spirit of this review, we discuss only the former scenario, referring the interested reader to Ref. [17] for discussions on power-law resetting. For exponential resetting, it was shown that the position probability distribution relaxes at long times to a stationary state [17]. P r, ss x ≃ r 2 α α + 1 α 4 K α r 1 / α + 1 | x − x 0 | 1 − α / α + 1 × exp − x − x 0 2 r α 4 K α 1 / α + 1 α 1 / α + 1 + α − α / α + 1 . (55)

Not surprisingly, for α = 1, the above equation reproduces correctly the result (14) for the standard BM. Corresponding to the stationary state (55), the MSD attains the value [17]. ⟨ x − x 0 2 ⟩ r, ss = 2 K α r α Γ α + 1 , (56) where Γ(x) is the Gamma function. Again, for α = 1, one gets back the result (15) for the standard BM. On the basis of the above discussions we thus see a drastic difference between the cases of nonrenewal and renewal resetting. While in the former, resetting fails to induce a stationary position distribution, see Eq. 53, it does suffice to induce a stationary distribution in the renewal case, see Eq. 55. Of course, for α = 1, when the diffusion coefficient has no dependence on time, there is no difference between the two cases of resetting, and one has a stationary state. In Ref. [17], the MFPT has been investigated for the case of renewal resetting. Considering a stationary target located at the origin and an SBM starting at x ₀ > 0 and resetting to x ₀ at exponentially-distributed random times, the MFPT through the target is given by [17]. ⟨ t f ⟩ r ≃ 1 r α + 1 2 α exp r α / α + 1 x 0 2 4 K α 1 / α + 1 α 1 / α + 1 + α − α / α + 1 − 1 . (57)

For α = 1, one recovers the result (25) for the standard BM. As a function of r, the above MFPT is minimized at r = r ^⋆ satisfying [17]. 1 − 2 α α + 1 exp − B r ⋆ c = B c r ⋆ c ; c ≡ α α + 1 , B ≡ x 0 2 4 K α 1 / α + 1 α 1 / α + 1 + α − α / α + 1 . (58)

The above equation for α = 1 reduces correctly to the result (26) for the standard BM.

We now discuss the case of fractional Brownian motion (FBM), another variant of the standard BM (1). In this case, the dynamics is given by [22, 23]. d x d t = η H t , (59) with η _H (t) a Gaussian noise with zero mean, which unlike the standard BM and the SBM is correlated in time: 〈 η H t 〉 = 0 , 〈 η H t η H t ′ 〉 ≃ K 2 H 2 H 2 H − 1 | t − t ′ | 2 H − 1 , (60) where H ∈ (0, 1) is the so-called Hurst exponent. The concept of FBM is invoked in discussions of subdiffusive dynamics [24], in investigating individual trajectories of fluorescently-labelled telomeres in the nucleus of living human cells [25], in discussing passive, thermally driven motion of micron-sized tracers in hydrogels of mucins, the main polymeric component of mucus [26], etc.

In absence of resetting, the MSD of the FBM behaves as ⟨ x − x 0 2 ⟩ t = 2 K 2 H t 2 H , (61) implying thereby normal diffusion for H = 1/2, subdiffusion for 0 < H < 1/2 and superdiffusion for 1/2 < H < 1. Resetting of FBM has been considered under the so-called fully-renewal scheme [27], whereby the memory of noise correlations given in Eq. 60 is completely erased at each resetting, and this is the case we consider here. In presence of exponential resetting, the MSD has the above behavior only for short times, while for times of order (1/r)[Γ(2H + 1)]^1/(2H) shows saturation to a plateau (pl) value given by ⟨ x − x 0 2 ⟩ pl ≈ 2 K 2 H Γ 2 H + 1 r 2 H . (62)

Comparing the above result with the one for renewal resetting of the SBM at exponentially-distributed times, Eq. 56, we find that the value is the same as that in the SBM case with α = 2H. In the same manner, the FBM with exponential resetting has a stationary state characterized by a time-independent position probability distribution, which at intermediate-to-large displacements is given by the result (55) for the renewal resetting of the SBM, with the substitution α = 2H.

While the definition (61) for the MSD involves averaging at a given time instant t over ensemble of statistically-independent values of x(t), it is interesting to ask for the behavior of the MSD when computed along a single trajectory of x(t) as a function of time t, and enquire about ensemble-averaged versus time-averaged mean-squared displacements whose non-equivalence implies breaking of ergodicity of the underlying dynamics. The single-trajectory-based averaging along the time series of particle position x(t) defines the time-averaged-MSD (TAMSD) as δ 2 Δ ̄ ≡ 1 T − Δ ∫ 0 T − Δ d t x t + Δ − x t 2 , (63) where Δ is the lag time and T is the length of the time series. On averaging over N statistically-independent TAMSD realizations, the mean TAMSD is obtained as ⟨ δ 2 Δ ̄ ⟩ = 1 N ∑ i = 1 N δ i 2 Δ ̄ , (64) where the angular brackets denote averaging over noise realizations. Reference [27] performed a detailed study of the behavior of the TAMSD versus MSD to conclude that for short times, one has the MSD and the TAMSD behaving respectively as t ^2H and Δ^2H for 0 < H < 1/2 and as t ^2H and Δ¹ for 1/2 < H < 1, so that time averaging and ensemble averaging do not coincide for 1/2 < H < 1, while they do coincide for 0 < H < 1/2. For long times, however, the two averages coincide for all values of H and have the behavior as in Eq. 62. While the two averages coincide for the FBM in the absence of resetting, the foregoing suggests that the resetting dynamics of originally ergodic FBM for superdiffusive H > 1/2 exhibits weak ergodicity breaking at short times. For a more detailed survey of the topic, the reader is referred to Ref. [27].