Extremal statistics for stochastic resetting systems
EExtremal statistics for stochastic resetting systems
Prashant Singh and Arnab Pal ∗ International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India and School of Chemistry, The Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 6997801, Israel (Dated: February 16, 2021)While averages and typical fluctuations often play a major role to understand the behavior of anon-equilibrium system, this nonetheless is not always true. Rare events and large fluctuations arealso pivotal when a thorough analysis of the system is being done. In this context, the statistics ofextreme fluctuations in contrast to the average plays an important role, as has been discussed infields ranging from statistical and mathematical physics to climate, finance and ecology. Herein, westudy Extreme Value Statistics (EVS) of stochastic resetting systems which have recently gained lotof interests due to its ubiquitous and enriching applications in physics, chemistry, queuing theory,search processes and computer science. We present a detailed analysis for the finite and large timestatistics of extremals (maximum and arg-maximum i.e., the time when the maximum is reached)of the spatial displacement in such system. In particular, we derive an exact renewal formula thatrelates the joint distribution of maximum and arg-maximum of the reset process to the statisticalmeasures of the underlying process. Benchmarking our results for the maximum of a reset-trajectorythat pertain to the Gumbel class for large sample size, we show that the arg-maximum densityattains to a uniform distribution regardless of the underlying process at a large observation time.This emerges as a manifestation of the renewal property of the resetting mechanism. The results areaugmented with a wide spectrum of Markov and non-Markov stochastic processes under resettingnamely simple diffusion, diffusion with drift, Ornstein-Uhlenbeck process and random accelerationprocess in one dimension. Rigorous results are presented for the first two set-ups while the lattertwo are supported with heuristic and numerical analysis.
I. INTRODUCTION
In many situations of physical relevance, extremeevents are of tremendous importance despite they oc-cur rarely. Starting from stock market crashes or largeinsurance losses in finance, records in Olympics to nat-ural calamities such as earthquake, heat waves, extremeevents or tsunamis – all are typical examples of extremeevents. Extreme Value Statistics (EVS) sits on the heartof a branch of statistics which deals with the probabili-ties generated by random processes responsible for suchunusual extreme events [1–8]. The study of EVS hasbeen extremely important in the field of disordered sys-tems [9, 10], fluctuating interfaces [11, 12], interactingspin systems [13], stochastic transport models [14, 15],random matrices [16–18], ecology [19], in binary searchtrees [20] and related computer search algorithms [21, 22]and even in material science [23]. We refer to these ex-tensive reviews which provide detailed account of recenttheoretical and application based progresses of EVS inscience. In EVS, the famous Gnedenko’s classical law ofextremes provides statistics of the maximum (or mini-mum) of a set of uncorrelated random variables (see e.g.[1–8]). However, there exists a myriad of systems forwhich the underlying random variables can be weakly orstrongly correlated due to the correlations [24–26] or aglobal conservation [27–30] among the random variables,see [8] for a comprehensive review. One of the centralgoals of the subject is then to understand the statistics ∗ [email protected]; [email protected] of extremes i.e., the maximum M ( t ) of a given trajectory x ( t ) which is observed upto time t and the time t m toreach the maximum for such correlated systems namelyBrownian motion and its generalizations [31–37], run andtumble motion [38, 39], fractional Brownian motion [40–42], random acceleration [37], anomalous walker [43] andfluctuating interfaces [44]. Surprising enough, there areonly a very limited exact results known on renewal pro-cesses (e.g., see [27] where EVS for the longest waitinginterval was analyzed and not the quantities of our inter-est). Moreover, prediction of limiting extreme value dis-tributions in renewal processes has been extremely chal-lenging (also see [45, 46] and the discussion below). Inthis paper, we set out to understand in details the ex-tremal statistics in a recently popularized renewal pro-cess namely stochastic resetting.Stochastic resetting is a renewal process in which thedynamics repeats by itself after random intervals con-trolled externally [45]. The subject has recently gainedconsiderable attention due to its vigorous applications instatistical physics [46–53], stochastic process [54–58] andother cross-disciplinary fields such as chemical and bi-ological process [59–62], computer science [63, 64] andsearch theory [65–67]. Brownian motion with reset-ting introduced by Evans and Majumdar in [46, 47] isthe paradigmatic example of the subject which essen-tially captures two central features: emergence of a non-equilibrium steady state with a non-zero probability cur-rent [46–52] and expedition of a first passage time process[68–76]. Over the years, a large volume of work has beendone to extend this simple model beyond diffusion toothers such as underdamped [77] and scaled [78] Brown-ian motion, random acceleration process [79], and active a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b particles [80–82]. Rigorous efforts have also been madeto understand non-Poissonian strategies [45, 52, 68, 72].The subject has also found interesting applications instochastic thermodynamics [83–85], quantum systems[86], many particle systems [87–89], and nonlinear sys-tems [90]. Very recently, resetting has also seen advancesin single particle experiments using optical traps [91, 92].We refer to this recent review [45] and references thereinfor more details on the subject.Notwithstanding that the subject has seen tremendousprogress in statistical physics, exact results on EVS ofstochastic resetting have been very limited. It has beenunderstood that the maximum M of a Brownian trajec-tory under stochastic resetting (i.e., Poissonian resetting)belongs to the Gumbel class for large sample size [45, 46].But only recently exact expressions for the first two mo-ments of the maximum M ( t ) have been obtained by Ma-jumdar et al in [93]. Using these, the mean perimeter andthe mean area of a convex hull of the 2D Brownian mo-tion with resetting were computed. On the other hand,when resetting times are taken from a power law den-sity (i.e., non-exponential or non-Poissonian resetting),it was shown that the distribution of the maximum ofthe reset process is given by the Fr´echet law when appro-priately centred and scaled [94]. A Weibull limit law wasderived for first passage time under restart with branch-ing [70]. But to the best of our knowledge, not muchis known on the statistics for the arg-max i.e., the time t m to reach this maximum which is also a very importantstatistical measure (recall the famous arcsine law of L´evyin classical probability theory [31]). Also exact resultsfor maximum beyond simple diffusion are not availableat this moment. This paper exactly aims to bridge thisvoid. First, we derive a renewal formula (5) for the jointdistribution of M ( t ) and t m ( t ) in the presence of restartin terms of their underlying joint distribution and otherstatistical quantities. This formula is valid for any under-lying stochastic process (Markov or non-Markov) as longas the memory is erased after each resetting event. Sec-ondly, utilizing the renewal formula, we derive exact andasymptotic n -th order moments of M ( t ) for diffusion anddrift-diffusion process. We also show the convergence tothe Gumbel limit law for both these cases. Third andimportantly, we obtain exact expression for the momentgenerating function for t m from which we show that thedensity of t m pertains to a universal /t form independentof the underlying process at large time with sub-leadingprocess dependent corrections. We first demonstrate thisresult exactly for diffusion and drift-diffusion process andthen generalize to arbitrary stochastic process.The remainder of the paper is structured as follows.We derive the joint distribution of M ( t ) and t m ( t ) in theLaplace space of t ( → s ) and t m ( → k ) in Sec. II. Thisrenewal formula becomes instrumental to investigate thestatistics of M ( t ) for simple diffusion and drift-diffusionprocess in Sec. III. In particular, we compute the exactmoments for the maximum M ( t ) and then present theasymptotic limiting distributions. In Sec. IV, we present x( τ ) ττ τ τ τ t m1 2 3 4 M t FIG. 1. Schematic of the maximum distance M and the time t m to reach M for a trajectory x ( τ ) of a given stochasticprocess observed upto time t . The vertical solid lines indicatethe resetting events which take place at times τ , τ , · · · takenfrom a distribution p ( τ ). The resetting coordinate is same asthe initial condition which is set to be the origin here. Totalnumber of resetting intervals in this realization is 4, and themaximum occurs during the third interval. our results for arg-max t m ( t ) for simple diffusion anddrift-diffusion process. We compute the moments andanalyze the large time behavior for the arg-max density.The large time limiting density is shown to be universalwhich is further proven in Sec. V for generic stochas-tic process augmented with numerical simulations. Wesummarize our results in Sec. VI. For brevity, many sup-plemented derivations of our results have been reservedto the Appendix. II. RENEWAL FORMULA FOR JOINTDISTRIBUTION OF M AND t m We begin with the derivation of the joint distributionfor M and t m in the presence of resetting. Consider atypical trajectory x ( τ ) of a particle governed by somestochastic law of motion and observed upto a fixed time t (see Fig. 1). Motion of the particle is also subjected toresetting that brings it back to the origin at a constantrate r . This essentially means that the waiting time be-tween any two resetting events is taken from the distri-bution p ( τ ) = re − rτ . Note that, in a fixed time window t , the number of resetting undergone by the particle isa random variable and varies from trajectory to trajec-tory. Let us assume that the trajectory is divided into N -intervals out of which the particle has experienced N − τ i with i = 1 , , ..., N −
1. However,in the last intervals τ N , the particle does not experienceany resetting event: the probability of which is given by (cid:82) ∞ τ N p ( τ ) dτ = e − rτ N . Since the observation time is fixed, N is a random variable and varies from trajectory to tra-jectory. Also, without any loss of generality, we assumethe starting point as a resetting event which gives N ≥ M, t m and N which is denoted by P r ( M, t m , N | t ). Notethat the maximum distance M can be attained dur-ing any of the N time intervals { τ i } = { τ , τ , ..., τ N } .For example, if the maximum M is attained at time τ during the the first interval then the contribution to P r ( M, t m , N | t ) is given by C = (cid:32) N (cid:89) i =1 (cid:90) ∞ dτ i (cid:33) (cid:90) ∞ dτ P ( M, τ | τ ) p ( τ ) × (cid:34) N − (cid:89) i =2 S (0 , τ i | M ) p ( τ i ) (cid:35) × (cid:2) e − rτ N S (0 , τ N | M ) (cid:3) × δ ( τ − t m ) δ ( t − N (cid:88) i =1 τ i ) , (1)where we introduce P ( M, t m | t ) as the joint distributionof M and t m for the underlying process (without reset-ting) till time t . Moreover, S ( x , τ j | M ) is the prob-ability that the particle starting from x always staysbelow M upto time τ j in the absence of resetting. Inother words, this is the survival probability that a par-ticle, starting from x , survives an absorbing boundaryat M ( > x ) till time τ j . Rationale behind various termsin C can be understood in the following way: the max-imum M is attained by the underlying process in thefirst interval that lasts for time τ which gives rise to P ( M, τ | τ ) p ( τ ) term in the first line of Eq. (1). Es-sentially, in the remaining N − M . As a consequence, we get N − δ -function asserts thecondition t m = τ which is true here while the second oneensures that the total observation time is t .Similarly, we can write the contribution to P r ( M, t m , N | t ) when the maximum M is attainedin the second interval at time τ while the particleremains below x = M during the other intervals. Thiscontribution is given by C = (cid:32) N (cid:89) i =1 (cid:90) ∞ dτ i (cid:33) (cid:90) ∞ dτ P ( M, τ | τ ) p ( τ ) × N − (cid:89) i =1 ,i (cid:54) =2 S (0 , τ i | M ) p ( τ i ) × (cid:2) e − rτ N S (0 , τ N | M ) (cid:3) × δ ( t − N (cid:88) i =1 τ i ) δ ( τ + τ − t m ) . (2)Following the same physical argument, one can also writethe contributions C , C .. for the maximum to be in thesecond, third,... interval respectively. In particular, when the maximum is in the last reset-free interval τ N , we have C N = (cid:32) N (cid:89) i =1 (cid:90) ∞ dτ i (cid:33) (cid:90) ∞ dτ P ( M, τ | τ N ) e − rτ N × (cid:34) N − (cid:89) i =1 S (0 , τ i | M ) p ( τ i ) (cid:35) × δ ( t − N (cid:88) i =1 τ i ) δ ( N − (cid:88) i =1 τ i + τ − t m ) . (3)Thus, the joint distribution P r ( M, t m , N | t ), can beobtained by summing over all the contributions C , C , ..., C N . Performing the sum, P r ( M, t m , N | t ) canbe formally written as P r ( M, t m , N | t ) = 1 r N (cid:88) j =1 (cid:90) ∞ dτ j dτ P ( M, τ | τ j ) p ( τ j ) × N (cid:89) j (cid:48) =1 ,j (cid:48) (cid:54) = j (cid:90) ∞ dτ j (cid:48) S (0 , τ j (cid:48) | M ) p ( τ j (cid:48) ) dτ j (cid:48) × δ (cid:32) j − (cid:88) i =1 τ i + τ − t m (cid:33) δ ( t − N (cid:88) i =1 τ i ) . (4)To proceed further, it is only natural to take theLaplace transformations with respect to t ( → s )and t m ( → k ). Denoting the Laplace transform of P r ( M, t m , N | t ) by Z r ( M, k, N | s ) and further perform-ing the sum over all the intervals N , we find (see Ap-pendix A) Z r ( M, k | s ) = Z ( M, k | r + s ) (cid:2) − r ¯ S (0 , s + r | M ) (cid:3) (cid:2) − r ¯ S (0 , s + r + k | M ) (cid:3) , (5) where Z r ( M, k | s ) is the Laplace transform of thejoint distribution P r ( M, t m | t ) (subscript 0 will indi-cate the same without resetting) and ¯ S (0 , s | M ) = (cid:82) ∞ dt e − st S (0 , t | M ) is the Laplace transformation ofthe underlying survival probability. Eq. (5) is the firstcentral result of our paper. Such a renewal formula isvery important since it relates the joint distribution of M and t m with resetting to the underlying joint distri-bution and the survival probabilities. For Markov pro-cesses, one can also use the path-decomposition method[33–35, 93] to arrive at Eq. (5). However, our derivationis more robust since it holds even when underlying pro-cess is non-Markov while the path-decomposition methodstrictly relies on the Markov property as was illustratedfor Brownian motion by Majumdar et al in [93]. Theonly assumption that goes into the derivation is that theprocess does not retain memory between the resettingintervals. Finally, we remark that although Eq. (5) hasbeen derived under a one-dimensional framework, it isvalid also in higher dimensions. Also see [39] where asimilar approach has been used to study the persistentproperties of run and tumble particles in arbitrary di-mensions.In what follows, we use the renewal formula in Eq.(5) to study statistics of M and t m for simple diffusionand drift-diffusion process in one dimension, respectivelydescribed by dxdτ = η ( τ ) , (6) dxdτ = v + η ( τ ) , (7)where η ( τ ) is the Gaussian white noise with mean (cid:104) η ( τ ) (cid:105) = 0 and correlation (cid:104) η ( τ ) η ( τ (cid:48) ) (cid:105) = 2 Dδ ( τ − τ (cid:48) ),and further assume the drift-velocity v >
0. Here, D is the diffusion constant (which will be set to withoutany loss of generality for the rest of the paper). We alsoconsider that the particle starts from the origin at t = 0and is reset to the origin at random times drawn fromthe distribution p ( τ ) = re − rτ . III. STATISTICS OF MAXIMUM M In this section, we use Eq.(5) to analyse the statisticalproperties of M for the above-mentioned stochastic pro-cesses. To compute the distribution of M , one needs tointegrate P r ( M, t m | t ) over all t m . This is equivalent toputting k = 0 in Z r ( M, k | s ) in Eq. (5). Let us considerthe case of simple diffusion first. A. Simple diffusion
It is instructive to first review some known results on M and t m without resetting which will be useful for sub-sequent studies. For simple diffusion, the joint distribu-tion of M and t m is given by [36] P ( M, t m | t ) = Mπt / m √ t − t m e − M tm . (8)Integrating over M , we get the distribution of t m as P ( t m | t ) = π √ t m ( t − t m ) or equivalently the cumulativedistributionProb[ t m ≤ T ] = (cid:90) T dt (cid:48) m P ( t (cid:48) m | t ) = 2 π sin − (cid:34)(cid:114) Tt (cid:35) , (9)which is the celebrated ‘Arc-sine’ law for the Brown-ian motion due to L´evy [31]. To use Eq.(5), we needto specify the Laplace transformations ¯ S (0 , s | M ) and Z ( M, k | s ). The latter can be computed by taking theLaplace transformation of Eq. (8) with respect to t and t m and this gives Z ( M, k | s ) = (cid:114) s e − √ s + k ) M . (10) On the other hand, survival probability for a Brownianparticle is a canonical result due to L´evy which reads S (0 , t | M ) = Erf (cid:16) M √ Dt (cid:17) , where Erf( z ) = √ π (cid:82) z e − y dy is the error-function. Taking the Laplace transform, onegets ¯ S (0 , s | M ) = 1 s (cid:16) − e −√ sM (cid:17) . (11)Inserting Eqs. (10) and (11) in the renewal Eq. (5) yieldsthe Laplace transformation Z r ( M, k | s ) from which onegets Z r ( M, k = 0 | s ) = √ r + s ) / e − √ r + s ) M (cid:16) s + re − √ r + s ) M (cid:17) . (12)Note that this was also obtained in [93] using the pathdecomposition method. To get the distribution of M inthe time domain, one has to perform the inverse Laplacetransform of Z r ( M, k = 0 | s ) with respect to s . Beforethat, we look at the moments of M to get the effect ofresetting on M . The n -th moment can be written interms of Z r ( M, k = 0 | s ) as (cid:90) ∞ dt e − st (cid:104) M n ( t ) (cid:105) = (cid:90) ∞ dM Z r ( M, k = 0 | s ) , = − n !2 n r ( r + s ) n − Li n (cid:16) − rs (cid:17) , = − n !2 n r ( r + s ) n − ∞ (cid:88) k =1 ( − r ) n k n s k +1 , (13)where Li n (cid:0) − rs (cid:1) is the PolyLog function in the second line[95] and while going to the third line, we have used theseries representation Li n ( − y ) = (cid:80) ∞ k =1 ( − y ) k k n . Next, toget the moments in the time domain, we use the followinginverse Laplace transformation: L − s → t (cid:34) r + s ) n − s k +1 (cid:35) = ¯ F (cid:0) − n , k + n , − rt (cid:1) t − k − n , (14)where ¯ F ( a, b, z ) stands for the regularized hypergeo-metric function [95]. Inserting this in Eq. (13), we findthat (cid:104) M n ( t ) (cid:105) possesses the scaling form (cid:104) M n ( t ) (cid:105) = 1( √ r ) n H n ( rt ) , (15)with the scaling function H n ( z ) given by H n ( z ) = ( − − n n ! ∞ (cid:88) k =1 1 ¯ F (cid:0) − n , k + n , − z (cid:1) k n ( − z ) − k − n . (16)Note again that the first two moments ( n = 1 ,
2) wererecently obtained in [93]. Our results are consistent with ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Theory ● Simulation 〈 M ( t ) 〉 ( a ) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Theory ● Simulation 〈 M ( t ) 〉 ( b ) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Theory ● Simulation 〈 M ( t ) 〉 ( c ) FIG. 2. Comparison of moments for the maximum (cid:104) M n ( t ) (cid:105) in Eq. (15) for the simple diffusion against the numerical simulationsfor (a) n = 1 , (b) n = 2 and (c) n = 3. We have fixed r = 1 for all the simulations. that in [93] and further extend to obtain exact expres-sions for all moments of M . In Figure 2, we have plotted (cid:104) M n ( t ) (cid:105) for n = 1 , , H n ( z ) for which we analyse Z r ( M, k = 0 | s )in Eq. (12) in various limits of sr − and then appropri-ately use Eq. (13) (first line) to obtain the moments of M . For the continuity of the presentation, we have rele-gated this derivation to appendix B and present only thefinal results here. The asymptotic forms read H n ( z ) (cid:39) log n z + O (cid:0) log n − z (cid:1) , as z → ∞ , (17) (cid:39) n !Γ (cid:0) n (cid:1) z n + B n z n +1 , as z → , (18)where the exact expression of B n is given in Eq. (B9).Finally, we insert Eqs. (17) and (18) in Eq. (15) to getthe behaviour of (cid:104) M n ( t ) (cid:105) at large and short times as (cid:104) M n ( t ) (cid:105) (cid:39) log n rt ( √ r ) n + O (cid:2) log n − rt (cid:3) , as t (cid:29) r , (19) (cid:39) n ! t n n Γ (cid:0) n (cid:1) + B n rt n +1 n/ , as t (cid:28) r . (20)For r = 0, the moments in Eq.(20) match, as expected,with that of the free Brownian motion [36]. It is worthnoting that there is a crossover in (cid:104) M n ( t ) (cid:105) from ∼ t n/ behaviour to ∼ log n rt behaviour at time scale t ∼ r with the crossover function given exactly in Eq. (16). Al-though a diffusing particle approaches a non-equilibriumsteady state in the presence of resetting [46], the max-imum M ( t ) still increases with time but rather slowly(with logarithmic growth) for t (cid:29) r . This same observa-tion was also made recently by Majumdar et al in [93].The logarithmic growth of the maximum in presence ofresetting can also be understood heuristically from theextreme value statistics of weakly correlated variableswhich we illustrate later.After looking at the moments, we now considerthe distribution of M for which we have to invert Z r ( M, k = 0 | s ) in Eq. (12) with respect to s . Perform- ing inversion for arbitrary t turns out to be challeng-ing. However for large t (or equivalently small s ), we canmake some analytic progress. For s (cid:28) r , we approximate r + s (cid:39) r and use this in Eq. (12) to get Z r ( M, k = 0 | s ) (cid:39) √ r / e −√ rM (cid:16) s + re −√ rM (cid:17) . (21)To get the distribution in the time domain, we use theinverse Laplace transform L − s → t (cid:104) s + a ) (cid:105) = te − at for a ≥ P r ( M | t ) (cid:39) √ r te −√ rM exp (cid:16) − rte −√ rM (cid:17) . (22)We emphasize that the approximate equality in Eq. (22)indicates that this equation is valid only for t (cid:29) r whenthe effect of resetting is highest. In Fig. 3 (top panel),we have compared P r ( M | t ) in Eq. (22) with the sameobtained from the numerical simulations. We observean excellent agreement between them. To understandEq. (22) heuristically, we remark that the survival prob-ability S r (0 , t | M ) for Brownian motion under reset pos-sesses the Gumbel form at large times. Based on theextreme value statistics of weakly correlated variables, itwas shown that S r (0 , t | M ) (cid:39) e − rte −√ rM for t (cid:29) r [46].Since S r (0 , t | M ) is the cumulative distribution of M , onecan appropriately differentiate it with respect to M toget the distribution P r ( M | t ) in Eq. (22). Furthermore,one can easily check that that using P r ( M | t ) from Eq.(22), we get the same form of moments at large time asgiven in Eq. (19). B. Diffusion with drift
We now consider a particle diffusing in one dimensionin presence of a constant drift v ( ≥ M for this process with dy-namics in Eq. (7). To this aim, we begin with Z r ( M, k | s )in Eq. (5) for which we need the following two quantities ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● Theory ● Simulation P r ( M | t ) ●●●●●●●●●●●●●●●●●●●●●●●●●● Theory ● Simulation t m P r ( t m | t ) FIG. 3.
Top panel: plot for the distribution of maximum P r ( M | t ) in Eq. (22) (in solid line) for the simple diffusionagainst the numerical simulations (in marker). For this plot,we fix: r = 2 and t = 20. Bottom panel: comparison betweenthe asymptotic form of P r ( t m | t ) in Eq. (49) for the simplediffusion with with the numerical distribution. Since the the-ory works only in the large t m limit, we see a deviation when t m is small. Parameters set for this plot are: r = 1 , t = 5. [35]:¯ S (0 , s | M ) = 1 s (cid:16) − e − M ( √ v +2 s − v ) (cid:17) , (23) Z ( M, k | s ) = √ v + 2 s − vs e − M (cid:16) √ v +2( s + k ) − v (cid:17) . (24)Substituting these two equations in Eq. (5) gives us Z r ( M, k | s ) which can then be suitably used to computethe distribution of M and t m .Let us first look at the statistics of M for drifted dif-fusion for which we put k = 0 in Z r ( M, k | s ) in Eq. (5)along with ¯ S (0 , s | M ) and Z ( M, k | s ) in Eqs. (23) and(24) respectively. This yields Z r ( M, k = 0 | s ) = ( s + r )[ (cid:112) v + 2( s + r ) − v ][ s + re − M (cid:16) √ v +2( s + r ) − v (cid:17) ] × e − M (cid:16) √ v +2( s + r ) − v (cid:17) . (25)As done for the Brownian motion, we first analyse themoments of M ( t ) followed by the distribution. UsingEq. (25), it is easy to see that the n -th order moment is given by (cid:90) ∞ dte − st (cid:104) M n ( t ) (cid:105) = (cid:90) ∞ dM Z r ( M, k = 0 | s ) , (26)= − n !( s + r ) Li n (cid:0) − rs (cid:1) sr [ (cid:112) v + 2( s + r ) − v ] n , (27)= − n !( s + r ) (cid:80) ∞ k =1 ( − r ) k k n s k sr [ (cid:112) v + 2( s + r ) − v ] n , (28)where Li n (cid:0) − rs (cid:1) is the PolyLog function and its seriesrepresentation Li n ( − y ) = (cid:80) ∞ k =1 ( − y ) k k n has been used inEq. (27). The Laplace transform in Eq. (28) can beinverted exactly as illustrated in the appendix F. The n -th moment obeys the scaling form (cid:104) M n ( t ) (cid:105) = (cid:18) t (cid:19) n/ H n (cid:18) rt, − v √ t √ (cid:19) , (29)where the scaling function H n ( z, y ) is given by H n ( z, y ) = ( − n nz ∞ (cid:88) k =1 ( − z ) k k n (cid:90) dwe − ( z + y ) w (1 − w ) k − Γ( k ) (cid:20) z − wk (cid:21) (cid:20) δ n, √ πw − d n − dy n − (cid:16) ye y w Erfc( y √ w ) (cid:17)(cid:21) , (30)where Erfc( z ) = 1 − Erf( z ) is the complementary errorfunction. The scaling function H n ( z, y ) can be simpli-fied further for some values of n . For example, one canperform the summation over k for n = 1 to get H ( z, y ) = | y | z [ γ E + Γ (0 , z ) + log z ]+ (cid:90) dwz J w ( z, y )(1 − w ) (cid:104) − e (1 − w ) z (cid:105) , with (31) J w ( z, y ) = e − ( z + y ) w √ πw + (cid:112) z + y Erf (cid:16)(cid:112) ( z + y ) w (cid:17) . (32)However, for arbitrary n , the scaling function is given byEq. (30). In Figure 4, we have plotted the first threemoments of M ( t ) and also compared with the numericalsimulations to find an excellent match. Next, we look atthe asymptotic forms of (cid:104) M n ( t ) (cid:105) to study the effect of re-setting. For small z , we consider Eq. (30) and perform adirect expansion in z to obtain the behaviour of H n ( z, y ).On the other hand, for large z , the scaling function goesas H n ( z, y ) (cid:39) log n z and thus, the asymptotic forms read H n ( z, y ) (cid:39) n ( − n C n ( y ) + O ( z ) , as z → , (33) (cid:39) log n z (cid:16)(cid:112) y + z − y (cid:17) n as z → ∞ . (34) ●●●●●●●●●●●●●●●●●●●●●●●●● Theory ● Simulation 〈 M ( t ) 〉 ( a ) ●●●●●●●●●●●●●●●●●●●●●●●●● Theory ● Simulation 〈 M ( t ) 〉 ( b ) ●●●●●●●●●●●●●●●●●●●●●●●●● Theory ● Simulation 〈 M ( t ) 〉 ( c ) FIG. 4. Comparison of (cid:104) M n ( t ) (cid:105) in Eq. (29) for the drift-diffusion process against the numerical simulations for (a) n = 1 , (b) n = 2 and (c) n = 3. For all the plots, we have chosen r = 1 , v = 0 . The function C n ( y ) in Eq. (33) is given by C n ( y ) = (cid:90) dwe − y w d n − dy n − (cid:104) ye y w Erfc( y √ w ) (cid:105) , (35)for n (cid:54) = 1 , = y Erfc( y ) − e − y √ π − Erf( y )2 y , for n = 1 . (36)Inserting the forms of H n ( z, y ) from Eqs. (33) and (34) inEq. (29), we find that (cid:104) M n ( t ) (cid:105) has the following asymp-totic forms (cid:104) M n ( t ) (cid:105) = ( − n n C n (cid:32) − v (cid:114) t (cid:33) t n/ + O ( rt ) , for t (cid:28) r − , (37)= log n rt (cid:0) √ v + 2 r − v (cid:1) n + O (cid:0) log n − rt (cid:1) , for t (cid:29) r − . (38)At leading order, (cid:104) M n ( t ) (cid:105) for t (cid:28) r − matches with thedrift-diffusion without resetting [35]. Also, for t (cid:29) r − we find that the maximum scales logarithmically withtime as M ( t ) ∼ log( rt ). Thus even though the posi-tion density reaches a non-equilibrium steady state atlate times [49], the maximum M ( t ) keeps growing albeitslowly in logarithmical scale. This behavior can also beunderstood heuristically from the extreme value statis-tics. To illustrate this, we look at the distribution of M ( t ) at t (cid:29) r − for which we analyse Z r ( M, k = 0 | s ) inEq. (25) for r (cid:29) s . Approximating s + r (cid:39) r in Eq. (25),we get Z r ( M, k = 0 | s ) (cid:39) r [ √ v + 2 r − v ] e − M ( √ v +2 r − v )[ s + re − M ( √ v +2 r − v )] . (39)Performing the inverse Laplace transformation with re-spect to s gives the distribution of M at large t as P r ( M | t ) = rtαe − Mα exp (cid:0) − rte − Mα (cid:1) , for t (cid:29) r − , (40)which is a Gumbel distribution and α = √ v + 2 r − v .In Fig. 5 (top panel), we have compared the distribution P r ( M | t ) in Eq. (40) with the same obtained from the nu-merical simulations. We observe an excellent agreement.Note that using this form of P r ( M | t ), it is straightfor-ward to reproduce the n -th moment as given in Eq. (38).Again, the appearance of the Gumbel distribution for M ( t ) in Eq. (40) can be understood from the EVS asdone for the simple diffusion. IV. STATISTICS OF ARG-MAX t m In this section, we will present the results for statisticsof the arg-max i.e., the time t m at which the maximum M occurs. The starting point would be again to considerthe joint distribution Z r ( M, k | s ) of M and t m in Laplacespace given in Eq. (5). Next, we would integrate out M to obtain an expression for the marginal distribution Z r ( k | s ). We will analyze this quantity to characterize t m for simple diffusion and then for drift-diffusion process. A. Simple diffusion
We now look at the statistical properties of t m for Brownian motion with resetting. For free Brow-nian motion, the distribution of t m is P ( t m | t ) = π √ t m ( t − t m ) whose cumulative exhibits the ‘Arc-sine’ law(see Eq. (9)). To investigate how the statistical prop-erties are influenced due to resetting, we first consider Z r ( M, k | s ) in Eq. (5) and integrate it over M . Theresultant function Z r ( k | s ) = (cid:82) ∞ dM Z r ( M, k | s ) givesthe double Laplace transformation of the distribution P r ( t m | t ) with respect to t m ( → k ) and t ( → s ). Inserting Z ( M, k | r + s ) and ¯ S (0 , s | M ) from Eqs. (10) and (11)in Z r ( M, k | s ) in Eq. (5) and performing the integrationover M , we get Z r ( k | s ) = (cid:90) ∞ dM (cid:112) r + s ) e − √ r + s + k ) M (cid:16) s + re − √ r + s ) M (cid:17) × ( r + s + k ) (cid:16) s + k + re − √ r + s + k ) M (cid:17) . (41) Doing a change of variable: w = e − √ r + s + k ) M in theabove equation gives Z r ( k | s ) = (cid:90) dw (cid:112) ( r + s )( r + s + k )( s + rw φ ) ( s + k + rw ) , (42)where φ = (cid:113) r + sr + s + k . We next use Z r ( k | s ) from Eq. (42)to compute the moments of t m explicitly at all time. The n -th order moment (cid:104) t nm ( t ) (cid:105) can be written in terms of Z r ( k | s ) as (cid:90) ∞ dt e − st (cid:104) t nm ( t ) (cid:105) = ( − n (cid:20) d n dk n Z r ( k | s ) (cid:21) k =0 . (43)Inserting Z r ( k | s ) from Eq. (42) in Eq. (43), one cananalyse the individual moments of t m ( t ) although gettinga closed form for the n -th order moment turns out to bedifficult. Below, we provide the exact expression for thefirst two moments of t m : (cid:104) t m ( t ) (cid:105) = 14 r (cid:2) rt + e − rt − γ E + Γ (0 , rt ) + log( rt ) (cid:3) , (44) (cid:104) t m ( t ) (cid:105) = 124 r (cid:34) e − rt (9 + 5 rt ) − rt (1 + 2 rt ) + 4 − γ E −
3Γ (0 , rt ) − rt )+4 rt (cid:110) − γ E + Γ (0 , rt ) + log( rt ) (cid:111) + e − rt (cid:110) − − γ E + 3Chi( rt ) + 3Shi( rt ) − rt ) (cid:111) + 2 e − rt (cid:90) rt dy y e y F (cid:16) { , , } , { , , } , {− rt } (cid:17)(cid:35) , (45)where Chi( z ) and Shi( z ) stands for hyperbolic cosineintegral and hyperbolic sine integral respectively and F (cid:16) { , , } , { , , } , { z } (cid:17) is the generalised hypergeo-metric function [95]. Expressions of (cid:104) t m ( t ) (cid:105) and (cid:104) t m ( t ) (cid:105) are derived in Appendix C. In Figure 6, we have plot-ted the first two moments of t m ( t ) and compared themagainst the same obtained from the numerical simula-tions. An excellent match is observed.To analyse the consequences of resetting, we study themoments of t m ( t ) at short and large times. It turns outthat for these times, one can write a closed form for the n -th order moment. Starting from Z r ( k | s ) in Eq. (42),we perform simplifications for the cases sr − → sr − → ∞ which in the time domain correspond to t (cid:29) r − and t (cid:28) r − respectively. We then use Eq. (43) tocompute the moments for these cases. In order to avoiddiscontinuity of the presentation, we have relegated thesedetailed calculations to appendix D and present only the final results here. The moments read (cid:104) t nm ( t ) (cid:105) (cid:39) (2 n − t n n !( n − n − + A n γt n +1 , for t (cid:28) r (46) (cid:39) t n n + 1 + t n − log( rt )2 r ( n + 1) , for t (cid:29) r , (47)where A n in the first line is given in Eq. (D6). For t (cid:28) r ,we recover the results of the free Brownian motion [31].Moreover, we observe that the leading order scaling ofthe n -th order moment with respect to t for both largeand small times are same i.e. (cid:104) t nm ( t ) (cid:105) ∼ t n . However,the corresponding prefactors are different. As we illus-trate later that the leading order behaviour of (cid:104) t nm ( t ) (cid:105) at t (cid:29) r is independent of the underlying stochastic processas long as every reset event renews the process. Contrar-ily, the sub-leading terms are sensitive to the underlyingstochastic process and resetting rate.In the remaining part of this section, we analyse thedistribution of t m which we denote by P r ( t m | t ). To com-pute this distribution, we have to perform the doubleinverse Laplace transformation of Z r ( k | s ) in Eq. (42)with respect to k and s . Performing inversion for arbi-trary values of s and k turns out to be difficult. How-ever, one can make some analytic progress by analysing Z r ( k | s ) in various limits of kr − and sr − . For r (cid:29) s and r (cid:29) k , we approximate φ (cid:39) Z r ( k | s ) in Eq. (42) to get Z r ( k | s ) (cid:39) (cid:90) dw (cid:112) ( r + s )( r + s + k )( s + rw ) ( s + k + rw ) . (48)Fortunately, one can now invert this double Laplacetransform to get the distribution P r ( t m | t ) [see appendixE]. Performing the inverse Laplace transformation, weget P r ( t m | t ) (cid:39) e − rt [ g ( rt m ) + g ( r ( t − t m )) − π (cid:112) t m ( t − t m )+ re − rt (cid:90) dw w e rtw Erf (cid:0) √ rwt m (cid:1) × Erf (cid:16)(cid:112) rw ( t − t m ) (cid:17) , (49)where g ( z ) = F ( { , } , { / , } , z ). Note that this ex-pression is valid only in the limits rt → ∞ and rt m → ∞ .In Fig. 3 (bottom panel), we have plotted P r ( t m | t ) andcompared it against the numerical simulations. Whileour analytic result is consistent with the simulation dataat large t m , the match is poor at small t m . This devia-tion stems from the fact that Z r ( k | s ) in Eq. (48) is valid ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● Theory ● Simulation P r ( M | t ) ●●●●●●●●●●●●●●●●●●●●●●●●●● Theory ● Simulation t m P r ( t m | t ) FIG. 5.
Top panel: comparison of the distribution P r ( M | t )in Eq. (40) for the drift-diffusion process against the numer-ical simulations. For this plot, we have taken: r = 2 , v = 1and t = 20. Bottom panel: here we have compared the dis-tribution P r ( t m | t ) in Eq. (61) for the drift-diffusion to thenumerical simulations. As before, the data fits the theoryperfectly in the large t m limit. Parameters chosen for thissimulation: r = 1 . t = 5. only for r (cid:29) k which translates to t m (cid:29) r − in the timedomain.We now look at P r ( t m | t ) when t m (cid:54) = t for whichEq. (49) can be simplified further. Approximating F ( { , } , { / , } , z ) (cid:39) (cid:112) πz e z and Erf( z ) (cid:39) z → ∞ and using them in (49), we find that the leadingorder behaviour of P r ( t m | t ) (cid:39) t . To get the sub-leadingterms of P r ( t m | t ), one has to consider higher order termson rs − while performing simplifications in Z r ( k | s ) inEq. (48). We refer to appendix D [see Eq. (D10)] for thederivation of the sub-leading corrections and present theresults here such that P r ( t m | t ) (cid:39) t + log( rt )2 rt . (50)Using this form of P r ( t m | t ), it is easy to verify that the n -th order moment of t m ( t ) is indeed given by Eq. (47).It is worth remarking that the t form of P r ( t m | t ) is quitedifferent than the form of distribution of t m without resetwhich is given by P ( t m | t ) = π √ t m ( t − t m ) . We later showthat t form of P r ( t m | t ) under resetting is independent ofthe underlying stochastic process as long as the processforgets about its prior history after every reset event. B. Diffusion with drift
We start again by inserting Z ( M, k | r + s ) and¯ S (0 , s | M ) for drift diffusion process from Eqs. (23)-(24) in Eq. (5) to obtain the following expression for Z r ( M, k | s ) given by Z r ( M, k | s ) = B s + r ( r + s + k ) e − M B s + r + k [ s + re − M B s + r ] [ s + k + re − M B s + k + r ] , (51)where B s = √ v + 2 s − v . Now performing the integra-tion over M and after some algebraic simplifications, wefind Z r ( k | s ) = (cid:112) v + 2( s + r ) − v (cid:112) v + 2( s + k + r ) − v × (cid:90) dw ( r + s + k )( s + rw φ d )( s + k + rw ) , (52)with φ d = √ v +2( s + r ) − v √ v +2( s + k + r ) − v . We can now use Eq. (43) toget the expression for the moments in the Laplace space.However, the inversion process to obtain the moments inreal time becomes quite tedious. Here, we just presentthe first moment for brevity. The first moment in Laplacespace reads (using Eq. (52) in Eq. (43))0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● SimulationTheory 〈 t m ( t ) 〉 ( a ) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● SimulationTheory 〈 t m ( t ) 〉 ( b ) FIG. 6. Comparison of the first two moments of t m in Eqs. (44) and (45) against the same obtained from the numericalsimulations for the simple diffusion. We have set r = 1 for the simulation purpose. (cid:90) ∞ dt e − st (cid:104) t m ( t ) (cid:105) = ˜ I ( s ) + ˜ I ( s ) , with (53)˜ I ( s ) = 2 r + s + s v √ v +2( r + s ) s ( r + s ) , (54)˜ I ( s ) = (cid:34) v (cid:112) v + 2( r + s ) (cid:35) rs log (cid:18) r + ss (cid:19) . (55) To get (cid:104) t m ( t ) (cid:105) , we use the following inverse Laplace trans-formation: I ( t ) = L − s → t (cid:104) ˜ I ( s ) (cid:105) = 14 r (cid:32) rt + e − rt Erfc (cid:20) v √ t √ (cid:21) + v √ v + 2 r Erf (cid:34) (cid:112) ( v + 2 r ) t √ (cid:35) − (cid:33) , (56) I ( t ) = L − s → t (cid:104) ˜ I ( s ) (cid:105) = J ( t ) + (cid:90) t dτ J ( τ ) ve − v r ( t − τ ) (cid:112) π ( t − τ ) , (57)where J ( t ) = 14 r [ γ E + Γ (0 , rt ) + log( rt )] . (58)Inserting these inverse Laplace transforms in Eq. (53),we get (cid:104) t m ( t ) (cid:105) = I ( t ) + I ( t ) . (59)One can also proceed to compute the higher moments ina similar manner, but the expressions are quite involved.Hence, we do not present them here. In Figure 7, wehave compared (cid:104) t m ( t ) (cid:105) in Eq. (59) with the numericalsimulations to find a perfect agreement between them.We now turn our attention to analyze P r ( t m | t ) for thedrift-diffusion process. Similar to the simple diffusion,performing an inversion for arbitrary values of s and k turns out to be difficult. Thus, we make the approxima-tions r (cid:29) s and r (cid:29) k to have φ d (cid:39)
1, and Eq. (52) simplifies to Z r ( k | s ) (cid:39) (cid:112) v + 2( s + r ) − v (cid:112) v + 2( s + k + r ) − v × (cid:90) dw ( r + s + k )( s + rw ) ( s + k + rw ) . (60)It is possible to perform the double Laplace inversion (seeAppendix G) to eventually arrive at P r ( t m | t ) (cid:39) I ( t m , t ) + I ( t m , t ) + I ( t m , t ) , (61)where I -functions are given in Eqs. (G8-G10). Noteagain that this expression is only valid in the limits of rt → ∞ and rt m → ∞ . We verify this result in Fig 5(bottom panel). One can again approximate the aboveexpression in large time (like we have shown in the case1of simple diffusion) to find the leading order behavior: P r ( t m | t ) (cid:39) t , which is again independent of the pro-cess details. However, one would expect that there willbe some correction (sub-leading) terms to this leadingbehavior. To find them, we first perform the small k expansion of φ d to obtain φ d (cid:39) − b ( r, v ) k , with b ( r, v ) = 1 √ v + 2 r ( √ v + 2 r − v ) . (62)Further, we approximate w φ d = e φ d log w (cid:39) e (1 − bk ) log w (cid:39) w (1 − bk log w ). Moreover, we also take the large t , large t m limit so that s + r (cid:39) r and s + r + k (cid:39) r and thus Z r ( k | s ) = r (cid:90) dw ( s + rw − rbwk log w )( s + k + rw ) . (63)We now first take the inverse Laplace transform withrespect to k → t m , and then we take the inverse Laplacetransform with respect to s → t to find P r ( t m | t ) = (cid:90) dw re − rtw brw log w (cid:20) − Θ (cid:18) t + t m brw log w (cid:19)(cid:21) . (64)Let us now do the following transform rtw = y and thenwe have w log w = yrt log yrt (cid:39) − yrt log rt for rt (cid:29) . (65)The argument inside Θ function in Eq. (64) becomes neg-ative, and hence this term does not contribute. Eventu-ally, we are left with P r ( t m | t ) = r (cid:90) rt dyrt e − y − a ( r, v ) y , (66)where a ( r, v ) = b ( r, v ) r log rtrt (cid:28) rt (cid:29)
1. Ex-panding in small a argument and also setting the upperlimit to + ∞ , we get P r ( t m | t ) (cid:39) t + a ( r, v ) t , (cid:39) t + b ( r, v ) log rtt , (67)where b ( r, v ) is given by Eq. (62). Comparing this expres-sion with that of the simple diffusion in Eq. (50), we seethat even though the leading order behaviour of P r ( t m | t )in both cases is 1 /t , the sub-leading terms are rather dif-ferent. The subleading term in Eq. (67) depends on thedrift v (limit for simple diffusion can be checked easily bynoting b ( r, v ) = 1 / r ). Finally, the limiting distributionin Eq. (67) also gives the moments at large t which read (cid:104) t nm (cid:105) (cid:39) t n n + b ( r, v ) t n − log rtr (1 + n ) , t (cid:29) r , (68) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Theory ● Simulation 〈 t m ( t ) 〉 FIG. 7. Comparison of (cid:104) t m ( t ) (cid:105) in Eq. (59) with the numer-ical simulations. We have chosen r = 1 and v = 1 for thesimulation purpose. where again the logarithmic correction to the universalform of the moments is observed. So far, we have pre-sented rigorous results for the statistics of t m for twocanonical models namely the simple diffusion followedby the drift-diffusion process. The large time universalform P r ( t m | t ) (cid:39) t is noteworthy in both cases. In thefollowing section, we present general arguments to showwhy this is indeed a robust and universal characteristicfor the arg-max for generic stochastic processes. V. LIMITING DISTRIBUTION OF t m FORGENERAL STOCHASTIC PROCESS
In the previous sections, we observed that the den-sity of arg-max converges to a uniform distribution ofthe form P r ( t m | t ) (cid:39) /t when t, t m (cid:29) r − and t m (cid:54) = t for both diffusion and drifted diffusion. Naturally, thequestion arises what are the ramifications of resetting toother stochastic processes. Here, we show that the 1 /t -form of P r ( t m | t ) is completely universal and independentof the underlying stochastic process as long as the pro-cess starts afresh after every reset event. To prove this,we analyse our main formula for Z r ( M, k | s ) in Eq. (5)in the limit of large rt . Note that the survival probabil-ity S (0 , M | t ) for any stochastic process is bounded as0 ≤ S (0 , M | t ) ≤ t . This bound suggests thatthe Laplace transform ¯ S (0 , M | s ) can be written as¯ S (0 , M | s ) = 1 s − s U ( M, s ) , (69)where U ( M, s ) is a general function with the constraintsthat we discuss in the following. Since ¯ S (0 , M | s ) ≥
0, wehave U ( M, s ) ≤
1. Furthermore ¯ S (0 , M | s ) ≤ /s , whichin terms of U ( M, s ) becomes U ( M, s ) ≥
0. Also, weexpect that the particle will survive without getting ab-sorbed at x = M → ∞ which implies ¯ S (0 , M → ∞| s ) (cid:39) /s or equivalently U ( M → ∞ , s ) (cid:39)
0. In what follows,we use Eq. (69) along with these constraints to analyze Z r ( M, k | s ) given in Eq. (5).2 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ t m P r ( t m | t ) FIG. 8. Comparison of P r ( t m | t ) in Eq. (75) (shown by solidgreen line) with the simulation results for five different un-derlying stochastic processes, namely (i) diffusion (cyan), (ii)diffusion with positive drift (blue), (iii) diffusion with neg-ative drift (red), (iv) random acceleration (black) and (v)Ornstein-Uhlenbeck process (magenta). While (i)-(iii) & (v)are Markov processes, (iv) is a non-Markov one. Simulationsare conducted for the parameters t = 1 and r = 20 in all thecases (see Sec. H for more details on the model systems). Inserting ¯ S (0 , M | s ) from Eq. (69) in Eq. (5), we get Z r ( M, k | s ) = ( r + s )( r + s + k ) Z ( M, k | r + s )[ s + r U ( M, r + s )] [ s + k + r U ( M, r + s + k )] . (70)For s → k →
0, we approximate Z r ( M, k | s ) as Z r ( M, k | s ) (cid:39) r [ s + r U ( M, r )] [ s + k + r U ( M, r )] d ¯ S (0 , M | r ) dM , (71)where we have used Z ( M, k = 0 | r ) = d ¯ S (0 ,M | r ) dM . Thisstems from the fact that S (0 , M | t ) is the cumulative dis-tribution of M ( t ) i.e., S (0 , M | t ) = Prob [ M ( t ) ≤ M ] . (72)We now perform the double inverse Laplace transforma-tion of Eq. (71) with respect to k and s to get the jointdistribution P r ( M, t m | t ) for t, t m (cid:29) r − and t m (cid:54) = t . Theexpression of P r ( M, t m | t ) reads P r ( M, t m | t ) (cid:39) r e − rt U ( M,r ) d ¯ S (0 , M | r ) dM . (73)Finally to get the distribution of t m , we integrate P r ( M, t m | t ) for all M to yield P r ( t m | t ) (cid:39) t (cid:104) e − rt U ( M →∞ ,r ) − e − rt U ( M → ,r ) (cid:105) . (74)Using U ( M → ∞ , r ) = 0 and U ( M → , r ) ≥
0, we get P r ( t m | t ) (cid:39) t . (75) In Figure 8, we have compared our analytic expressionfor P r ( t m | t ) (given in Eq. (75)) to the results of thenumerical simulations for five different stochastic pro-cesses namely diffusion, diffusion with positive and neg-ative drift, random acceleration and Ornstein Uhlenbeckprocess. It is important to remark that random acceler-ation process (RAP) is a non-Markov process while theothers are Markovian in nature. We observe an excellentagreement between our analytical prediction and the sim-ulations in all the cases (we have consigned the detailsof simulation in Appendix H). As stressed before, thisresult (75) is indeed independent of the nature of theunderlying stochastic process.We end this section by discussing the origin of thisuniversal limiting distribution for t m . Consider a longtrajectory x ( τ ) with 0 ≤ τ ≤ t with many resetting in-tervals. However, these intervals are statistically inde-pendent since the entire configuration (all the variables)is renewed after each resetting event, and hence there areno correlations between the intervals. Hence, we can mapour problem to a simple example of a discrete time inter-vals { τ , τ , · · · , τ N − } of N − p ( τ ). As the observation time t becomeslarge, effect of the last interval τ N becomes negligiblei.e., t − τ N ≈ t . Note that the maximum M can bein any one of these N intervals with equal probability1 /N for a given N . However, N is a random variablein a fixed time interval t . In fact, N is a Poisson pro-cess with P ( N ) = ( rt ) N N ! e − rt . Taking the average, we findthat arg-max t m converges to a uniform distribution atlarge t namely Eq. (75). Thus, this result is completelyuniversal, i.e., independent of the PDF p ( τ ) and the un-derlying stochastic motion at large time. The essentialkey point in this derivation is that resetting makes the in-tervals completely independent to each other. Thus, eventhough for simple process like Brownian motion (withoutresetting), computation of P ( t m | t ) can not be made us-ing this simple argument, resetting simplifies the problemelegantly in many folds and the resulting density pertainsto a uniform distribution as we have already shown. VI. CONCLUSION
In summary, we have extensively studied statistics ofthe maximum distance M ( t ) and the time t m ( t ) taken toreach this maximum distance (upto an observation time t ) by a stochastic process which is subject to a resettingmechanism. Resetting occurs at a constant rate r whichreinstates the particle back to its initial position inter-mittently. The process is renewed after each resettingevent, and the memory from the previous trial is erased.Utilizing this key property, we derive a renewal formula(Eq. (5)) for the joint distribution of M ( t ) and t m in thepresence of resetting ( r >
0) in terms of the same butwith r = 0. Our derivation is quite generic and holds forboth Markov and non-Markov underlying process.Next, we use Eq. (5) and marginalize it to study statis-3tics of M ( t ) and t m ( t ) respectively. We choose simplediffusion and diffusion with drift as the underlying pro-cess and add resetting to them. In the case of simplediffusion, we explicitly computed all moments of M ( t )from which we showed that they have logarithmic timedependence at large time. Surprisingly, although the po-sition density converge to time independent steady state, (cid:104) M n ( t ) (cid:105) grows with time but rather slowly with logarith-mic dependence (see also [93]). Our results are consistentto demonstrate that the limiting distribution of M ( t ) be-longs to a Gumbel class which was recently understoodfrom the EVS theory of weakly interacting random vari-ables [8, 45, 46]. For the drift-diffusion case, we havealso computed all moments of M ( t ). As in the case ofsimple diffusion, we again observe the logarithmic growthfor (cid:104) M n ( t ) (cid:105) despite the fact that drift-diffusion processreaches a steady state in the presence of resetting. Fi-nally, we do a consistency check to show that the distri-bution of the maximum reproduces the Gumbel law asexpected from the EVS theory.We then turn our attention to the statistics of the arg-max t m . We first derived the moments generating func-tion for t m ( t ) in the Laplace space for simple diffusionwith reset in Eq. (42). This allowed us to compute thefirst two moments exactly for all t . We next show thatthe arg-max density at large time converges to a uni-form distribution which only depends on the observationtime t but not on the specifics of the underlying process.The sub-leading correction terms are shown to be processdependent. For the drift-diffusion process, computationof the higher order moments beyond the first one be-comes quite tedious however at large time we extractedthe leading contributions with logarithmic sub-leadingterms. The density at large time is again uniform withprocess dependent correction terms.Borrowing wisdom from these exact results, we nextanalyzed the density of t m for generic stochastic pro-cesses subject to stochastic resetting with a rate r . Fol-lowing an asymptotic expression for the joint distribu-tion of M ( t ) and t m ( t ), we show that indeed at largetime P r ( t m | t ) (cid:39) t which is independent of the underly- ing process but the sub-leading terms are naturally pro-cess dependent as demonstrated for simple diffusion anddrift-diffusion. We provide a probabilistic interpretationof the result based on the renewal property of the reset-ting phenomena. Numerical simulations covering manyof the above-mentioned processes are in perfect agree-ment with our analytical predictions.Estimating the probability of extreme events is an im-portant problem in statistics as well as in statistical,mathematical and condensed matter physics and in otherinterdisciplinary subjects. Despite its paramount rele-vance, EVS poses serious computational challenges andexact results are scarce. In this paper, we showcase onesuch example for a renewal process namely stochasticresetting which has gained immense interest in recenttimes. Although the subject has been very dynamic fromthe perspective of non-equilibrium transport propertiesor the first passage estimation, obtaining exact resultsand limiting distributions related to EVS has been verylimited. Finding such results can be often subtle due tothe complexity inherent to the systems. However, draw-ing inspirations from the canonical model systems suchas simple diffusion and diffusion with drift, we have beenable to unravel some of the universal features of extremalsin stochastic resetting systems. We conclude by statingthat the scope of the current formalism is not restrictedonly to instantaneous resetting but also can accommo-date scenarios when returns are spatio-temporally corre-lated [65, 96–100]. ACKNOWLEDGMENTS
Prashant Singh acknowledges useful discussions dur-ing ICTS-TIFR programs ‘BSSP X’ and ‘Fluctuationsin Nonequilibrium Systems: Theory and Applications’.Arnab Pal gratefully acknowledges support from the Ray-mond and Beverly Sackler Post-Doctoral Scholarship andthe Ratner Center for Single Molecule Science at Tel-Aviv University.
Appendix A: Derivation of the renewal formula in Eq. (5)
In this section, we present the derivation of the renewal formula in Eq. (5) which was presented in the main text.For brevity, we recall that the maximum M can occur in any of the N -intervals. Following the main text, these4contributions are given by C = (cid:32) N (cid:89) i =1 (cid:90) ∞ dτ i (cid:33) (cid:90) ∞ dτ P ( M, τ | τ ) p ( τ ) (cid:34) N − (cid:89) i =2 S (0 , τ i | M ) p ( τ i ) (cid:35) (cid:2) e − rτ N S (0 , τ N | M ) (cid:3) δ ( t − N (cid:88) i =1 τ i ) δ ( τ − t m ) , C = (cid:32) N (cid:89) i =1 (cid:90) ∞ dτ i (cid:33) (cid:90) ∞ dτ P ( M, τ | τ ) p ( τ ) N − (cid:89) i =1 ,i (cid:54) =2 S (0 , τ i | M ) p ( τ i ) (cid:2) e − rτ N S (0 , τ N | M ) (cid:3) δ ( t − N (cid:88) i =1 τ i ) δ ( τ + τ − t m ) , C = · · ·C = · · ·· · ·C N = (cid:32) N (cid:89) i =1 (cid:90) ∞ dτ i (cid:33) (cid:90) ∞ dτ P ( M, τ | τ N ) e − rτ N (cid:34) N − (cid:89) i =1 S (0 , τ i | M ) p ( τ i ) (cid:35) δ ( t − N (cid:88) i =1 τ i ) δ ( N − (cid:88) i =1 τ i + τ − t m ) . (A1)To obtain joint distribution P r ( M, t m , N | t ), one has to sum all contributions C , C , ..., C N . Inserting the contributionsfrom Eqs. (A1) and noting that e − rτ N = r p ( τ N ) with p ( τ i ) = re − rτ i , the joint distribution P r ( M, t m , N | t ) can beformally written as P r ( M, t m , N | t ) = 1 r N (cid:88) j =1 (cid:90) ∞ dτ j dτ P ( M, τ | τ j ) p ( τ j ) N (cid:89) j (cid:48) =1 ,j (cid:48) (cid:54) = j (cid:90) ∞ dτ j (cid:48) S (0 , τ j (cid:48) | M ) p ( τ j (cid:48) ) dτ j (cid:48) × δ (cid:32) j − (cid:88) i =1 τ i + τ − t max (cid:33) δ ( t − N (cid:88) i =1 τ i ) , (A2)where the second delta function ensures that the total observation time is t . To simplify this expression, it is usefulto perform Laplace transformations with respect to t ( → s ) and t m ( → k ). Denoting the Laplace transformation of P r ( M, t m , N | t ) by Z r ( M, k, N | s ), we take the Laplace transformation of Eq. (A2) to yield Z r ( M, k, N | s ) = N (cid:88) m =1 Z ( M, k | r + s ) (cid:2) r ¯ S (0 , s + r + k | M ) (cid:3) m − (cid:2) r ¯ S (0 , s + r | M ) (cid:3) N − m , (A3)= Z ( M, k | r + s ) r (cid:2) ¯ S (0 , s + r | M ) − ¯ S (0 , s + r + k | M ) (cid:3) (cid:104)(cid:0) r ¯ S (0 , s + r | M ) (cid:1) N − (cid:0) r ¯ S (0 , s + r + k | M ) (cid:1) N (cid:105) . (A4)In the first line, we have used the notation ¯ S (0 , s, M ) as the Laplace transformation of S (0 , t | M ). To get the jointdistribution of M and t m in the Laplace space of k and s , we sum Z r ( M, k, N | s ) for all values of N from 1 to ∞ which results in Z r ( M, k | s ) = ∞ (cid:88) N =1 Z r ( M, k, N | s ) , = Z ( M, k | r + s ) (cid:2) − r ¯ S (0 , s + r | M ) (cid:3) (cid:2) − r ¯ S (0 , s + r + k | M ) (cid:3) , (A5)where in going to the second line from the first line, we have substituted Z r ( M, k, N | s ) from Eq. (A4). This concludesthe proof for Eq. (5) in the main text. Appendix B: Asymptotic forms of (cid:104) M n ( t ) (cid:105) forBrownian motion In this appendix, we derive the asymptotic behaviourof (cid:104) M n ( t ) (cid:105) for Brownian motion whose exact expressionis given in Eq. (15). Below, we look at the behaviour of (cid:104) M n ( t ) (cid:105) for large and short times separately.
1. Case I: (cid:104) M n ( t ) (cid:105) for t (cid:29) r To begin with, we consider the Laplace transformationof P r ( M | t ) from Eq. (12) and rewrite here as Z r ( M, k = 0 | s ) = √ r + s ) / e − √ r + s ) M (cid:16) s + re − √ r + s ) M (cid:17) . (B1)5For small r (cid:29) s (which corresponds to large r (cid:28) t − ),we approximate r + s (cid:39) s in Eq. (B1) to yield Z r ( M, k = 0 | s ) (cid:39) √ r / e −√ rM (cid:16) s + re −√ rM (cid:17) . (B2)To get the distribution in the time domain, we performthe inverse Laplace transform of Eq. (B2) by using L − s → t (cid:104) s + a ) (cid:105) = te − at for a ≥ P r ( M | t ) (cid:39) √ r te −√ rM exp (cid:16) − rte −√ rM (cid:17) . (B3)We emphasise that the approximate equality in this equa-tion indicates that it is valid only for t (cid:29) r . We use thisform of the distribution to get the moments at large t as (cid:104) M n ( t ) (cid:105) = (cid:90) ∞ dM M n P r ( M | t ) , (B4) (cid:39) r ) n/ log n ( rt ) + O (cid:0) log n − ( rt ) (cid:1) , (B5)where in going to the second line, we have inserted P r ( M | t ) from Eq. (B3) and performed the integrationover M . Comparing Eq. (B5) with the scaling form inEq. (15), we find that the scaling function H n ( z ) at large z is given by H n ( z ) (cid:39) log n z + O (cid:0) log n − z (cid:1) . (B6)This result has been quoted in Eq. (17) of the main text.
2. Case II: (cid:104) M n ( t ) (cid:105) for t (cid:28) r To find the moments at times t (cid:28) r , we need the formof the distribution P r ( M | t ) at these times as indicatedby Eq. (B4). In the Laplace space, this will correspondto the r (cid:28) s behaviour of Z r ( M, k = 0 | s ) in Eq. (B1).By direct expansion of Eq. (B1) in r , we find Z r ( M, k = 0 | s ) (cid:39) (cid:114) s e −√ sM + r s / (cid:104)(cid:16) √ − M √ s (cid:17) × e −√ sM − √ e − √ sM (cid:105) . (B7)We next use Eq.(13) (first line) to get the moments in theLaplace space and then perform inverse Laplace inversionto yield (cid:104) M n ( t ) (cid:105) (cid:39) n !2 n Γ (cid:0) n (cid:1) t n + B n r n/ t n +1 , with (B8) B n = Γ( n + 1) (cid:16) √ − − n (cid:17) − √ n + 2)2 √ (cid:0) n +42 (cid:1) . (B9) Finally comparing this equation with the scaling form inEq. (15), we find that the scaling function H n ( z ) at small z is given by H n ( z ) (cid:39) n !Γ (cid:0) n (cid:1) z n + B n z n +1 , (B10)which is Eq. (18) in the main text. Appendix C: Derivation of (cid:104) t m ( t ) (cid:105) and (cid:104) t m ( t ) (cid:105) forBrownian motion In this appendix, we derive the exact expressions of (cid:104) t m ( t ) (cid:105) and (cid:104) t m ( t ) (cid:105) which are presented in Eqs. (44) and(45) respectively. To derive these results, we insert theexpression of Z r ( k | s ) from Eq. (42) in Eq. (43) for n = 1 , (cid:90) ∞ dte − st (cid:104) t m ( t ) (cid:105) = 2 r + 3 rs + s s ( r + s ) + log (cid:0) r + ss (cid:1) rs , (C1) (cid:90) ∞ dte − st (cid:104) t m ( t ) (cid:105) = 16 r + 36 rs + 15 s s ( r + s ) − Li (cid:0) − rs (cid:1) rs ( r + s )+ (4 r + s ) log (cid:0) r + ss (cid:1) rs ( r + s ) . (C2)Let us invert Eq. (C1) with respect to s . To this aim,we use the following inverse Laplace transformations: L − s → t (cid:34) log (cid:0) r + ss (cid:1) s (cid:35) = γ E + Γ(0 , rt ) + log rt, (C3) L − s → t (cid:20) r + 3 rs + s s ( r + s ) (cid:21) = 2 rt + e − rt − r . (C4)Using these two inverse Laplace transforms in Eq. (C1),we recover the expression of (cid:104) t m ( t ) (cid:105) as written in Eq.(44). We next look at Eq. (C2) to get (cid:104) t m ( t ) (cid:105) for whichwe need three inverse Laplace transformations. InverseLaplace transform of the first term in the RHS of Eq.(C2) is given by L − s → t (cid:20) r + 36 rs + 15 s s ( r + s ) (cid:21) = 124 r (cid:2) e − rt (9 + 5 rt ) − rt + 8 r t (cid:3) . (C5)Next, we turn to the second term in the RHS of Eq. (C2).Note that this term is the product of r + s and s Li (cid:0) − rs (cid:1) which implies that the convolution theorem for Laplacetransforms can be directly used. Doing so, we find L − s → t (cid:34) Li (cid:0) − rs (cid:1) s ( r + s ) (cid:35) = (cid:90) t dt e − r ( t − t ) f ( t ) , where(C6) f ( t ) = L − s → t (cid:34) Li (cid:0) − rs (cid:1) s (cid:35) , (C7)= − rt F (cid:16) { , , } , { , , } , {− rt } (cid:17) . (C8)6We now look at the third term in the R.H.S. of Eq. (C2).Once again, we notice that it has the product form be-cause of which we use the convolution theorem. Theinverse Laplace transform then reads L − s → t (cid:34) (4 r + s ) log (cid:0) r + ss (cid:1) rs ( r + s ) (cid:35) = (cid:90) t dt f ( t − t ) f ( t ) , (C9)where the functions f ( t ) and f ( t ) are given by f ( t ) = L − s → t (cid:20) log (cid:18) r + ss (cid:19)(cid:21) , (C10)= 1 − e − rt t , (C11) f ( t ) = L − s → t (cid:20) (4 r + s )24 rs ( r + s ) (cid:21) , (C12)= 4 rt − e − rt r . (C13)Finally inserting Eqs. (C5), (C6) and (C9) in Eq. (C2),we recover the form of (cid:104) t m ( t ) (cid:105) as quoted in Eq. (45). Appendix D: Asymptotic forms of (cid:104) t nm ( t ) (cid:105) forBrownian motion This appendix deals with the derivation of the forms of (cid:104) t nm ( t ) (cid:105) for Brownian motion when t (cid:28) r − and t (cid:29) r − .These asymptotic forms are presented in Eqs. (46) and(47). To begin with, we analyse Z r ( k | s ) in Eq. (42)in various limits of sr − using which we compute thedistribution P r ( t m | t ) for different regimes of t . We thenuse P r ( t m | t ) to compute moments. Below we considerthe cases t (cid:28) r − and t (cid:29) r − separately.
1. Case I: t (cid:28) r In the Laplace space of s , the limit t (cid:28) r correspondsto rs − → s means r →
0. Taking thelimit r → Z r ( k | s ) in Eq. (42), we get Z r ( k | s ) (cid:39) (cid:112) s ( s + k ) + rs / (cid:20) √ s + k − √ s + √ s + k (cid:21) . (D1)We next invert this Laplace transform to obtain the dis-tribution P r ( t m | t ) using which we compute moments.Note that we have to perform double inverse Laplacetransformations: one from s → t and the other from k → t m . To invert Eq. (D1), we use the following double inverse Laplace transformations: L − s → t L − k → t m (cid:34) (cid:112) s ( s + k ) (cid:35) = 1 π (cid:112) t m ( t − t m ) , (D2) L − s → t L − k → t m (cid:20) s − / ( √ s + √ s + k ) (cid:21) = 2 π (cid:20)(cid:114) t − t m t m − cos − (cid:114) t m t (cid:35) . (D3)Using these two equations in Z r ( k | s ) in Eq. (D1), weget P r ( t m | t ) (cid:39) π (cid:112) t m ( t − t m ) + rπ (cid:32) − (cid:114) t m t − (cid:114) t − t m t m (cid:33) . (D4)We emphasize that this expression is valid only in thelimit rt →
0. Finally, we use this form of P r ( t m | t ) tocompute the moments of t m which then read (cid:104) t nm ( t ) (cid:105) (cid:39) (2 n − t n n !( n − n − + A n γt n +1 , with (D5) A n = 1Γ( n + 2) (cid:20) Γ (3 / n ) √ π ( n + 1) − (2 n − n ( n − (cid:21) , (D6)which is Eq. (46) in the main text.
2. Case II: t (cid:29) r We next look at the moments when t (cid:29) r . Onceagain we begin with the expression of Z r ( k | s ) in Eq.(42) and analyse it in the limit of large r . For large r , weapproximate r + s (cid:39) r and w (cid:113) r + sr + s + k (cid:39) w − wk log w r andinsert them in Eq. (42) to yield Z r ( k | s ) (cid:39) r (cid:90) dw ( s + k + rw ) (cid:16) s + rw − wk log w (cid:17) . (D7)To get the distribution in the time domain from Eq.(D7), we now use the following standard inverse Laplacetransformation L − s → t (cid:20) e − bs s + a (cid:21) = e − a ( t − b ) Θ( t − b ) , with a, b ≥ , (D8)in Eq. (D7) to get P r ( t m | t ) (cid:39) r (cid:90) dw e − rtw w log w (cid:20) − Θ (cid:18) t + 2 t m w log w (cid:19)(cid:21) . (D9)We are now left with the integration over w . This canbe done by making the transformation rtw = y in Eq.7(D9). To proceed further, we approximate log w = log y − log rt (cid:39) − log rt and rt log rt → ∞ for rt → ∞ . Withthese approximations, the integration in Eq. (D9) canbe performed explicitly to get P r ( t m | t ) (cid:39) t + log rt rt . (D10)Using this form of P r ( t m | t ) for t (cid:29) r − , it is straightfor-ward to show the moment is given by (cid:104) t nm ( t ) (cid:105) (cid:39) t n n + 1 + t n − log( rt )2 r ( n + 1) , (D11)which has been mentioned in Eq. (47). Appendix E: Derivation of P r ( t m | t ) in Eq. (49) forBrownian motion In this appendix, we perform the double inverseLaplace transformation of Z r ( k | s ) in Eq. (48) to getthe distribution P r ( t m | t ) in Eq. (49). We first performthe inversion of Eq. (48) with respect to k for which weuse the following: L − k → t m (cid:20) √ k + bk + a (cid:21) = e − bt m √ πt m + √ b − a e − at m × Erf (cid:16)(cid:112) ( b − a ) t m (cid:17) , (E1)with a, b ≥
0. Using this in Eq. (48) by reading appro-priately a and b , we get¯ P r ( t m | s ) = ¯ Q ( t m | s ) + ¯ Q ( t m | s ) , (E2)where ¯ P r ( t m | s ) stands for the Laplace transformation of P r ( t m | t ). Also, the functions ¯ Q ( t m | s ) and ¯ Q ( t m | s ) aregiven by¯ Q ( t m | s ) = (cid:90) dw √ r + s e − ( r + s ) t m √ πt m ( s + rw ) , (E3)¯ Q ( t m | s ) = (cid:90) dw (cid:112) r ( r + s )(1 − w ) e − ( s + rw ) t m s + rw × Erf (cid:16)(cid:112) rt m (1 − w ) (cid:17) . (E4)To obtain the distribution P r ( t m | t ), we have to performinverse Laplace transformation in Eq. (E2). Let us firstperform the inversion for ¯ Q ( t m | s ) in Eq. (E3) for whichwe use the inverse Laplace transformation in Eq. (E1)with k replaced by s . Denoting the inverse Laplace trans-formation of ¯ Q ( t m | s ) by Q ( t m | t ), we get Q ( t m | t ) = e − rt π (cid:112) t m ( t − t m ) + e − rt m √ πt m (cid:90) dw (cid:112) r (1 − w ) × e − rw ( t − t m ) Erf (cid:16)(cid:112) r (1 − w )( t − t m ) (cid:17) . (E5) To perform the integration in the second term, we makethe following change in variable: y = r (1 − w )( t − t m ),which, in turn, yields Q ( t m | t ) = e − rt π (cid:112) t m ( t − t m ) (cid:20) √ πr ( t − t m ) × (cid:90) r ( t − t m )0 dy √ y e y Erf( √ y ) (cid:35) . (E6)The integration over y in the second line can be explicitlyperformed using Mathematica in terms of the generalizedhypergeometric functions. The final result reads Q ( t m | t ) = e − rt F ( { , } , { / , } , r ( t − t m )) π (cid:112) t m ( t − t m ) . (E7)We next turn to ¯ Q ( t m | s ) in Eq. (E2). For ¯ Q ( t m | s ), itturns out that one has to follow similar steps as done for¯ Q ( t m | s ). To avoid repetition, we only present here thefinal expression of Q ( t m | t ) which reads Q ( t m | t ) = e − rt [ F ( { , } , { / , } , r ( t − t m )) − π (cid:112) t m ( t − t m )+ re − rt (cid:90) dw w e rtw Erf (cid:0) √ rwt m (cid:1) × Erf (cid:16)(cid:112) rw ( t − t m ) (cid:17) . (E8)Finally using Eqs. (E7) and (E8) in Eq. (E2), we recoverthe result for P r ( t m | t ) in Eq. (49). Appendix F: Derivation of (cid:104) M n ( t ) (cid:105) for drift-diffusionprocess This appendix provides the derivation of the scalingrelation in Eq. (29) for (cid:104) M n ( t ) (cid:105) . We begin with theLaplace transform of (cid:104) M n ( t ) (cid:105) in Eq. (28). Looking atthis equation, we find that the Laplace transform is prod-uct to two terms which leads us to use convolution prop-erty for Laplace transforms. Using this property, we get (cid:104) M n ( t ) (cid:105) = − n !2 n/ ∞ (cid:88) k =1 ( − r ) k k n (cid:90) t dτ f ( t − τ ) , f n ( τ )(F1)where the functions f ( t ) and f ( t ) are given below. f ( t ) = L − s → t (cid:34) (cid:0) √ s − β + γ (cid:1) n (cid:35) , = ( − n − ( n − (cid:20) e βt δ n, √ πt − d n − dγ n − (cid:110) γe ( β + γ ) t Erfc (cid:16) γ √ t (cid:17)(cid:111)(cid:21) f ( t ) = L − s → t (cid:20) s + rs k +1 (cid:21) , = t k +1 Γ( k ) + rt k Γ( k + 1) , (F2)8where β = − (cid:0) r + γ (cid:1) and γ = − v/ √
2. Inserting theforms of f ( t ) and f ( t ) in the expression of (cid:104) M n ( t ) (cid:105) in Eq. (F1) and then changing the variable τ = tw , we getthe scaling relation in Eq. (29). Appendix G: Derivation of P r ( t m | t ) in Eq. (61) for drift-diffusion process In this section, we present derivation for the density P r ( t m | t ) namely Eq. (61) which was presented in the maintext. We recall from the main text (Eq. (60)) that Z r ( k | s ) (cid:39) (cid:112) v + 2( s + r ) − v (cid:112) v + 2( s + k + r ) − v (cid:90) dw ( r + s + k )( s + rw ) ( s + k + rw ) . (G1)To invert this expression, we do the inverse Laplace transforms with respect to k and s respectively. Next, we performthe integral. Following the first inversion, we rewrite Z r ( t m | s ) (cid:39) (cid:104)(cid:112) v + 2( s + r ) − v (cid:105) (cid:90) dw ( s + rw ) L − k → t m s + r + ks + rw + k (cid:112) v + 2( s + k + r ) − v (cid:124) (cid:123)(cid:122) (cid:125) I ( k ) . (G2)We can now rearrange I ( k ) and then do the inversion to find L − k → t m [ I ( k )] = L − k → t m (cid:34) vs + rw + k + 12 (cid:112) v + 2( s + k + r ) s + rw + k (cid:35) = e − ( v + s + r ) t m √ πt m + ve − ( s + rw ) t m (cid:113) v + r (1 − w ) √ e − ( s + rw ) t m Erf (cid:34)(cid:115)(cid:18) v r (1 − w ) (cid:19) t m (cid:35) . (G3)Now, we need to perform the Laplace inversion wrt s i.e., P r ( t m | t ) = L − s → t [ Z r ( t m | s )] , (G4)where we have Z r ( t m | s ) = (cid:90) dw e − ( v + s + r ) t m √ πt m (cid:112) v + 2( s + r ) − vs + rw + (cid:90) dw ve − ( s + rw ) t m (cid:112) v + 2( s + r ) − vs + rw + (cid:90) dw (cid:113) v + r (1 − w ) √ (cid:34)(cid:115)(cid:18) v r (1 − w ) (cid:19) t m (cid:35) e − ( s + rw ) t m (cid:112) v + 2( s + r ) − vs + rw . (G5)Substituting Eq. (G5) into Eq. (G4) and performing the Laplace inversions, we obtain P r ( t m | t ) (cid:39) e − ( v + r ) t m √ πt m (cid:90) dw J (cid:96) ( t, w ) + v (cid:90) dw e − rwt m J (cid:96) ( t, w )+ (cid:90) dw (cid:113) v + r (1 − w ) √ (cid:34)(cid:115)(cid:18) v r (1 − w ) (cid:19) t m (cid:35) e − rwt m J (cid:96) ( t, w ) , (cid:39) I ( t m , t ) + I ( t m , t ) + I ( t m , t ) , (G6)9where we have used the following inverse Laplace transform to arrive at Eq. (G6) J (cid:96) ( t, w ) = L − s → t (cid:34) (cid:112) v + 2( s + r ) − vs + rw e − st m (cid:35) = − ve − rw ( t − t m ) Θ( t − t m )+ √ (cid:34) e − ( t − t m )( v + r ) (cid:112) π ( t − t m ) + e − rw ( t − t m ) (cid:114) v r (1 − w ) Erf (cid:32)(cid:114) ( t − t m )( v r − rw ) (cid:33)(cid:35) Θ( t − t m ) . (G7)The I -functions introduced in Eq. (G6) are formally defined in the following. They can also be simplified occasionally.The first component in Eq. (G6) reads I ( t m , t ) = e − ( v + r ) t m √ πt m (cid:90) dw J (cid:96) ( t, w ) , = e − ( v + r ) t π (cid:112) t m ( t − t m ) − ve − ( v + r ) t m √ πt m − e − r ( t − t m ) r ( t − t m ) + e − ( v + r ) t π (cid:112) t m ( t − t m ) (cid:90) v r +1 v r dz e r ( t − t m ) z √ rz Erf (cid:104)(cid:112) r ( t − t m ) √ z (cid:105) , = e − ( v + r ) t π (cid:112) t m ( t − t m ) − v e − ( v + r ) t m √ πt m − e − r ( t − t m ) r ( t − t m )+ e − ( v + r ) t π ( t − t m ) √ πt m − v F (cid:104) , , ( t − t m ) v (cid:105) r + (cid:18) v r + 1 (cid:19) F (cid:20) ,
1; 12 , r ( t − t m ) (cid:18) v r + 1 (cid:19)(cid:21) − . (G8)Similarly, the second component in Eq. (G6) gives I ( t m , t ) = v (cid:90) dw e − rwt m J (cid:96) ( t, w ) , = ve − ( v + r )( t − t m ) (cid:112) π ( t − t m ) 1 − e − rt m rt m − v − e − rt rt + ve − ( v + r ) t √ (cid:90) v r +1 v r dz e rtz √ rz Erf (cid:104)(cid:112) r ( t − t m ) √ z (cid:105) . (G9)Finally, simplifying the third component in Eq. (G6), we get I ( t m , t ) = (cid:90) dw (cid:113) v + r (1 − w ) √ (cid:34)(cid:115)(cid:18) v r (1 − w ) (cid:19) t m (cid:35) e − rwt m J (cid:96) ( t, w ) , = e − ( v + r ) t (cid:112) π ( t − t m ) (cid:90) v r +1 v r dz e rt m z √ rz Erf (cid:2) √ rt m √ z (cid:3) − ve − ( v + r ) t √ (cid:90) v r +1 v r dz e rtz √ rz Erf (cid:2) √ rt m √ z (cid:3) , + re − ( v + r ) t (cid:90) v r +1 v r dz z e rtz Erf (cid:2) √ rt m √ z (cid:3) Erf (cid:104)(cid:112) r ( t − t m ) √ z (cid:105) = e − ( v + r ) t πt m (cid:112) π ( t − t m ) − v F (cid:104) , , t m v (cid:105) r + (cid:18) v r + 1 (cid:19) F (cid:20) ,
1; 12 , rt m (cid:18) v r + 1 (cid:19)(cid:21) − − ve − ( v + r ) t √ (cid:90) v r +1 v r dz e rtz √ rz Erf (cid:2) √ rt m √ z (cid:3) + re − ( v + r ) t (cid:90) v r +1 v r dz z e rtz Erf (cid:2) √ rt m √ z (cid:3) Erf (cid:104)(cid:112) r ( t − t m ) √ z (cid:105) . (G10)Joining all the I -functions results in the expression (61) for P r ( t m | t ) which was announced in the main text. Appendix H: Details of the model systems used in the simulation in Sec. V
In this section, we present details of the processes used in the simulation in Sec. V. We have used four differentmodel systems as underlying processes. Three of them are Markovian in nature while one is a non-Markov process. All0of them are subjected to resetting at a rate r which means that the time intervals between the resetting events weretaken from an exponential distribution namely p ( τ ) = re − rτ . We observe trajectories governed by these processes fora fixed observation time t , and compute the maximum displacement M that it undertook by this time. Moreover, wealso note down the time t m at which this maximum took place. Details of the model systems are as follows:1. Simple diffusion:
Motion of the particle for a simple diffusing particle is given by dxdτ = η ( τ ) , (H1)where η ( τ ) is the Gaussian white noise with mean zero and variance 2 Dτ . This is a Markov process.2. Diffusion with drift:
Here, we consider the diffusing particle in the presence of a drift velocity v > dxdτ = v + η ( τ ) , (H2)which is also a Markov process.3. Random acceleration process:
In this case, the position x ( t ) of the particle evolves via dxdτ = v, dvdτ = η ( τ ) , (H3)so that the process becomes non-Markov in x -variable.4. Ornstein-Uhlenbeck process:
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