Large Deviation in Continuous Time Random Walks
LLarge Deviations in Continuous Time Random Walks
Adrian Pacheco-Pozo and Igor M. Sokolov
1, 2 Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany IRIS Adlershof, Humboldt-University of Berlin, Newtonstraße 15, 12489 Berlin, Germany
We discuss large deviation properties of continuous-time random walks (CTRW) and present ageneral expression for the large deviation rate in CTRW in terms of the corresponding rates forthe distributions of steps’ lengths and waiting times. In the case of Gaussian distribution of steps’lengths the general expression reduces to a sequence of two Legendre transformations applied to thecumulant generating function of waiting times. The discussion of several examples (Bernoulli andGaussian random walks with exponentially distributed waiting times, Gaussian random walks withone-sided L´evy and Pareto-distributed waiting times) reveals interesting general properties of suchlarge deviations.
I. INTRODUCTION
Continuous time random walk (CTRW) introduced byMontroll and Weiss [1] is a generalization of a simplerandom walk model, in which the steps follow inhomoge-neously in time. In the standard variant of the CTRW,times of steps follow a renewal process in which the wait-ing times for subsequent steps are independent and iden-tically distributed (i.i.d.) random variables. The modelis then fully defined by specifying the probability den-sity function (PDF) of waiting times and of spacial dis-placements in single steps (also being i.i.d. random vari-ables) [2]. In physics, CTRW is used to model systemsshowing anomalous diffusion, e.g. situations when themean square displacements (MSD) (cid:104) x ( t ) (cid:105) does not growlinearly in time, as predicted by the Fick’s law, espe-cially in the cases of subdiffution, when (cid:104) x ( t ) (cid:105) ∝ t α with α <
1. Such anomalous subdiffusion often arisesdue to the lack of the first moments of the distributionof waiting times. The anomaly in MSD is accompaniedby a non-Gaussian shape of PDF of displacements dur-ing a given time t . This line of modelling, following thepioneering work [3], is discussed in the review articles [4]and [5], with some newer developments discussed in chap-ters of collective monographs [6] and [7]. Many recentworks have reported weaker type of anomaly, in whichthe time-dependence of the MSD is linear, but the formof the PDF is pronouncedly non-Gaussian, especially atshorter times. The decay of the tails of the PDFs is of-ten exponential, see e.g. [8–10] as well as [11–13] andreferences therein.In Ref. [12], Barkai and Burov showed that such expo-nential decay is a universal behavior of far tails in CTRW,presenting the approach based on subordination of ran-dom processes and large deviation theory. In Ref. [14],Wang, Barkai, and Burov presented explicit calculationsfor the case of exponential and Gamma- (Erlang) distri-butions of waiting times. In the present work we buildon this approach and present a general expression for thelarge deviations’ rate function in CTRW, which we usefor discussing several examples beyond the ones consid-ered in [12, 14], where we consider both the case whenthe mean WTD is finite and the case when it diverges. For the case of Gaussian step length distribution, as in[14], the analysis is especially simple, and the generalexpression reduces to a sequence of two Legendre trans-formations of a cumulant generating function of waitingtimes.The structure of the present work is as follows: In Sec.II we conduct a preliminary discussion of the model andintroduce notation used throughout the work. In Sec.III we derive the main result, which we use in the dis-cussion of several examples; first of them already givenin this section. In Sec. IV we consider the case withGaussian step length distribution, for which the expres-sions are especially simple. Here different further exam-ples with and without mean waiting times are considered,which are enough for understanding the physics behindthe large deviations in CTRW. Sec. VI concludes thework. One “derivation” of a known mathematical re-sult on a physical level of rigor, which we consider to beuseful for understanding the approach, is placed in theAppendix. II. PRELIMINARIES AND NOTATION
Continuous time random walk (CTRW) is a randomprocess which ensues due to subordination of a simplerandom walk (RW) process x ( n ) to another random pro-cess n ( t ), which gives us the number of steps done bythe random walk up to time t , and is called a directingprocess , or operational time of the CTRW scheme [2].The process x ( n ) is called with this respect a parent pro-cess of the CTRW (see Refs. [15, 16] for a systematicdiscussion and terminology). The “inverse” process t ( n )corresponding to the sum of independent, identically dis-tributed waiting times in CTRW is called a leading pro-cess .Let the PDF of the parent process be p n ( x ), and theprobability to make exactly n steps up to time t be χ n ( t ).Then the PDF of displacements in CTRW is [2] p ( x, t ) = ∞ (cid:88) n =0 p n ( x ) χ n ( t ) . For long t , the typical number of steps is very large, and a r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec the operational time τ = n can be taken to be continuous.In this limit p ( x, t ) = (cid:90) ∞ f ( x, τ ) T ( τ, t ) dτ, (1)where f ( x, τ ) is a continuous approximation to (fluidlimit of) p n ( x ) and T ( τ, t ) is the same kind of approx-imation to χ n ( t ). Eq. (1) is called the integral formulaof subordination. In what follows we will only work inthis limit.The parent and the leading processes of the CTRWscheme are processes with independent and identicallydistributed (i.i.d.) increments, i.e. correspond to sumsof i.i.d. random variables. Exactly such sums constitutethe elementary (and elemental) class of objects studiedin the theory of large deviations. Let x n be a sequence ofindependent random variables, s n = ( x + x + ...x n ) /n bethe empirical mean of the first n elements in the sequence,and P n ( s ) be the probability distribution of s n . Then,under known conditions, there exists the limit C ( s ) = − lim n →∞ n ln P n ( s )called the rate function (RF), or Cram´er’s function, oflarge deviations for s n [17–19]. The distribution of s n forsufficiently large n is then given, up to a normalizationconstant, by P n ( s ) ∼ exp [ − n C ( s )] . When making the inverse variable transformation from s n to the sum S n = x + x + ... + x n = ns n of the first n random variables, one sees that its PDF takes the form p n ( S ) ∼ exp (cid:20) − n C (cid:18) Sn (cid:19)(cid:21) , (2)up to an n -dependent normalization constant. Eq. (2)will be called the large deviation form for the PDF of thecorresponding sum. To avoid ambiguity, we will denotethe rate functions by scriptized letters.At the beginning, only two PDFs are known: the oneof the single step lengths, and the one of the waitingtimes. Therefore, in our discussion, we will start fromthe rate functions of sums of independent and identi-cally distributed random variables (step times and steplengths), and derive the properties of all other rate func-tions from two rate functions: I ( t ) and R ( x ), the onesfor the leading process and for the parent process (simpleRW), which can be readily calculated. The large devia-tion forms for the PDFs of the corresponding sums arethen f ( x, τ ) ∼ exp (cid:104) − τ R (cid:16) xτ (cid:17)(cid:105) , (3)and p ( t, τ ) ∼ exp (cid:20) − τ I (cid:18) tτ (cid:19)(cid:21) . (4) To easily manipulate the variables we have to keep thearguments of all non-power-law functions dimensionless.To do so, we introduce the natural units of length andtime. As a natural unit of length one may choose themean squared displacement in a single step, and as anatural unit of time a characteristic time of a single step.If the mean waiting time for a step exists, we will choosethis mean waiting time as the characteristic time. Inthe cases when the mean waiting time diverges, for thepurpose of comparison of different situations, one shouldchoose a characteristic time such that the mean numberof steps (cid:104) n ( t ) (cid:105) made up to time t is the the same for allcases compared. Note that this requirement is fulfilledwhen choosing the mean waiting time, provided it exists,as a characteristic time, as above. Thus, from here on,all our quantities are dimensionless. The units can beeasily restored in the final expressions when necessary.The two derived rate functions, the ones for the direct-ing process and the one for the whole CTRW (a functionwe seek to know) will be denoted by T ( τ ) and X ( x ), re-spectively. The large deviation forms for the PDFs of τ and x are: T ( τ, t ) ∼ exp (cid:104) − t T (cid:16) τt (cid:17)(cid:105) , (5)and p ( x, t ) ∼ exp (cid:104) − t X (cid:16) xt (cid:17)(cid:105) . (6)Before starting our derivation, let us go through someimportant properties of the rate functions [17–19]. Therate functions for the mean of i.i.d. random variablesare convex [18]. Cram´er’s theorem states that therate function C ( x ) is the convex conjugate (a Legendre-Fenchel transformation) of the cumulant generation func-tion L ( q ) = ln (cid:104) e xq (cid:105) of the distribution of x . For thesake of brevity we will call this transformation simply aLegendre transformation, as it is done, say, when trans-forming a Lagrange to a Hamilton function in mechanics.For strictly convex rate functions (i.e. for all cases dis-cussed in the present work) this transformation is invert-ible. The inverse of a Legendre transformation is givenby a Legendre transformation again (an involution prop-erty). Two other properties of rate functions, which areimportant for what follows, are that the rate function isnon-negative, and that it vanishes identically at the valueof x = µ corresponding to the mean of the distribution ofthe single variable. The rate function can be defined onthe whole real line (e.g. for a Gaussian distribution of x ),on a half-line (like it is for waiting times), or on a finiteinterval. Outside of its domain of definition (i.e. outsideof the domain available for the initial random variables)it is taken to be “literaly infinite”, so that the probabil-ity to have such values (the ones outside the domain ofdefinition) is strictly zero.To keep equations concise we will introduce the follow-ing nonstandard notation for the Legendre transforma-tion: Let g ( x ) be a convex function defined on the cor-responding interval, and let a function f ( z ) of variable z be its Legendre transformation defined as g ( z ) = sup x { zx − f ( x ) } . The Legendre transformation is an operator which makesa mapping f ( x ) (cid:55)→ g ( z ). Our notation will keep track ofthe variable’s names in the corresponding functions g ( z ) = (cid:98) L { z ← x } f ( x ) . The direction of the arrow is connected with the orderof the symbols: the operator (cid:98) L acts on the function of x standing right from it, and transforms the function f ofvariable x into a function g of variable z (the names ofthe variables are changed from right to the left). Since wechoose to denote the variable’s names in the notation forthe operator, we don’t have to put them in the functionsat all (but we will).The main result of the present article is as follows: X ( z ) = − sup ξ (cid:20) − R ( zξ ) + I ( ξ ) ξ (cid:21) , (7)with I ( z ) = (cid:98) L { z ← q } L ( q ) and R ( z ) = (cid:98) L { z ← q } D ( q ) where L ( q ) = (cid:104) e qt (cid:105) = (cid:82) ∞ e qt ψ ( t ) dt with ψ ( t ) being the PDF ofwaiting times (waiting time density, WTD), and D ( q ) = (cid:104) e qx (cid:105) = (cid:82) ∞−∞ e qx λ ( x ) dx with λ ( x ) being the PDF of dis-placements in a single step (steps’ lengths density, SLD).For a Gaussian distribution of single steps’ lengths dis-cussed in [12, 14], the result can be put in a closed forminvolving only the Legendre transformations X ( x ) = − (cid:98) L {− x / ← ξ } (cid:104) ξ − (cid:98) L { ξ ← q } L ( q ) (cid:105) . (8) III. THE LARGE DEVIATIONS RATE FORCTRW
Let us now derive the forms, Eq. (7) and (8). Substi-tuting the large deviation forms for PDFs of the parentand directing processes (Eqs.(3) and (5), respectively)into the integral formula of subordination, Eq. (1), onegets p ( x, t ) ∼ (cid:90) ∞ exp (cid:104) − τ R (cid:16) xτ (cid:17) − t T (cid:16) τt (cid:17)(cid:105) dτ. On the l.h.s., we introduce the large deviation form ofthe whole CTRW (Eq.(6))exp (cid:104) − t X (cid:16) xt (cid:17)(cid:105) ∼ (cid:90) ∞ exp (cid:104) − τ R (cid:16) xτ (cid:17) − t T (cid:16) τt (cid:17)(cid:105) dτ and change to the new variables z = x/t and ξ = t/τ :exp [ − t X ( z )] ∼ (cid:90) ∞ tξ exp (cid:26) − t (cid:20) R ( zξ ) ξ + T (cid:18) ξ (cid:19)(cid:21)(cid:27) dξ. Now, we use the relation between the RF for the leadingand directing processes, which is given by T ( τ ) = τ I (cid:18) τ (cid:19) , (9)see Ref. [20]. The proof of this relation (and the discus-sion of conditions under which it holds) implies a longerchain of mathematical discussions. Although this rela-tion is known, we sketch a simple explanation on thephysical level of rigor is given in Appendix A.Using this relation one getsexp [ − t X ( z )] ∼ (cid:90) ∞ tξ exp (cid:26) − t (cid:20) R ( zξ ) ξ + I ( ξ ) ξ (cid:21)(cid:27) dξ. This integral can then be solved by the Laplace’s method[21]. Thus, we can writeexp [ − t X ( z )] ∼ exp (cid:26) − t inf ξ (cid:20) R ( zξ ) ξ + I ( ξ ) ξ (cid:21)(cid:27) , where the pre-exponential term is disregarded, since itdoes not contribute to the large deviations when takingthe limit lim t →∞ ln[ p ( x, t )] /t . Equating the arguments ofthe exponentials, we get X ( z ) = − sup ξ (cid:20) − R ( zξ ) + I ( ξ ) ξ (cid:21) , which is our main result, Eq. (7). Note that, the infi-mum was changed for the supremum of the negative ofthe expression, which will be useful to later relate thisquantity to a Legendre transformation in the case of aGaussian SLD. Since the large deviation rates R ( x ) and I ( t ) are readily given by the Legendre transformationsof the corresponding cumulant generating functions, thecorresponding supremum can be easily calculated for thecases of interest. In what follows, we consider in somedetail the case brought to our attention by the works[12, 14], namely, the case of Gaussian distribution of sin-gle step lengths. For this case, the RF X ( z ) follows fromthe cumulant generating function of waiting times by twoLegendre transformations (see below). However, first, asan example, let us analyse the simplest random walk, i.e.,the Bernoulli one. A. CTRW with a fixed step length
As a first example, let us consider the Bernoulli randomwalk, with steps of fixed length ± /
2, respectively, and the simplest possible leading pro-cess, a Poisson process with rate 1, i.e. with ψ ( t ) = e − t .The Bernoulli random walk is essentially the first exam-ple of Refs. [17] and [18]. For this case p N ( S ) = 2 − N − N ! (cid:0) N − S (cid:1) ! (cid:0) N + S (cid:1) ! . Applying Stirling formula we get in the first order in N ln p N ( s ) = − N − s ) ln(1 − s ) + (1 + s ) ln(1 + s )]with s = S/N , so that R ( x ) = 12 [(1 − x ) ln(1 − x ) + (1 + x ) ln(1 + x )] . This function is parabolic around 0 and diverges for x → ± − , I ( ξ ) = (cid:98) L { ξ ← q } L ( q ) correspond-ing to a Poisson process can be readily calculated (andessentially is known since long ago), and reads I ( ξ ) = ξ − − ln ξ. (10)The supremum in Eq.(7), for fixed z is achieved at ξ =1 / √ z , and X ( z ) is readily evaluated: X ( z ) = 1 + (cid:112) z (cid:110) − (cid:112) z + 12 (cid:20)(cid:18) − z √ z (cid:19) ln (cid:18) − z √ z (cid:19) + (cid:18) z √ z (cid:19) ln (cid:18) z √ z (cid:19)(cid:21)(cid:27) , (11)which is the function with a quadratic behavior close to z = 0, X ( z ) (cid:39) z / O ( z ), and with the large- z asymp-totics being X ( z ) (cid:39) | z | (1 − ln 2 + ln | z | ) (note that thefunction is not “cut” at a finite value of z , at differenceto the large deviation rate of the parent process). IV. CTRW WITH GAUSSIAN DISTRIBUTIONOF STEPS LENGTHS
For the Gaussian distribution of step lengths the largedeviations rate for the parent process is given by R ( x ) = x . Introducing this result into Eq. (7), one obtains X ( x ) = − sup ξ (cid:20) − x ξ − I ( ξ ) ξ (cid:21) , which is given by a Legendre transformation of the func-tion I ( ξ ) /ξ taken at the value of the Legendre variable z = − x /
2. Using our notation for the Legendre transfor-mations and the fact that I ( ξ ) = (cid:98) L { ξ ← q } L ( q ) we arriveat the final result in a closed form X ( x ) = − (cid:98) L {− x / ← ξ } (cid:104) ξ − (cid:98) L { ξ ← q } L ( q ) (cid:105) , Eq. (8). Now we use this formula for discussing sev-eral important examples and then make some conclusionsabout the behavior of the large deviation rate functionsof CTRW with Gaussian step lengths’ distribution forvery large deviations ( x (cid:29) A. Exponential waiting times
Let us assume that the waiting times follow a Poissonprocess, with ψ ( t ) = e − t and a rate function given by Eq. (10). Now, we define the function f ( ξ ) = I ( ξ ) ξ = 1 − ξ − ln ξξ , and perform the Legendre transformation z = ddξ f ( ξ ) = ln ξξ so that ξ = exp (cid:20) − W ( − z ) (cid:21) , where W ( · ) is the Lambert function. Then, according toEq. (8), we change the variable to z = − x / X ( x ) = 1 − exp (cid:20) W ( x ) (cid:21) + x (cid:20) − W ( x ) (cid:21) + 12 W ( x ) exp (cid:20) W ( x ) (cid:21) . (12)Using the properties of the Lambert function, the twolimits, of small and of large x , can be found. For x >> W ( x ) ∼ ln x + ln ln x , and X ( x ) ∼ | x | (cid:112) | x | .On the other hand, for x << W ( x ) ∼ x , andperforming a Taylor expansion around zero, one has X ( x ) ∼ x / O ( x ) corresponding to a Gaussian. Fig.1 shows a comparison between the rate functions for theBernoulli case with exponential WTD, Eq. (11), and forthe Gaussian SLD with exponential WTD, Eq. (12). Onecan see that, as x →
0, both curves coinside and showa parabolic behavior which is a consequence of the Cen-tral Limit Theorem (CLT). For x > -10 -5 0 5 10 x X ( x ) BernoulliGaussian
FIG. 1. A comparison of the rate functions X ( x ) for aBernoulli CTRW and CTRW with Gaussian distribution ofstep lengths, both with exponential WTD, Eq. (11). For x (cid:28)
1, both curves have the same parabolic behavior, whichis a consequence of the CLT, see the text for details.
B. One-sided L´evy law
Now we take ψ ( t ) to follow a one-sided L´evy law withexponent α . Its Laplace characteristic function f ( s ) = (cid:104) e − st (cid:105) reads: f ( s ) = (cid:82) ∞ ψ ( t ) e − st dt = exp( − σs α ) with0 < α < s >
0) and σ being the scale parameter.To allow for further comparison with the behavior forthe Pareto-distributed waiting times, we fix this to be σ = 1 / Γ( α + 1). This fixes the characteristic time of astep. The mean number of steps (cid:104) n ( t ) (cid:105) done during thetime t is then (cid:104) n ( t ) (cid:105) = t α . (13)Let us denote C L = Γ( α + 1) /α . Thus, f ( s ) =exp[ − ( s/C L ) α ], so that f ( q ) = (cid:26) exp[ − ( − q ) α /C αL ] q ≤ ∞ q > L ( q ) = (cid:26) − ( − q ) α /C αL q ≤ ∞ q > . Now, we perform the Legendre transform: t = ddq L ( q ) = α ( − q ) α − C αL −→ q = − (cid:18) C αL tα (cid:19) α − , which leads to I ( t ) = (1 − α ) (cid:18) C L tα (cid:19) − α − α . Now f ( ξ ) = I ( ξ ) ξ = (1 − α ) (cid:18) αC L (cid:19) α − α ξ − − α , and now the second Legendre transformation can be per-formed: z = ddξ f ( ξ ) = − (cid:18) αC L (cid:19) α − α ξ − − α − α , with ξ = (cid:18) C L α (cid:19) − α − α ( − z ) − − α − α . Finally, the rate function of displacements reads X ( x ) = (2 − α )2 − − α (cid:18) αC L (cid:19) α − α | x | − α . (14)For α → α → C. Pareto distributions
Now let us consider two other WTDs, namely thePareto distributions of types I and II with parameter α (for the sake of brevity these will be sometimes referredto as Pareto I and Pareto II distributions). Both dis-tributions have the same asymptotic behavior for longtimes, ψ ( t ) ∝ t − − α but differ with respect to their shorttime behavior. The distributions with 0 < α ≤ α > α ∈ (0 , α >
1, both Pareto distributions possessa mean waiting time, so the scaling factor is this lastquantity. For 0 < α ≤
1, we scale the distributions insuch a way that the mean number of steps (cid:104) n ( t ) (cid:105) for agiven t is the same for the both distributions and exactlythe same as for the L´evy case (Eq. (13)). Thus, thePareto type I WTD ψ ( t ) is defined as ψ ( t ) = ≥ t > C I αC αI t α +1 t ≥ C I (15)where C I = [Γ(1 + α )Γ(1 − α )] − /α if α ≤
1, or C I =( α − /α if α >
1. Its Laplace transform reads f ( s ) = αC αI s α Γ( − α, C I s ) , (16)with Γ( x, y ) being an upper incomplete Γ-function. Onthe other hand, the Pareto type II WTD can be definedas ψ ( t ) = αC αII ( C II + t ) α +1 , (17)where C II = [Γ(1 + α )Γ(1 − α )] − /α if α ≤
1, and C II =( α −
1) if α >
1. Its Laplace transform reads f ( s ) = αC αII s α e C II s Γ( − α, C II s ) . (18)Due to the presence of incomplete Gamma functions, theLegendre transforms have to be performed numerically,by solving algebraic equations.Fig. 2 shows a comparison between the rate functions X ( x ) of displacements in the CTRW with Gaussian STDand the three following WTDs which do not possess meanwaiting time: one-sided L´evy law with α = 0 .
5, andPareto I and II distributions, both with α = 0 .
5, all withthe same (cid:104) n ( t ) (cid:105) as given by Eq. (13). As in the caseof the distributions with finite mean waiting times, thecurves coinside in the central domain (although now theyare not Gaussian but are given by Eq. (14)), but deviatefor x large. -10 -5 0 5 10 x X ( x ) L α = 0 .
5P II α = 0 .
5P I α = 0 . FIG. 2. A comparison of the rate functions X ( x ) for theCTRW with Gaussian SLD and the WTDs following a one-sided L´evy law, and Pareto type I and type II distributions,all with with α = 0 . (cid:104) n ( t ) (cid:105) as given by Eq.(13). Fig. 3 shows the corresponding comparison betweenthe Pareto type I and Pareto type II WTDs for the cases α = 1 . α = 2 . x isuniversally parabolic, as it should be, but the RFs forPareto type I WTDs universally grow faster at large x than those for Pareto type II WTDs with the same α .We now discuss in more detail these asymptotic growthproperties discussing the domain of very large deviations.The cases of the Pareto distributions give enough physi-cal intuition to understand the general behavior of verylarge deviations. V. VERY LARGE DEVIATIONS FOR PARETOWAITING TIME DENSITIES
Physically, the behavior of very large deviations ( x →∞ ) is dominated by realizations in which the number ofsteps is unusually large, and thus is governed by the be-havior of ψ ( t ) for very short t . Therefore the two Pareto -10 -5 0 5 10 x X ( x ) P II α = 1 .
5P II α = 2 .
5P I α = 1 .
5P I α = 2 . FIG. 3. A comparison of the rate function X ( x ) for the CTRWwith Gaussian SLD and the following WTDs: Pareto I dis-tributions with α = 1 . , .
5, and Pareto II with α = 1 . , . x is the quotient between the position andthe time. All these WTDs have a mean waiting time. Notethe difference in the asymptotic behavior, which for the caseof Pareto I is quadratic (Eq. (20)), whereas for the Pareto IIis linear with a slowly varying correction (Eq. (22)). cases serve as examples for the cases when ψ ( t ) tends toa constant limit for t → t → ψ (0) = 0, but shooting up fasterthan any power of t for non-vanishing but small t , showsthe behavior in-between of these two extrema.Note that the expressions for both Pareto distributions(Eqs. (15) and (17)) are the same for all values of theparameter α , the only difference being the values of thescaling parameters C I and C II . Hence, all subsequentresults hold independent of the value of α . A. Pareto Type I WTD
First, let us consider the Pareto I WTD (Eq. (15)),and let us work with its Laplace transform as given byEq. (16). The small time behavior ( t →
0) of the WTD ismirrired in the asymptotic behavior of its Laplace trans-form for s → ∞ . For the Pareto I PDF this asymptoticbehavior is given by f ( s ) ∼ αC − I s − e − C I s , (19)which form can alternatively be obtained either by us-ing the asymptotic expansion of the incomplete Gammafunction, or by evaluating the corresponding integral forthe Laplace transform using the Laplace method. Fromthis form it follows that L ( q ) = ln α − ln C I − ln( − q ) + C I q. Performing the Legendre transformation we get t = d L ( q ) dq = C I − q → q = 1 C I − t , and finally obtain I ( t ) = − ln( t − C I )in the leading order. Now, I ( ξ ) ξ = − ln( ξ − C I ) ξ , and the second Legendre transform can be performed: − x df ( ξ ) dξ = − ξ ( ξ − C I ) + ln( ξ − C I ) ξ . Making the change of variable u = ξ − C I , it can berewritten as − x − u ( u + C I ) + ln u ( u + C I ) Very large and negative values of the l.h.s. correspond to u →
0, so that − x − uC I + ln uC I To invert this expression, one can apply de Bruijn’s The-orem for slowly varying functions, see [22]. Hence, oneends up with u = 2 C I,i x (cid:20) C I x ) C I x (cid:21) , and, going back to the variable ξ : ξ = 2 C I x (cid:20) C I x ) C I x (cid:21) + C I . Finally, the asymptotic behavior ( | x | → + ∞ ) of the ratefunction for the CTRW has the form X ( x ) ∼ C I x + 2 C I ln | x | , (20)which is basically a quadratic behavior with a correctiongiven by a slowly varying function. Note that, apart fromthe value of the constant C I being a function of α , Eq.(20) does not depends on the parameter α . B. Pareto Type II WTD
Let us now consider the case of a Pareto II WTD (Eq.(17)), which in the Laplace domain is given by Eq. (18).Following the same procedure as for the Pareto I WTD, let us consider the asymptotic behavior ( s → ∞ ) of Eq.(18) given by (cid:101) ψ ( s ) ∼ αC − II s − , (the difference with Eq. (19) is the absence of the expo-nential cutoff for very large s ), which allow us to obtainthe asymptotics of the cumulant generating function: L ( q ) = ln α − ln C II − ln( − q ) . (21)Applying the Legendre transformation we get: t = d L ( q ) dq = − q → q = − t , so that the function I ( t ) in the leading order is given by I ( t ) = − ln t. Then we construct I ( ξ ) ξ = − ln ξξ , and perform the second Legendre transformation − x df ( ξ ) dξ = − ξ ) ξ . The values of | x | → ∞ correspond to ξ →
0, so that x − ln ξξ . To invert this expression, one applies again the deBruijn’s Theorem. Hence, the inverse reads ξ = | x | − (cid:112) | x | . Finally, the asymptotic behavior ( | x | → + ∞ ) of the ratefunction for the CTRW is X ( x ) ∼ | x | (cid:112) | x | , (22)which is an essentially linear behavior, with a correctiongiven by a slowly varying function. This result does notdepend on the value of the parameter α at all (and notnot only up to the parameter values, like in the Pareto Icase). C. Erlang distributions
A similar analysis can be performed for Gamma-distributions (Erlang distributions) discussed in [12, 14] ψ ( t ) = λ n t n − e − λt ( n − , with n ∈ { , , , . . . } , and λ ∈ (0 , ∞ ). The Laplacecharacteristic function of the Erlang distributions has thefollowing form f ( s ) = λ n ( λ + s ) − n . Then the asymptotic form of L ( q ) differs from Eq.(21)only by an additional proportionality factor in front ofln( − q ): L ( q ) = − n ln( − q ) , in the limit q → −∞ . Therefore, the essentially linearbehavior with a slowly varying correction ensues. Thedifference between the L´evy and Pareto type I cases onone hand and Pareto type II and Erlang cases on theother hand is the fact that for the first class of distri-butions the WTD vanishes at zero together with all itsderivatives, while in the second situation this is no morethe case. The full classification of possible behaviors willbe discussed elsewhere. The lesson learned from theseexamples is that the essentially linear behavior of thelarge deviation rate function in CTRW is not universalbut is pertinent to the specific classes of the waiting timedistributions. The large deviation behavior of displace-ments probes the WTD for very short waiting times andis therefore a test for microscopic dynamics of the system. VI. SUMMARY
In this paper, we presented a general procedure to com-pute the rate function for large deviations of displace-ments in CTRW in a general setting, i.e. for any steplength distribution and any waiting time distribution.The situation with Gaussian step length distribution isespecially simple. In this case the rate function for dis-placement is given by a sequence of Legendre transformsof the cumulant generating function for waiting times.The general discussion is accompanied by analysing im-portant particular examples like the one-sided L´evy andthe Pareto-distributed waiting times. This discussionshows that the large deviation in displacement probe thewaiting time density for very short times. The essentiallylinear behavior of the rate function for very large devia-tions is specific only for situations in which waiting timedensity does not vanish too fast when the waiting timesapproach zero.
ACKNOWLEDGMENTS
The work of APP was financially supported by “Doc-toral Programmes in Germany” funded by the DeutscherAkademischer Austauschdienst (DAAD) (Programme ID57440921).
Appendix A: Relation between the rate functionsfor the leading and directing processes
The relation between the large deviation rates for theleading and the directing process of the CTRW schemefoots on the known relation between the cumulative dis-tribution functions (CDFs) for the leading and directingprocesses [15, 23]. Let p ( τ | t ) = T ( τ, t ) be probabilitydensity to have exactly τ steps up to time t (i.e. theprobability density of τ conditioned on t ), and p ( t | τ ) thecorresponding density of the time of the last step condi-tioned on the number of steps. Then, from the fact that t is monotonically non-decreasing function of τ , the inte-gral relation between the two follows. In the continuouslimit this relation takes the form (cid:90) ∞ τ p ( τ (cid:48) | t ) dτ (cid:48) = (cid:90) t p ( t (cid:48) | τ ) dt (cid:48) . (A1)We note that the rate function I ( z ) is already known.From the general properties of the rate functions it fol-lows that I ( z ) is a convex function, monotonically non-growing (in our case, essentially monotonically decaying)for z < z with z being equal to the mean waiting time,and monotonically non-decaying (in our case growing)for z > z . If the mean diverges, the function is alwaysmonotonically decaying. From Eq. (A1) it follows that p ( τ | t ) = − ddτ (cid:90) t p ( t (cid:48) | τ ) dt (cid:48) , or, equivalently, p ( τ | t ) = ddτ (cid:20) − (cid:90) t p ( t (cid:48) | τ ) dt (cid:48) (cid:21) = ddτ (cid:90) ∞ t p ( t (cid:48) | τ ) dt (cid:48) . Substituting the large deviation forms we getexp (cid:104) − t T (cid:16) τt (cid:17)(cid:105) ∼ − ddτ (cid:90) t exp (cid:20) − τ I (cid:18) t (cid:48) τ (cid:19)(cid:21) dt (cid:48) = (cid:90) t (cid:20) I (cid:18) t (cid:48) τ (cid:19) − t (cid:48) τ I (cid:48) (cid:18) t (cid:48) τ (cid:19)(cid:21) exp (cid:20) − τ I (cid:18) t (cid:48) τ (cid:19)(cid:21) dt (cid:48) (A2)or exp (cid:104) − t T (cid:16) τt (cid:17)(cid:105) ∼ ddτ (cid:90) ∞ t exp (cid:20) − τ I (cid:18) t (cid:48) τ (cid:19)(cid:21) dt (cid:48) = − (cid:90) ∞ t (cid:20) I (cid:18) t (cid:48) τ (cid:19) − t (cid:48) τ I (cid:48) (cid:18) t (cid:48) τ (cid:19)(cid:21) exp (cid:20) − τ I (cid:18) t (cid:48) τ (cid:19)(cid:21) dt (cid:48) (A3)The expression in the non-exponential part of both Eqs.(A2) and (A3), f ( x ) = I ( x ) − x I (cid:48) ( x ) (with x = t (cid:48) /τ ) is nonnegative for x < z and non-positive for x > z . The first statement (for x ≥ I ( x ) is nonneg-ative and its derivative for x < z non-positive. Thesecond statement is slightly finer and follows from thethe relation g ( z ) ≥ g ( y ) + g (cid:48) ( y )( z − y ) (A4)for convex differentiable functions, which we rewrite as g ( y ) − yg (cid:48) ( y ) ≤ g ( z ) − g (cid:48) ( y ) z. Now one takes g ( y ) = I ( y ) and z = z , so that g ( z ) = I ( z ) vanishes, and notes that for y > z the derivative g (cid:48) ( y ) = I (cid:48) ( y ) is nonnegative. Therefore the prefactors ofthe exponentials in both integrals on the r.h.s. of Eqs.(A2) and (A3) are non-negative (essentially, positive fornon-degenerated cases). For z < z the function I ( z ) ismonotonically decaying ( I (cid:48) ( z ) < z > z thefunction I ( z ) is monotonically growing ( I (cid:48) ( z ) > I and T , the argument ofthe function on the l.h.s. has to be kept constant. Fixing x = t/τ and changing the variable of integration on ther.h.s. to x (cid:48) = t (cid:48) /τ one obtains:exp (cid:20) − τ x T (cid:18) x (cid:19)(cid:21) ∼ τ (cid:90) ba | f ( x (cid:48) ) | exp [ − τ I ( x (cid:48) )] dx (cid:48) , where the limits of integration { a, b } are { , x } in case ofEq.(A2) and { x, ∞} in case of Eq.(A3). Assuming that inthe vicinity of x , I ( x (cid:48) ) ≈ I ( x ) + I (cid:48) ( x )( x (cid:48) − x ) + o ( x − x (cid:48) ),both integrals can be estimated as | f ( x (cid:48) ) ||I (cid:48) ( x ) | exp [ − τ I ( x )] , i.e. in the exponential order of magnitudeexp (cid:20) − τ x T (cid:18) x (cid:19)(cid:21) ∼ exp [ − τ I ( x )] , from which Eq.(9) follows. [1] E. W. Montroll and G. H. Weiss, Random walks on lat-tices. II, J. Math. Phys. , 167 (1965).[2] J. Klafter and I. Sokolov, First Steps in Random Walks:From Tools to Applications (OUP Oxford, 2011).[3] H. Scher and E. W. Montroll, Anomalous transit-timedispersion in amorphous solids, Phys. Rev. B , 2455(1975).[4] J. Haus and K. Kehr, Diffusion in regular and disorderedlattices, Phys. Rep. , 263 (1987).[5] R. Metzler and J. Klafter, The random walk’s guide toanomalous diffusion: a fractional dynamics approach,Phys. Rep. , 1 (2000).[6] R. Klages, G. Radons, and I. Sokolov, Anomalous Trans-port: Foundations and Applications (Wiley, 2008).[7] J. Klafter, S. C. Lim, and R. Metzler,
Fractional Dynam-ics (World Scientific, 2011).[8] P. Chaudhuri, L. Berthier, and W. Kob, Universal na-ture of particle displacements close to glass and jammingtransitions, Phys. Rev. Lett. , 060604 (2007).[9] B. Wang, S. M. Anthony, S. C. Bae, and S. Granick,Anomalous yet brownian, P. Natl. Acad. Sci. , 15160(2009).[10] A. C. Silva, R. E. Prange, and V. M. Yakovenko, Ex-ponential distribution of financial returns at mesoscopictime lags: a new stylized fact, Physica A , 227 (2004).[11] A. V. Chechkin, F. Seno, R. Metzler, and I. M. Sokolov,Brownian yet non-gaussian diffusion: From superstatis-tics to subordination of diffusing diffusivities, Phys. Rev.X , 021002 (2017).[12] E. Barkai and S. Burov, Packets of diffusing particlesexhibit universal exponential tails, Phys. Rev. Lett. ,060603 (2020).[13] E. B. Postnikov, A. Chechkin, and I. M. Sokolov, Brow-nian yet non-gaussian diffusion in heterogeneous media: from superstatistics to homogenization, New J. Phys. ,063046 (2020).[14] W. Wang, E. Barkai, and S. Burov, Large deviations forcontinuous time random walks, Entropy (2020).[15] R. Gorenflo, F. Mainardi, and A. Vivoli, Continuous-timerandom walk and parametric subordination in fractionaldiffusion, Chaos Soliton Frac. , 87 (2007).[16] R. Gorenflo and F. Mainardi, Subordination pathwaysto fractional diffusion, Eur. Phys. J-Spec. Top. , 119(2011).[17] F. Cecconi, M. Cencini, A. Puglisi, D. Vergni, andA. Vulpiani, From the law of large numbers to large de-viation theory in statistical physics: An introduction,in Large Deviations in Physics: The Legacy of the Lawof Large Numbers , edited by A. Vulpiani, F. Cecconi,M. Cencini, A. Puglisi, and D. Vergni (Springer BerlinHeidelberg, Berlin, Heidelberg, 2014) pp. 1–27.[18] S. R. S. Varadhan, Large deviations, in
Proceedings of theInternational Congress of Mathematicians 2010 (ICM2010) (2011) pp. 622–639.[19] H. Touchette, The large deviation approach to statisticalmechanics, Phys. Rep. , 1 (2009).[20] P. W. Glynn and W. Whitt, Large deviations behaviorof counting processes and their inverses, Queueing Syst. , 107 (1994).[21] R. W. Butler, Saddlepoint Approximations with Applica-tions , Cambridge Series in Statistical and ProbabilisticMathematics (Cambridge University Press, 2007).[22] R. Bojanic and E. Seneta, Slowly varying functionsand asymptotic relations, J Math. Anal. Appl. , 302(1971).[23] A. Baule and R. Friedrich, Joint probability distributionsfor a class of non-markovian processes, Phys. Rev. E71