Let R be the set of real numbers, we denote by R + and R 0 + the set of positive real numbers and the set of nonnegative real numbers, e.g., the half-line of spatial excursions excluded the origin R + = { x ∈ R | x > 0 } and the elapsed times R 0 + = { t ∈ R | t ≥ 0 }. Analogously, we denote by R − and R 0 − the set of negative real numbers and the set of nonpositive real numbers. Let furthermore N be the set of natural numbers and N 0 = N ∪ { 0 }. Let S n = X 1 + ⋯ + X n be the sum of n ∈ N i.i.d. random variables X n ∈ R. Then the first ladder epoch { T = n } = { S 1 ≥ 0 , … , S n − 1 ≥ 0 , S n < 0 } is the epoch of the first entry of the walker into the negative semiaxis R − . By adding a constant x 0 ∈ R 0 + to all terms, we obtain a random walk with initial position at x 0 . The probability for a walker started at S 0 = x 0 to remain in the initial half-axis after n steps, that is, the probability for T to be larger than n, is called survival probability conditioned on the initial position and the site x = 0 is termed as the location of an absorbing barrier. Thus, let the survival probability conditioned on the initial position be denoted by ϕ n : R 0 + → [ 0 , 1 ] , ∀ n ∈ N. Then for a walker starting at S 0 = x 0 we have ϕ n ( x 0 ) = P ( T > n ), and it yields (1) ∫ 0 + ∞ k ( z − x 0 ) ϕ n ( z ) d z = ϕ n + 1 ( x 0 ) , x 0 ∈ R 0 + , n ∈ N 0 , with initial condition ϕ 0 ( ξ ) = 1 , ∀ ξ ∈ R 0 + , where k : R → R 0 + is the probability density function of the i.i.d. random variables X n , ∀ n ∈ N. Within the framework of the CTRW, the density function k ( x ) turns into the jump-size distribution.

Integral equation (1) is known as the Wiener–Hopf equation [12]. A first connection between random walks with an absorbing barrier and the Wiener–Hopf integrals was pointed out by D.V. Lindley in 1952 [28]. For a more general relations between the Wiener–Hopf integrals and probabilistic problems, see, e.g., reference [13, Sections XII.3a and XVIII.3]. However, as observed by W. Feller, the connections between first-passage problems and the Wiener–Hopf integrals “are not as close as they are usually made to appear” [13, Introduction to Chapter XII]. For more recent papers, see, e.g., references [21, 15, 35, 34, 3, 36, 7]. A general solution to the Wiener–Hopf equation (1) is known in the literature as the Pollaczek–Spitzer formula [3, formula (12)]. This name refers to a formula (see [49, theorem 5, formula (4.6)]) derived by F. Spitzer in 1957 [49, theorem 3, formula (3.1)] on the basis of an auxiliary formula obtained by F. Pollaczek in 1952 [40, formula (8)] but through a different method. It is possible to show that problem (1) is equivalent to a Sturm–Liouville problem with proper boundary conditions [8], and this is also an alternative and easier approach with respect to the Pollaczek–Spitzer formula for the calculation of the survival probability. In particular, one of the required boundary conditions for the Sturm–Liouville system emerged due to the Sparre Andersen theorem. Moreover, the Sparre Andersen theorem can be derived on the basis of formula (1) under the assumption that ϕ n + 1 ( · ) is of bounded variation [14]. Actually, the Sparre Andersen theorem [46] is a fundamental result in the study of the first-passage time problems.

Originally established in 1954 [46], the Sparre Andersen theorem states (2) ϕ n ( 0 ) = 2 − 2 n 2 n n ∼ 1 n π , n → ∞ . Thereby, the Sparre Andersen formula (2) is valid for arbitrary but symmetric jump-size distribution, i.e., k ( ξ ) = k ( − ξ ) with ξ ∈ R. In other words, the Sparre Andersen theorem [46] provides the exact survival probability conditioned on the initial position for symmetric Markovian random walkers starting in the same location of the absorbing barrier and states that it is independent of the jump-size distribution k ( x ) (whenever it is assumed to be symmetric). See, for example, F. Spitzer [48] and W. Feller [13, Section XII.7]. This independency uncovers a universal nature of the Sparre Andersen theorem which is reflected also in other results, see, e.g., [4, 24, 25, 34, 10, 38, 27]. Thus, we refer to this independency as the universality of the Sparre Andersen theorem. Sparre Andersen formula (2) is indeed one of the many results emerging from a deep corpus studiorum on random walks based on combinatorial arguments by E. Sparre Andersen [44, 45, 47, 46] and F. Spitzer [48–50]. For more recent and general results on the basis of probabilistic arguments, see, e.g., [2]. The general formulation of the first-passage time problem for processes with discrete-time is available in [3, 36].

The survival probability problem can be formulated also in the continuous-time setting. In particular, we consider the CTRW on homogeneous and continuous space in the uncoupled formulation as originally introduced by Montroll and Weiss in the year 1965 [37]. This means that the i.i.d. random waiting-times and the i.i.d. random jump-sizes are, at any epoch, independent of each other and also of the current position and time. Therefore, in analogy with the discrete-time setting, the walker’s position after n iterations, with n ∈ N, is given by the sum of n i.i.d. random variables X n ∈ R distributed according to the jump-size distribution k ( x ), i.e., S n = x 0 + X 1 + ⋯ + X n where x 0 ∈ R 0 + is the initial position. The actual time t ≥ 0 is given by the sum of n i.i.d. random waiting-times τ j ∈ R 0 + between two consecutive jumps, i.e., t = ∑ j = 1 n τ j with initial instant t = 0. Moreover, let the survival probability conditioned on the initial position in the continuous-time setting be denoted by Λ : R 0 + × R 0 + → [ 0 , 1 ]. Thus, by using the same approach adopted to compute the walker’s density function in the CTRW theory [37], the conditional survival probability Λ is given by the weighted superposition of all the possible discrete-time counterparts ϕ n ( · ), and it results in the series (3) Λ ( x 0 , t ) = ∑ n = 0 ∞ ϕ n ( x 0 ) Ψ n ( t ) , x 0 , t ∈ R 0 + , where the index n counts the number of occurred jumps and Ψ n ( · ) is the probability to have an elapsed time equals to t ∈ R 0 + after n ∈ N 0 jumps such that (4) Ψ n ( t ) = ∫ 0 t Ψ n − 1 ( t − τ ) ψ ( τ ) d τ , Ψ 0 ( t ) = Ψ ( t ) = 1 − ∫ 0 t ψ ( τ ) d τ , and ψ : R 0 + → R 0 + is the distribution of the waiting-times.

For the present purposes, we pass to the Laplace domain and obtain for formula (3) that (5) L { Λ ( x 0 , t ) ; s } = Λ ˜ ( x 0 , s ) = Ψ ˜ ( s ) ∑ n = 0 ∞ ϕ n ( x 0 ) ψ ˜ ( s ) n , where L { g ( t ) ; s } : = g ˜ ( s ) = ∫ 0 + ∞ e − s t g ( t ) d t is the Laplace transform of a sufficiently well-behaving function g ( t ). Hence, after splitting formula (5) in (6) Λ ˜ ( x 0 , s ) = Ψ ˜ ( s ) ϕ 0 ( x 0 ) + Ψ ˜ ( s ) ∑ n = 1 ∞ ϕ n ( x 0 ) ψ ˜ ( s ) n , by recalling that ϕ 0 ( x 0 ) = 1 , ∀ x 0 ∈ R 0 + , and expressing ϕ n ( · ) through the Wiener–Hopf integral equation (1), from formula (6) we get (7) Λ ˜ ( x 0 , s ) = Ψ ˜ ( s ) + ψ ˜ ( s ) ∫ 0 + ∞ k ( ξ − x 0 ) Λ ˜ ( ξ , s ) d ξ . Finally, after the inverse Laplace transformation g ( t ) = L − 1 { g ˜ ( s ) ; t }, the equation for the survival probability is (8) Λ ( x 0 , t ) = Ψ ( t ) + ∫ 0 t ψ ( t − ζ ) ∫ 0 + ∞ k ( ξ − x 0 ) Λ ( ξ , ζ ) d ξ d ζ , x 0 , t ∈ R 0 + , which is the analogue of formula (1) in continuous time, see also [7, 8, 42], and we recover Λ ( x 0 , 0 ) = 1 , ∀ x 0 ∈ R 0 + , because Ψ ( 0 ) = 1 by definition (4).

The Sparre Andersen theorem can also be put in the framework of the uncoupled CTRW, in particular, through formula (3). In fact, if we set x 0 = 0 in formula (5), and from the Sparre Anderson formula (2) for the discrete-time setting we take the expression of ϕ n ( 0 ), then (9) Λ ˜ ( 0 , s ) = Ψ ˜ ( s ) ∑ n = 0 ∞ 2 − 2 n 2 n n ψ ˜ ( s ) n = Ψ ˜ ( s ) 1 − ψ ˜ ( s ) = s Ψ ˜ ( s ) s = 1 s 1 − ψ ˜ ( s ) , where in the second line we used the series (10) ∑ n = 0 ∞ 2 − 2 n 2 n n z n = 1 1 − z , | z | < 1 . Formula (9) was already reported to some extent in [1, formula (7)] but not formally derived, yet. By applying the initial and final value theorems we have (11a) Λ ( 0 , 0 ) = lim s → ∞ s Λ ˜ ( 0 , s ) = 1 , (11b) Λ ( 0 , ∞ ) = lim s → 0 s Λ ˜ ( 0 , s ) = 0 , because ψ ˜ ( 0 ) = 1 and ψ ˜ ( ∞ ) = 0. After the Laplace antitransformation of the first line of formula (9), the Sparre Andersen theorem reads (12) Λ ( 0 , t ) = ∑ n = 0 ∞ 2 − 2 n 2 n n Ψ n ( t ) .

For Markovian random walks, we have that ψ ( t ) = ψ M ( t ) = e − t [53, 33], such that (13) ψ ˜ M ( s ) = Ψ ˜ M ( s ) = 1 1 + s . and then formula (9) reads (14) Λ ˜ M ( 0 , s ) = 1 s ( 1 + s ) . Consequently, formula (12) for the Markovian case gives in the original domain [42] (15) Λ M ( 0 , t ) = e − t / 2 I 0 ( t / 2 ) , where I 0 ( · ) is the modified Bessel function of the first kind of order 0 defined by the series (16) I 0 ( ζ ) = ∑ j = 0 ∞ ( ζ 2 / 4 ) j ( j ! ) 2 , ζ ∈ R 0 + , with (17) I 0 ( 0 ) = 1 , d I 0 ( ζ ) d ζ ζ = 0 = 0 , d 2 I 0 ( ζ ) d ζ 2 ζ = 0 = 1 2 . From formula (14) we have in the long-time limit, i.e., s → 0, that Λ ˜ M ( 0 , s ) ∼ 1 / s and after the Laplace inversion it gives Λ M ( 0 , t ) ∼ 1 / t for t → + ∞. This last limit is the continuous-time counterpart of limit (2).

For non-Markovian random walks, we consider the specific model [20] (18) ψ ( t ) = t β − 1 E β , β ( − t β ) , Ψ ( t ) = E β , 1 ( − t β ) = E β ( − t β ) , where E β , α ( z ) is the Mittag-Leffler function [29, Appendix E] (19) E β , α ( z ) = ∑ j = 0 ∞ z j Γ ( β j + α ) , Re { β } > 0 , α ∈ C , z ∈ C , which in the Laplace domain correspond to (20) ψ ˜ ( s ) = 1 1 + s β , Ψ ˜ ( s ) = s β − 1 1 + s β . Hence, formula (9) reads (21) Λ ˜ ( 0 , s ) = s β / 2 − 1 1 + s β . We observe that (22) Λ ˜ ( 0 , s ) = s β − 1 Λ ˜ M ( 0 , s β ) , and by applying the Efros formula [11, 51], i.e., (23) L − 1 v ( s ) W ˜ [ q ( s ) ] = ∫ 0 ∞ W ( ζ ) L − 1 v ( s ) e − ζ q ( s ) d ζ , with v ( s ) = s β − 1 and q ( s ) = s β , in the original domain we have (24) Λ ( 0 , t ) = t − β ∫ 0 ∞ e − ζ / 2 I 0 ( ζ / 2 ) M β ( ζ / t β ) d ζ , where M β ( · ) is the Mainardi/Wright function defined by the series [29, Appendix F] (25) M ν ( z ) = ∑ j = 0 ∞ ( − 1 ) j j ! z j Γ [ − ν j + ( 1 − ν ) ] , 0 < ν < 1 , z ∈ C , whose Laplace transform is known in the literature [30, (4.26)]. From formula (21) we have that Λ ˜ ( 0 , s ) ∼ s β / 2 − 1 , when s → 0, and after the Laplace inversion Λ ( 0 , t ) ∼ 1 / t β / 2 for t → + ∞.

Moreover, after reminding that the unconditional survival probability is determined by the integral ∫ 0 ∞ ρ ( ξ ) Λ ( ξ , t ) d ξ, where ρ : R 0 + → R 0 + is the distribution of the initial position, we can state the following theorem. Theorem 1.

The unconditional survival probability of a symmetric CTRW is independent of the jump-size distribution if this distribution is equal to the distribution of the initial position.

The duration of excursions of a random walker until its first comeback to the starting position is called first-return time (FRT). This observable can be understood by an example from the animal kingdom in terms of the return of an animal to a previously occupied area, such as the duration of the flight of birds when they leave and then return to their nests. This is a wide-spread behaviour associated with a number of ecological processes as site fidelity, breeding, social associations, optimal foraging [18] and is successfully modelled also by fractional kinetics [16, 52].

The FRT for the uncoupled CTRW reported at the beginning of Subsection 2.2 can be exactly calculated by using integral formula (27). In particular, we can exactly calculate the FRT for the fractional kinetics emerging from a CTRW with Mittag-Leffler distributed waiting-times (18). In this section, we consider a random walk starting at the origin x = 0 and, provided that the first jump away is of length, say, ξ, we are interested in how much time it takes to the random walker to pass through the origin for the first time after the departure. As a matter of fact, the problem is equivalent to a first-passage time problem when the random walker starts at the initial instant τ 0 from the random first-landing position ξ and the associated absorbing barrier is located at the origin x = 0. Actually, the first-landing position ξ is distributed according to the jump-size distribution k ( ξ ) and τ 0 is the delay elapsed between the two locations x = 0 and x = ξ. The CTRW formalism allows for two different formulations of the problem: first jump then wait (jw), such that τ 0 = 0, and first wait then jump (wj), such that τ 0 is a random variable distributed according to ψ ( t ). Thus, by following the literature [9, 6, 22], we define the FRT as follows: (28) FRT ≡ T ℓ ( ξ ) + τ 0 , ξ ∈ R − , T r ( ξ ) + τ 0 , ξ ∈ R + , where T is the first-passage time and, in particular, T ℓ is the first-passage time when the first jump away from the origin is towards the left into the position ξ < 0 and T r when it is towards the right into the position ξ > 0.

Let λ ( ξ , t ) = − ∂ Λ ∂ t be the first-passage time density conditioned to the starting position ξ and satisfying the normalisation condition ∫ 0 ∞ λ ( ξ , t ) d t = 1. Then the density function of the first-passage time weighted over all the possibile first-jump landing positions is (29) f ( t ) = ∫ − ∞ + ∞ k ( ξ ) λ ( ξ , t ) d ξ = 2 ∫ 0 + ∞ k ( ξ ) λ ( ξ , t ) d ξ , where the symmetry property of the jump-size distribution, k ( ξ ) = k ( − ξ ), is used.

Thus, since T and τ 0 are statistically independent, we have that the density function of their sum, namely the FRT (28), is (30a) P jw ( t ) = f ( t ) , when τ 0 = 0 , (30b) P wj ( t ) = ∫ 0 t ψ ( t − ζ ) P jw ( ζ ) d ζ , when τ 0 ∼ ψ ( t ) .

An alternative formulation is possible by starting from the discrete-time setting with the extension to the continuous-time setting in the fashion of formula (3) [1]. In particular, by defining the discrete-time first-passage time density λ n ( ξ ) = ϕ n − 1 ( ξ ) − ϕ n ( ξ ), with n ≥ 1, and exploiting the Sparre Andersen theorem (2) at ξ = 0, we have [41] (31) λ n ( 0 ) = 2 − 2 ( n − 1 ) 2 ( n − 1 ) n − 1 − 2 − 2 n 2 n n = 2 − 2 ( n − 1 ) 2 n n n 4 n − 2 − 1 4 = 2 − 2 ( n − 1 ) 2 n n 1 2 ( 4 n − 2 ) = 2 − 2 n 2 n − 1 2 n n , which correctly gives ∑ n = 1 ∞ λ n ( 0 ) = 1. Moreover, by definition (29), in the discrete-time setting we have (32) f n = 2 ∫ 0 ∞ k ( ξ ) λ n ( ξ ) d ξ = 2 ∫ 0 ∞ k ( ξ ) ϕ n − 1 ( ξ ) − ϕ n ( ξ ) d ξ = 2 ϕ n ( 0 ) − ϕ n + 1 ( 0 ) = 2 λ n + 1 ( 0 ) , n ≥ 1 , where in the third line we used (1), and it holds that f 0 = 0.

We remind that in this case P jw ( t ) = f ( t ) (see (30a)), thus in the following we study the density function f ( t ).

In particular, thanks to the relation in the Laplace domain between the first-passage time density and the corresponding survival probability, i.e., λ ˜ ( ξ , s ) = 1 − s Λ ˜ ( ξ , s ), from (29) we have (33) f ˜ ( s ) = 2 ∫ 0 + ∞ k ( ξ ) λ ˜ ( ξ , s ) d ξ = 1 − 2 s ∫ 0 ∞ k ( ξ ) Λ ˜ ( ξ , s ) d ξ = 1 − 2 1 − ψ ˜ ( s ) − 1 + ψ ˜ ( s ) ψ ˜ ( s ) = 2 − ψ ˜ ( s ) − 2 1 − ψ ˜ ( s ) ψ ˜ ( s ) , where formula (27) has been used in the third line. It can be checked that the normalisation condition f ˜ ( 0 ) = 1 is met because ψ ˜ ( 0 ) = 1. Since formula (27) from Theorem 1 has been used for deriving (33), we underline the following remark.

Moreover, if we consider the non-Markovian setting (20), from formula (33) we finally obtain that the Laplace transform of the density function of the FRT for a CTRW of (jw)-type is (34) f ˜ ( s ) = 1 + 2 s β − 2 s β s β + 1 . When β = 1, the system is Markovian. Therefore, if f M ( t ) denotes the corresponding density of the FRT, the following equality holds in the Laplace domain: (35) f ˜ ( s ) = f ˜ M ( s β ) . Hence, we can state the next theorem.

We observe that from (36) the Markovian case is recovered when β = 1 because ℓ 1 ( t ) = δ ( t − 1 ).

In the Markovian case with β = 1, from (34) we have (37) f ˜ M ( s ) = ( 1 + s ) 1 + 2 s − 2 s s + 1 1 + s = 1 1 + s + 2 s 1 1 + s − 1 s ( 1 + s ) + s 1 1 + s + 2 s 1 1 + s − 1 s ( 1 + s ) . Thus, by remembering the properties of the Laplace transform, and also the Laplace transform pair (14) and (15), we can calculate the exact result (38) f M ( t ) = e − t / 2 I 0 ( ζ ) − d I 0 ( ζ ) d ζ ζ = t / 2 − e − t + d d t e − t / 2 I 0 ( ζ ) − d I 0 ( ζ ) d ζ ζ = t / 2 − e − t = 1 2 e − t / 2 I 0 ( ζ ) − d 2 I 0 ( ζ ) d ζ 2 ζ = t / 2 , and then the integral formula (36) is fully determined.

From (38), by using the values given in (17), we have that (39) f M ( 0 ) = 1 4 , and, when plugged in (29), it gives the following universal result independent of k ( ξ ) (40) f M ( 0 ) = ∫ − ∞ + ∞ k ( ξ ) λ M ( ξ , 0 ) d ξ = 1 4 , in opposition to the first-passage time density which is indeed dependent on k ( ξ ) also at t = 0 according to λ M ( ξ , 0 ) = ∫ ξ ∞ k ( y ) d y. In the non-Markovian case (41) f ( 0 ) = + ∞ in analogy with the first-passage time density λ ( ξ , 0 ). In fact, after the change of variable ζ = ( t / χ ) β in (36), we have (42) f ( t ) = β t β − 1 ∫ 0 ∞ ℓ β ( χ ) f M ( t β / χ β ) d χ χ β ∼ C t β − 1 , t → 0 , with C = β 4 ∫ 0 ∞ ℓ β ( χ ) d χ χ β .

From (34) we can derive the asymptotic behaviour for large elapsed times. Actually, we have f ˜ ( s ) ∼ 1 − 2 s β / 2 when s → 0 which is consistent with the behaviour of e − 2 s β / 2 for small s and gives (43) f ( t ) ∼ 1 2 2 / β ℓ β / 2 ( t / 2 2 / β ) ∼ 2 t β / 2 + 1 , t → + ∞ . From (43) it follows that the mean FRT in the (jw)-case is infinite both in the Markovian and non-Markovian setting but the right tail of the density function decreases slower in non-Markovian systems. Because of formula (29), we have that scaling (43) is also the scaling of the corresponding first-passage time density λ ( ξ , t ) [22], notwithstanding the exact functions may differ.

Tools of fractional calculus can be used for studying the integral formula (36). In fact, let J μ , with μ > 0, be the Riemann–Liouville fractional integral defined by the Laplace symbol s − μ [17], then in the Laplace domain (44) L { J 1 − β f ( t ) ; s } = f ˜ ( s ) s 1 − β = ∫ 0 ∞ e − ζ s β s 1 − β f M ( ζ ) d ζ , which, by using the formula L { t − β M β ( ζ t − β ) ; s } = s β − 1 e − ζ s β , with Re { s } > 0, [30, (4.26)], gives (45) J 1 − β f ( t ) = ∫ 0 ∞ 1 t β M β ( ζ / t β ) f M ( ζ ) d ζ . Since from the study of time-fractional diffusion equations [30] we have that lim t → 0 t − β M β ( ζ / t β ) = δ ( ζ ), then (46) J 1 − β f ( t ) t = 0 = f M ( 0 ) = 1 4 . Furthermore, we can also have the Laplace transform of the cumulative density function F jw ( t ) = ∫ 0 t f ( ζ ) d ζ which, by using (35), leads to the equality (47) F ˜ jw ( s ) = f ˜ ( s ) s = s β − 1 f ˜ M ( s β ) s β = s β − 1 F ˜ M jw ( s β ) . Therefore, we can state the following theorem.

To conclude, we remind the formula [32, (6.3)] (49) t − β M β ( ζ / t β ) = t − ν ∫ 0 ∞ M η ( ζ / y η ) M ν ( y / t ν ) d y y η , β = ν η , and then from Theorem 3 we have the following corollary.

A large number of other formulae can be obtained using the existing results for the Mainardi/Wright function, e.g., [30, 29, 31, 39].

The FRT density function in the (wj)-case (30b) can be studied in the Laplace domain. In particular, by using (29) and (33), from (30b) we have (52) P ˜ wj ( s ) = ψ ˜ ( s ) f ˜ ( s ) = 2 − ψ ˜ ( s ) − 2 1 − ψ ˜ ( s ) . It can be checked that the normalisation condition P ˜ wj ( 0 ) = 1 is met and by applying the initial and the final value theorems one has (53a) P wj ( 0 ) = lim s → + ∞ s P ˜ wj ( s ) = 0 , (53b) P wj ( + ∞ ) = lim s → 0 s P ˜ wj ( s ) = 0 , because ψ ˜ ( 0 ) = 1 and ψ ˜ ( + ∞ ) = 0. In analogy with Remark 1, we underline also in this case that, since Theorem 1 lays behind the derivation of (52), we have the following remark.

Formulae (53a) and (53b) hold both in the Markovian and non-Markovian case. More explicitly, from (20) we have that the density function of the FRT for a CTRW of (wj)-type is (54) P ˜ wj ( s ) = 1 + 2 s β − 2 s β s β + 1 s β + 1 . When β = 1, the system is Markovian and therefore, denoting by P M wj ( t ) the corresponding density of the FRT, in analogy with (35), in the Laplace domain one has (55) P ˜ wj ( s ) = P ˜ M wj ( s β ) . Formula (55) can be inverted by applying the Efros formula (23), and we have the following theorem.

In the Markovian case with β = 1, from (54) we have (57) P ˜ M wj ( s ) = 1 1 + s + 2 s 1 1 + s − 1 s ( 1 + s ) , and, by remembering the properties of the Laplace transform together with the Laplace transform pair (14) and (15), we can calculate the exact result (58) P M wj ( t ) = e − t / 2 I 0 ( ζ ) − d I 0 ( ζ ) d ζ ζ = t / 2 − e − t , P M wj ( 0 ) = 0 , so that the integral formula (56) is fully determined.

We can quantify the difference between the two cases (jw) and (wj). In fact, first by comparing (38) and (58) we have the proposition.

Later, by using (36) and (56), we find (60) P wj ( t ) = 2 P jw ( t ) − ∫ 0 ∞ ζ − 1 / β ℓ β ( t / ζ 1 / β ) e − ζ d ζ − ∫ 0 ∞ ζ − 1 / β ℓ β ( t / ζ 1 / β ) e − ζ / 2 d I 0 d χ − d 2 I 0 d χ 2 χ = ζ / 2 d ζ . The second term in the r.h.s. can be solved and put in a more clear form. In fact, by using some formula concerning Lévy density function [30, (4.26)], Mainardi/Wright function [30, (4.32)] and Mittag-Leffler function [29, (1.45)], we have the equalities (61) ∫ 0 ∞ ζ − 1 / β ℓ β ( t / ζ 1 / β ) e − ζ s β d ζ = β t ∫ 0 ∞ ζ t β M β ( ζ / t β ) e − ζ s β d ζ = − s 1 − β t ∂ ∂ s ∫ 0 ∞ M β ( y ) e − t β s β y d y = − s 1 − β t ∂ ∂ s E β ( − t β s β ) = t β − 1 E β , β ( − t β s β ) . Thus, by remembering (18) and setting s = 1, one has (62) ∫ 0 ∞ ζ − 1 / β ℓ β ( t / ζ 1 / β ) e − ζ d ζ = ψ ( t ) . And so we have the following proposition.

From (54) we can derive the asymptotic behaviour for large elapsed times and, in analogy with (43), it is (64) P wj ( t ) ∼ 1 2 2 / β ℓ β / 2 ( t / 2 2 / β ) ∼ 2 t β / 2 + 1 , t → + ∞ , from which it follows that the mean FRT also in the (wj)-case is infinite both in the Markovian and non-Markovian settings, and the right tail of the density function decreases slower in non-Markovian systems.

The analogies between the (jw)- and the (wj)-cases also include formulae (45) and (46) that now read as (65) J 1 − β P wj ( t ) = ∫ 0 ∞ 1 t β M β ( ζ / t β ) P M wj ( ζ ) d ζ , (66) J 1 − β P wj ( t ) t = 0 = P M wj ( 0 ) = 0 , as well as the analogue of Theorem 3, given below.