Aging uncoupled continuous time random walk limits

Aging is a prevalent phenomenon in physics, chemistry and many other fields. In this paper we consider the aging process of uncoupled Continuous Time Random Walk Limits (CTRWL) which are Levy processes time changed by the inverse stable subordinator of index $0<\alpha<1$. We apply a recent method developed by Meerscheart and Straka of finding the finite dimensional distributions of CTRWL, to obtaining the aging process's finite dimensional distributions, self-similarity-like property, asymptotic behavior and its Fractional Fokker-Planck equation.


Introduction
Continuous time random walks(CTRW) are widely used in physics and mathematical finance to model fractional diffusions which in many cases better describe phenomena in these fields. Continuous time random walk limits (CTRWL) are used to model anomalous diffusion, where the squared averaged distance of the process from the origin is no longer proportional to the time index t. Nevertheless, CTRWL are usually not Markovian a fact that makes the calculation of their finite dimensional distributions not so simple. It is therefore that the distribution of the increments (which can be obtained by the finite dimensional distributions) of the CTRWL are not well understood.
In this paper we wish to study the distribution of the increments of a large class of CTRWL that consists of all processes of the form Y t = A Et where A t is a Levy process that is time changed by the inverse of an independent stable subordinator of index 0 < α < 1; we denote this class by S . Section 2 is devoted to a brief review of the theory and method introduced by M. Meerschaert and P. Straka in [16] and [17] upon which be base our results. In section 3 we give the main result of this paper, that the distribution of the increments of the process Y t ∈ S can be obtained by a convolution in time of the one dimensional distribution of Y t and a generalized beta prime distribution. In section 4 we obtain results on the asymptotics of the distribution of increments of Y t that are far from the origin and in some cases the governing equation for the process starting from t 1 > 0.
One example for a process that lies in S is the Fractional Poisson Process(FPP) which we denote by N α t . The FPP is a renewal process with interarrival times W n such that P ( is the Mittag-Leffler function. Since the interarrival times are not exponentially distributed the process W i are the arrival times, is not Markovian and the calculation of the finite dimensional distributions of N α t is no longer straightforward. The FPP was first studied in [10], [8] and [11,12]. In [3] an integral representation of the one dimensional distribution of the FPP was given and was used in [19] to find and simulate the finite dimensional distribution of the FPP. In [13], it was shown that N α t = N Et were N t is a Poisson process and E t is the inverse of a standard stable subordinator of index 0 < α < 1. Since the distribution of the increments of the CTRWL are closely related with the two dimensional distributions, the study of the increments is quite cumbersome. In a recent paper [16], Meerscheart and Straka found a way of embedding Continuous Time Random Walks Limits (CTRWL) in a larger state space that renders the process Markovian. We use this method to find an interesting representation of the increments distribution. We then proceed in finding asymptotic behavior of the increments distributions and their governing equations.

Finite dimensional distribution of CTRWL
Although the method in [16] is very general we focus only on uncoupled CTRWL. In order to facilitate reading of this section and referring to the original paper we retain most of the notation in [16]. Uncoupled CTRW consist of two independent sequences of iid r.vs, {W c n } and {J c n } . The parameter c is the convergence parameter as in [14] which allows us to construct infinitesimal triangular arrays. Here, {J c n } represents the size of the jumps of a particle in space, while {W c n } represents the waiting times between jumps. Hence, the time of the particle's k'th jump is T c k = D c 0 + k i=1 W c i and the position of the particle is S c Assume we have where ⇒ denotes weak convergence in the Skorokhod J 1 topology. Now, let E t = inf {s : D s > t} be the entrance time of the subordinator D t , also called the inverse of the stable subordinator D t . By [23,Theorem 2.4.3], for independent space and time jumps we have where convergence is in the Skorokhod J 1 topology, see also [24, theorem 3.6]. Since the CTRWL are semi-Markov processes, it is possible to embed them in a process of larger state space that includes the time to regeneration, the remaining life time process R t . More precisely, let D [0, ∞), R 2 be the space of cadlag functions f : [0, ∞) → R 2 with the J 1 Skorokhod topology which is endowed with transition operators T u , u > 0 and hence a probability measure P χ,τ . Thus, we have a stochastic basis (Ω, F ∞ , F u , P χ,τ ), where each element of Ω is a trajectory of the process (A, D) t , F u = σ ((A u (ω) , D u (ω))) and F ∞ = ∨ u>0 F u . The process (A, D) t has a generator of the form The occupation time measure of the process (A, D) t is the average time spent by the process in a given Borel set in When U χ,τ (dx, dt) is absolutely continuous with respect to the Lebesgue measure we write u χ,τ (x, t) for its density. Let us now define the remaining life time process R t It was proven in [16,Theorem 2.3] that It turns out that when the coefficients b (x, t) , γ (x, t) , a (x, t) and K (x, t; dy, dw) do not depend on t, the process (Y t , R t ) is a homogeneous Markov process. More precisely, we define One can use the Chapman-Kolmogorov's equation to obtain the finite dimensional distribution of the process Y t . For example, for the two dimensional distribution of the process Y t at times t 1 < t 2 we have Remark 1. In [16] a result stronger than (2.5) was shown. Indeed, the process (Y t , R t ) is a strong Markov process with respect to a filtration larger than the natural filtration. For the sake of brevity and the fact that the Markov property is adequate for our work we brought the result in a weaker form.

Stationarity kernels of increments
Let us assume (2.1) holds, and so A t is a Levy process with CDF P t (x) = P (A t ∈ (−∞, x]) and with Levy triplet (µ, A, φ), i.e and is known to be absolutely continuous with respect to the Lebesgue measure [25, section 2.4].
Since (A, D) t is a Levy process the coefficients in (2.3) are independent of t and therefore the process (A, D) t is a Markov additive process [16, section 4] and the occupation measure is of the form U y (dx, dt) =´∞ 0 P u (dx − y) g (t, u) dudt. We say that the r.v X has beta distribution with parameters α, β > 0 if it has pdf of the form is the Beta function and we write X ∼ B (α, β) . We say that the r.v X has beta prime distribution with parameters α, β > 0 if it has pdf of the form x > 0 and we write X ∼ B ′ (α, β). It was noted in [7,II.4] that if X ∼ B (α, β) then X 1−X ∼ B ′ (α, β). The distribution (3.2) can be further generalized to the so called generalized Beta prime distribution also known as the general Beta of the second kind distribution whose pdf is with h, α, β > 0 . If X has generalized Beta prime distribution then we write X ∼ B ′ (α, β, h). We shall need the following lemma.
Lemma 2. Let 0 < α < 1, t, r > 0. Then Proof. Integrate the left side of equation ( with respect to r and use Tonelli's theorem to obtain where we used the identity I Now, apply the change of variables u = r t+r to see that where A t is a Levy process and E t is the inverse of a stable subordinator of index 0 < α < 1 with E e −uDt = e −tcu α independent of A t . Then for B ⊆ R a Borel measurable subset such that 0 / ∈ B , where p t1 (x) is the pdf of a r.v X and X ∼ B ′ (1 − α, α, t 1 ).
Proof. Note that for a Borel set B and some t 1 , t > 0 the increment of the process Y t is where E ⊆ R 4 is a measurable set in B R 4 , the Borel sigma algebra on R 4 . The set E can be written explicitly as The relevant indicator function is where the last equality follows from the fact that 0 / ∈ B. Then, by plugging (3.8) in (3.7) we have By [21, eq. 37.12] if D t is a stable subordinator of index 0 < α < 1 with E e −uXt = e −tcu α and probability distribution P (D t ∈ dx) = g (x, t) dx then its potential density is given by Substitute (3.10) in (3.9) and apply the change of variables r = w ′ − (t 1 − s ′ ) to obtain and it can easily be seen that p t1 (x) is the pdf of X where X ∼ B ′ (1 − α, α, t 1 ).
Remark 4. It follows from theorem 3 that Therefore, the distribution of Y t1+t − Y t1 has an atom at the origin for every t. More remarkable is the fact that if P (A t = 0) = 0(this is true for all processes with pdf) than P (Y t1+t − Y t1 = 0) does not depend on the choice of the process A t . One the other hand it can be easily seen that for every Henceforth we shall denote by B a Borel set in R such that 0 / ∈ B.
It is possible to obtain the asymptotics of P (Y t1+t − Y t1 ∈ B) as t 1 goes to infinity as suggested by the following corollary.
Corollary 5. Let B ⊆ R be a Borel measurable subset such that 0 / ∈ B, then Proof. By theorem3 and lemma 2 where we have used the identity is the Pochammer symbol. We now let t 1 → ∞ and by noting lim Remark 6. When the process Y t is a renewal process, that is the case for the FPP, the convergence of P (Y t1+t − Y t1 ∈ B) to zero is expected by the renewal theorem [7, XI.1] and the fact that the interarrival times have the Mittag-Leffer distribution with infinite expectation. Interestingly, it was shown by Erickson in [6,Theorem 1] , that if Y t is a renewal process with interarrival times T n with F (t) = P (T 1 ≤ t) such that 1 − F (t) ∼ t −α as t → ∞ for 0 < α < 1 and F is not arithmetic, then where C is a constant. In view of 3.13 it would be interesting to verify whether (3.12) can be made strict, that is, While it is known that generally CTRWL lose their stationarity property for 0 < α < 1 [5, section 3.6], theorem 3 suggests a way of measuring the stationarity of a process in the class S. The FPP for example has no stationary increments for 0 < α < 1, however, for α = 1 we obtain the Poisson process which is of course stationary as being a Levy process. We proceed with a useful lemma that states that processes in S are stochasticly continuous.
where h (x, t) is the pdf of the process E t . Then It was noted in [18, eq. (2.9) is the pdf of D 1 , the subordinator D t evaluated at t = 1. Since by [25] g (x) is smooth it follows that h (x, t) is continuous on t, x > 0 . Trivially we have Hence, a basic result in analysis [20,Theorem 7] implies that and the results follows.
The next theorem states that as α → 1 the process Y t , in some sense, becomes more stationary.
Theorem 8. Let Y t ∈ S, then for every t, t 1 > 0 Proof. In [22, eq. 3.1.19] it was shown that where U (a, b, s) is a hypergeometric function that solves the confluent hypergeometric equation, also known as Kummer's equation By (2.1) and a simple change of variables we find that the Laplace transform of the generalized Beta prime distribution is given by Particularly, define and note that by [4] since the subordinator D t has no drift and has a Levy measure φ (dy) = cα Γ(1−α) y −1−α 1 {y>0} dy such that φ (0, ∞) = ∞, it is a strictly increasing subordinator. Therefore, we have E 0 = 0 which implies P (Y 0 ∈ dx) = δ 0 (dx) and P (Y t−r ∈ B) = 0 at r = t. Consequently, lemma 7 suggests that f (r) is continuous. By the fact that we also have and by Portmanteau theorem the proof is completed.