Limit properties of Lévy walks

In this paper we study properties of the diffusion limits of three different models of Lévy walks (LW). Exact asymptotic behavior of their trajectories is found using LePage series representation. We also prove an existing conjecture about total variation of LW sample paths. Based on this conjecture we verify martingale properties of the limit processes for LW. We also calculate their probability density functions and apply this result to determine the potential density of the associated non-symmetric α-stable processes. The obtained theoretical results for continuous LW can be used to recognize and verify this type of processes from anomalous diffusion experimental data. In particular they can be used to estimate parameters from experimental data using maximum likelihood methods.


Introduction
Lévy walks (LW) were introduced in pioneering papers written by Shlesinger et al [29] in 1982 and Klafter et al [14] in 1987. Since then these stochastic processes have quickly become useful models in physics and biology describing superdiffusion [11,23]. Among the most spectacular applications of LW one should mention: modeling of migration of swarming bacteria [1], light transport in a special optical material known as Lévy glass [2], foraging patterns of animals [5] or blinking nanocrystals [22]. There is also a recent article in Nature where authors show that LW can be used to describe transport of certain proteins (neuronal mRNP)-see [31]. For more applications we refer the curious reader to the review paper [34].
LW can be defined in the framework of continuous time random walks (CTRW) [6,13,30]. This approach is used in our manuscript. It should be added that LWs can also be defined using subordinated Langevin equations, see [7,17] for the details. Let T i be a sequence of independent, identically distributed (IID) waiting times such that P(T i > x) ≈ Cx −α when x → ∞, α ∈ (0, 1), C > 0. Here by f(x) ≈ g(x) we mean lim x→∞ f (x) g(x) = 1. The jumps are defined as J i = V i T i where V i govern the direction of the jump. It is assumed that V i are IID with P(V i = 1) = p and P(V i = −1) = q, q = 1 − p, p ∈ [0, 1]. Notice that waiting times T i and jumps J i are dependent. This coupling allows the CTRW process to have heavy-tailed jumps-a phenomenon which is often observed in experimental data-but still the process can have a finite mean square displacement (variance) which makes it a suitable physical model [8].
Note that the parameter α is restricted here to the case α ∈ (0, 1). This is due to the fact that for α > 1 the diffusion limit of LW is the well-known and studied α-stable Lévy motion (Lévy flight) [16]. This is in sharp contrast with the case α ∈ (0, 1), for which the diffusion limit is a completely different, much more complicated process [16,17].
The renewal process is given by N (t) = max n 0 : n i=1 T i t . In mathematical and physical literature there appear three different types of LW [8,16]. Wait-first Lévy walk (also known as undershooting Lévy walk) is defined as: The second model-jump-first Lévy walk (overshooting Lévy walk) differs from wait-first LW by changing the order of waiting and jumping: Jump-first LWR (t) seems to be similar to the wait-first model R(t), but has completely different properties-for instance its second moment diverges. This makes it less suitable for physical applications but still is very interesting in the context of limit diffusion processes. Finally Lévy walk (or continuous LW) is given by L(t) is the most useful process from a physical point of view-it has a finite second moment and continuous trajectories. Figure 1 shows sample trajectories of these three different LW.
In this article we determine the exact asymptotic behavior of trajectories of the limit processes and their total variation (proving an already existing conjecture [16]) using LePage series representation for stable process (M α (t), U α (t)). Using the proven conjecture we also verify martingale properties of the limit processes. Moreover in section 3 we determine the distributions of all limit processes X(t), Y(t) and Z(t) for all values of p ∈ (0, 1) and α ∈ (0, 1). Although the limit processes are composition of α-stable processes their probability density functions (PDFs) are given by elementary functions. In the last section we take advantage of the calculated densities and obtain formulas for the potential densities of the associated non-symmetric α-stable processes.
The obtained theoretical results for continuous LW can be used to recognize and verify this type of processes from anomalous diffusion experimental data. In particular, the explicit density functions can be used to estimate parameters from experimental data using maximum likelihood methods.

Asymptotic properties of trajectories
We are going to analyze the limit processes X(t), Y(t) and Z(t) for different values of the parameter p ∈ [0, 1] which governs the probability of jumping up and down. To underline the dependence on p sometimes we are going to write it in the subscript: X p (t), Y p (t) and Z p (t). Let us begin with the non symmetric case p = 1. The lower bound asymptotic behavior for the process X 1 (t) is known:

Theorem 1 (Theorem 3.4 in [4]).
Let f : (0, ∞) → (0, ∞) be a continuous strictly increasing function with lim x→∞ f (x) Then for wait-first Lévy walk X 1 (t) we have almost surely In particular it follows that and almost surely for ε > 0. The following easy to prove fact concerns the upper bound asymptotics.

Proposition 1.
For the process X 1 (t) we have Proof. The process X 1 (t) is bounded, that is X 1 (t) t almost surely. When p = 1 we also have X 1 (t) = G(t) = (U − α (S α (t))) + . Moreover there exists a sequence of jumping times t n of U α (t) such that t n → ∞. Now since X 1 (t n ) = G(t n ) = t n the proof is completed.
Notice that the case p = 1 is trivial for continuous Lévy walk since Z 1 (t) = t. Due to the symmetry analogous results can be obtained immediately for p = 0. Let us move to the more complicated case p ∈ (0, 1).

Theorem 2.
Wait-first Lévy walk X p (t) with p ∈ (0, 1) has the following asymptotic behavior Proof. Fourier transform of the law of M α (t) can be written as where β = 2p − 1. Therefor the limit process M α (t) admits the following LePage series representation (see [15] and example 3.10.3 in [27]) where D α = 1−α Γ(2−α) , d = denotes equality of all finite-dimensional distributions, (γ i ) i∈N is a sequence of IID random variables satisfying (Γ i ) i∈N is a sequence of arrival times of a Poisson process with unit arrival rate and (W i ) i∈N is a sequence of IID random variables uniformly distributed on [0, 1]. These sequences are assumed to be independent. This representation holds for t ∈ [0, 1], but since Lévy processes have independent and stationary increments we can extend it for all t ∈ [0, ∞) is a sequence of arrival times of a Poisson process N j (t) with unit arrival rate for each j ∈ N (the processes N j (t) are independent) and [W i, j ] is a matrix of independent random variables such that W i, j is uniformly distributed on [ j − 1, j] for each i ∈ N. We assume that the matrices and sequences are independent. Intuitively describe magnitude of the jumps, γ i, j govern their direction and W i, j correspond to moments of their occurrence. We can apply this representation also to the limit Lévy process (M α (t), S α (t)) obtaining This is a consequence of the fact that |ΔM α (t)| = ΔU α (t) for all t almost surely. Now if we remove all the direction components γ i, j from the series representation (we change the direction of all jumps to the upward one), we obtain the series representation of the process (U α (t), U α (t)). theorem 1 assures us that This in turn means that infinitely often (i.o.) as t → ∞ we can observe 'large' jumps of the process U α (t) compared to the value of this process itself at time point t. More precisely let ε > 0. Then We are going to show that for p ∈ (0, 1) we also observe both 'large' positive and 'large' negative jumps of the process M α (t) compared to the value of the process U α (t) i.o. as t → ∞. To achieve this goal we number the elements of the matrix [i, j] i, j∈N -let P n = (i n , j n ). We choose the sequence P n such that it satisfies the condition j n → ∞. This can be done for instance by numbering the elements of the matrix using diagonals. Now we use the series representation and define two sequences of events-(A n ) n∈N and describes the event that at time W i n , j n there was a 'large' jump. Since W i n , j n ∈ [ j n − 1, j n ], j n → ∞ from equation (24) we get that P( lim sup n→∞ A n ) = 1 almost surely. Next let B n = {γ (i n , j n ) = 1}. From Borel-Cantelli lemma we get that P( lim sup n→∞ B n ) = 1 because p ∈ (0, 1). Since (A n ) n∈N and (B n ) n∈N are independent it follows that P( lim sup n→∞ (A n ∩ B n )) = 1. This Therefore almost surely i.o. as t → ∞. The choice of ε > 0 was arbitrary thus we conclude that The proof of the second part of theorem (lim inf t→∞ X p (t) t = −1 almost surely) is analogous. The next corollaries concern continuous Lévy walk Z p (t).

Corollary 1.
Let p ∈ (0, 1). Continuous Lévy walk Z p (t) has the following asymptotic behavior Proof. The values of X p (t) and Z p (t) coincide for t ∈ [0, ∞] : ΔX p (t) = 0. Notice that equation (26) holds also i.o. as t → ∞ for t such that X p (t) = 0. Thus we deduce that i.o. as t → ∞ almost surely for arbitrary ε > 0. The proof of the lower bound asymptotics is analogous.
Proof. This is an immediate consequence of the previous corollary combined with the fact that Z p (t) has continuous trajectories.
It remains an open problem to find the asymptotic behavior of trajectories of jump-first Lévy walk Y p (t).

Total variation
In this section we calculate the total variation for the process X(t), Y(t) and Z(t) on a time interval [0, T], T > 0 using LePage series representation. The result for the limit process of continuous Lévy walk proves the hypothesis from [16]. In the next section we also use these findings to verify martingale properties of the limit processes.
where supremum is taken over all partitions

Proposition 2. The total variation of the process X p (t) on a time interval
where Now we take advantage of the series representation given by equation (22) and use the fact that changing direction of all jumps of X p (t) to positive we obtain the jumps of the process The proof of the following result for the process Y p (t) is analogous.

Proposition 3. The total variation of the process Y p (t) on a time interval
where H(T) = (U α (S α (T))) + .
We can now move to the continuous case-the theorem below appears as hypothesis in [16].
Proof. Recall that where R(U α ) is a (countable) set of values of the process U α (t), The trajectory of Z p (t) is continuous Moreover during calculation ] is a straight line the variation calculated based on P 0 is not smaller than the variation calculated based on P 0 . The values of the processes Z p (t) and X p (t) coincide for t ∈ R(U α ). Also notice that the values of the process X p (t) are constant on the interval [G(t 0 ), H(t 0 )), so the total variation for the process X p (t) can also be calculated based only on partitions which contain only points from the set R(U α ). Therefore from proposition 2 we get Therefore Remark 1. The above proof remains valid also in case of a multidimensional Lévy walk [16]. The only difference is in the series representation given by equation (22). Instead of random variables γ i, j we have unit random vectors.

Martingale properties
In this section we will verify if the limit processes are martingales with respect to the natural filtration. Recall that the natural filtration for the process (W(t), t 0) is defined as F W t = σ(W(s) : s t) and the process W(t) is a martingale with respect to the natural filtration if Here σ(W(s) : s t) is the sigma algebra generated by W up to time t.

Proposition 4. The limit process X p (t) for wait-first Lévy walk is a martingale if and only if p = 1/2 (symmetric case).
Proof. First assume that p = 1/2. Then EX p (t) = tEX p (1) since X p (t) is self-similar with an index of self-similarity equal to 1 and EX p (1) = 0 which follows from the lack of symmetry (the exact distribution of X p (t) is given in theorem 4). Thus the expected value is not constant over time and X p (t) is not a martingale in this case. Next assume p = 1/2. We will show that in this case X p (t) is a martingale. Since |X p (t)| t we have E |X p (t)| < ∞. It remains to prove the second martingale condition. Let 0 s < t, then almost surely Once again we use the series representation (equation (22)). Notice that the directions of all jumps of X p (t) on an interval (s, t] are defined by the random variables γ i, j which are independent of the previous history of the process on a time interval [0, s]. The length of the next jump after time s may depend on this history but the direction does not. Since for all i, j ∈ N we have and therefore E[X p (t)|F X p s ] = X p (t) almost surely for p = 1/2. The situation is different for the jump-first process.

Proposition 5. The process Y p (t) is not a martingale for all p ∈ [0, 1].
Proof. The first martingale condition is not fulfilled-E |Y p (t)| = ∞-this can be calculated directly using the distribution of Y p (t) which is given in theorem 5.
Finally we look at the continuous process Z p (t). It is not a martingale similarly as Y p (t) but for different reasons (contrary to Y p (t), the process Z p (t) is bounded).

Proposition 6. The process Z p (t) is not a martingale for all p ∈ [0, 1].
Proof. The limit process Z p (t) has continuous trajectories. We also proved that it is of bounded variation (v T 0 (Z p (t) = t). However from martingale theory we now that the only continuous process of bounded variation is the constant process (theorem 30.4 in [26]). Thus Z p (t) is not a martingale.

Probability distributions of the limit processes
In this section we are going to find explicit formulas for PDFs of the limit processes X p (t), Y p (t) and Z p (t) for all values of parameter p ∈ (0, 1) and α ∈ (0, 1). It turns out that these densities are given by elementary functions. This extends results from [8] where PDFs were obtained only in the symmetric case p = q = 1/2. Also the results from this section simplify findings from [19] where PDFs of the jump processes X p (t) and Y p (t) were expressed using integrals of special functions. ∈ (0, 1). The PDF p(x, t) of the process X p (t) equals

Theorem 4. Let p
where Proof. In the proof we will use techniques from [8,9]. Fourier-Laplace transform of the density p(x, t) of the process where Our process X(t) is self-similar with an index of self-similarity equal to 1 so the PDF has the form p(x, t) = 1 t φ x t for a certain scaling function φ. In such scenario it is possible to invert Fourier-Laplace transform using Sokhotski-Weierstrass theorem. We have Therefore The density of φ(x) of the random variable X(1) can be written as From Sokhotsky-Weierstrass theorem we get [32] lim where P.V. denotes the principal value. Thus After combining the equations (48) and (50) we obtain (51) Therefore Now we apply the above formula for the function g(ξ) given by equation (45) φ For y ∈ (0, 1) we have When y ∈ (−1, 0) we perform analogous calculations or alternatively take advantage of the following symmetry p p (x, t) = p q (−x, t) where p p (x, t) and p q (x, t) denote the PDFs of the processes X p (t) and X q (t) respectively . Next when |y| > 1 and φ(y) = 0.
The PDF of the jump-first process Y p (t) is calculated in the theorem below.
Theorem 5. Let p ∈ (0, 1). The PDF p(x, t) of the process Y p (t) is given by where Proof. We proceed similarly as in the proof of the previous theorem. Fourier-Laplace transform of the density p(x, t) of the process where Therefore For y ∈ (0, 1) this yields For y > 1 the result is different Finally we move to the continuous process Z(t).
Theorem 6. Let p ∈ (0, 1). The density p(x, t) of the process Z p (t) equals where Proof. The proof is analogous as in the case of jump processes X(t) and Y(t). where This implies that When y ∈ (0, 1) we obtain For y > 1 we get φ(y) = 0.
It is also worth to mention here that for d-dimensional rotationally invariant LW when d is odd, the densities of the limit processes are also given by elementary functions [20,21].

Potentials of non-symmetric α-stable processes
It turns out that the results for densities we laboriously derived in the previous section can be applied to get a formula for potential of the α-stable process (M α (t), U α (t)) defined in equation (6). Before we proceed further let us recall some basic facts from potential theory. The potential measure U(dx) of the d-dimensional stochastic process W(t) is defined as For a symmetric (rotationally invariant) stable process W(t) with Fourier transform E exp −i k, W (t) = exp −t k α the potential measure U(dx) is absolutely continuous with respect to Lebesgue measure and its density u(x) is given by Riesz potential [28] u where C α,d is a constant which depends on α and d. When W(t) is non-symmetric, the situation is more complicated. In a general case Fourier transform of a d-dimensional α-stable process W(t) with α ∈ (0, 1) is given by [28,33] where λ is a finite measure on the unit sphere S = {x ∈ R d : x = 1}. It is the spherical part of the Lévy measure corresponding to W(t). When λ is a uniform measure we recover the symmetric case. In [33] potential theoretic properties of W(t) were studied under the assumption that λ has a density with respect to the surface measure which is bounded and bounded away from zero. In this context estimates for the density of the potential measure were obtained.
Here we focus on the process (M α (t), U α (t)) for which λ is an atomic measure . This corresponds to the Lévy measure given by equation (7). Interestingly, it turns out that in this setting it is possible to obtain explicit expressions for the potential density u(x) and the results are non-comparable with the symmetric Riesz potential as it was the case in [33]. Subsection 4.2 is devoted to the case where M α (t) is a d-dimensional rotation invariant α-stable process. We apply similar methods to derive the potential density of the d + 1-dimensional process (M α (t), U α (t)) which is associated with ddimensional Lévy walk (see [16]). In this scenario the spherical part λ of Lévy measure from equation (69) equals where x = (x 1 , . . . , x d , x d+1 ) and ν is a uniform measure on a hypersphere S = {y ∈ R d : y = √ 2/2}. The potential density has a particularly elegant form when d = 3 and α = 1/2 as corollary 3 shows.

Two-Dimensional process
Let us begin with the two-dimensional process (M α (t), U α (t)) and denote its potential density as u(x, t).

Theorem 7. The potential density density u(x, t) is given by the formula
where the scaling function ψ(y) is given by dw for − 1 < y < 0 0 otherwise. and integrands η + and η − are elementary functions η + (y) = p sin (απ) α πΓ (α) (1 − y) 1−α y 1−α p 2 (1 − y) 2α + q 2 (1 + y) 2α + 2pq 1 − y 2 α cos (απ) 2 and η − (y) = q sin (απ) α πΓ (α) Proof. The main idea behind finding the potential density is its relation with the density p(x, t) of the wait-first limit process X p (t). More precisely the density p(x, t) can be expressed in terms of the density u(x, t) of the potential measure U in the following way [25] p( which was also obtained using different methods in [19]. In the symmetric case p = q = 1/2 (and α = 1/2) the function ψ is particularly simple ψ (y) = 1 √ π 1 (−1,1) (y) and the potential equals The scaling function ψ(y) is supported on the interval (−1, 1). It turns out that ψ displays an interesting behavior near the endpoints −1 and 1 which depends heavily on the value of α.
Proposition 7. The scaling function ψ(y) has the following asymptotics Proof. Due to the symmetry ψ c (y) = ψ 1−c (−y) (c in the subscript denotes here the value of the parameter p) it is enough to prove to asymptotic behavior when y → 1 − . From theorem 7 we have for 0 < y < 1 We have the following equivalence ψ (y) ≈ We can notice that η + (w) ≈ g(w) when w → 1 − . Therefore The left panel of figure 2 presents a shape of the scaling function ψ(y) for different values of α. Clearly α = 0.5 is the borderline case. The behavior of ψ changes drastically when α > 0.5 and α < 0.5. The right panel shows the plot of the potential u(x, t) as a function of 2 variables. Recall that u (x, t) = t α−2 ψ x t . We can notice how the shape of ψ is propagated when t increases.

Remark 2.
In [19] the potential density of (M α (t), U α (t) was obtained using Meier Gfunction. Combining the two different formulas yields an interesting integral identity.