On Self-Similar Bernstein Functions and Corresponding Generalized Fractional Derivatives

We use the theory of Bernstein functions to analyze power law tail behavior with log-periodic perturbations which corresponds to self-similarity of the Bernstein functions. Such tail behavior appears in the context of semistable Lévy processes. The Bernstein approach enables us to solve some open questions concerning semi-fractional derivatives recently introduced in Kern et al. (Fract Calc Appl Anal 22(2):326–357, 2019) by means of the generators of certain semistable Lévy processes. In particular, it is shown that semi-fractional derivatives can be seen as generalized fractional derivatives in the sense of Kochubei (Integr Equ Oper Theory 71:583–600, 2011) and generalized fractional derivatives can be constructed by means of arbitrary Bernstein functions vanishing at the origin.

Its first derivative f = ψ ′ is a completely monotone function, i.e.
Due to a celebrated result of Bernstein, the completely monotone function f is the Laplace transform of a unique Borel measure µ on [0, ∞) As a consequence, the Bernstein function admits the representation Key words and phrases.Power law tails, log-periodic behavior, Laplace exponent, Bernstein functions, self-similarity, discrete scale invariance, semistable Lévy process, semi-fractional derivative, semi-fractional diffusion, Sonine kernel, Sibuya distribution, space-time duality.
In Section 2 we will introduce a self-similarity property for Bernstein functions which is intimately connected to the following class of functions.We call a function θ : R → R admissable with respect to the parameters α ∈ (0, 2) \ {1} and c > 1 if the following three conditions are fulfilled: θ(x) > 0 for all x ∈ R, (1.5) the mapping t → t −α θ(log t) is non-increasing for t > 0, (1.6) θ is log(c 1/α )-periodic.(1.7)In case α ∈ (0, 1) we use an admissable function θ to define (1.8) φ(t, ∞) := t −α θ(log t) for all t > 0 as the positive tail of a Lévy measure φ concentrated on (0, ∞) which belongs to a semistable distribution ν with log-characteristic function (1.9) ψ(x) = ∞ 0 e ixy − 1 dφ(y) for all x ∈ R given uniquely by the Fourier transform ν(x) = R e ixy dν(y) = exp(ψ(x)).The corresponding Lévy process (X t ) t≥0 given by E[exp(ix • X t )] = exp(t • ψ(x)) for all t ≥ 0 and x ∈ R is called a semistable subordinator.For details on semistable distributions and Lévy processes we refer to the monographs [20,29].The power law of order α ∈ (0, 1) is given by the non-local operator where L is the generator of the corresponding semistable Lévy process and at least functions f in the Sobolev space W 2,1 (R) belong to the domain of the semi-fractional derivative.In terms of the Fourier transform we can equivalently rewrite (1.10) as (1.11) where the Fourier transform of f is given by f (x) = R e ixy f (y) dy; see [12] for details.In Section 2, we will start with the elementary observation that for α ∈ (0, 1) there is a one-to-one correspondence between self-similar Bernstein functions given as the Laplace exponent ψ(x) = −ψ(ix) for x > 0 and semistable Lévy measures φ of the form (1.8).In particular, this enables us to show that semi-fractional derivatives of order α ∈ (0, 1) can be seen as a special case of generalized fractional derivatives in the sense of [16].A constant function θ corresponds to the complete Bernstein function ψ(x) = x α and an ordinary fractional derivative of order α ∈ (0, 1).For details on classical fractional derivatives we refer to the monographs [15,25,27].
In Section 3 we will prove a discrete approximation formula of the generator in (1.10) involving a generalized Sibuya distribution given in terms of the self-similar Bernstein function.
In case α ∈ (1, 2) we use an admissable function θ to define (1.12) φ(−∞, −t) := t −α θ(log t) for all t > 0 as the negative tail of a Lévy measure φ concentrated on (−∞, 0) which belongs to a different semistable distribution ν with log-characteristic function Given an admissable function θ with respect to α ∈ (1, 2) and c > 1 with corresponding semistable Lévy measure φ given by (1.12), by Definition 2.5 in [12] the negative semi-fractional derivative ) is given by the non-local operator where L is again the generator of the corresponding semistable Lévy process and at least functions f in the Sobolev space W 2,1 (R) belong to to the domain of the semifractional derivative.For α ∈ (1, 2) the function x → ψ(−ix) cannot be a Bernstein function, but we will show in Section 4 that it has an inverse which is a self-similar Bernstein function.This will enable us to solve an open question from [11] concerning space-time duality for semi-fractional differential equations.
Definition 2.1.We call a Bernstein function ψ self-similar with respect to α ∈ (0, 1) and c > 1 if it admits the discrete scale invariance The following elementary observation is our key result.
Remark 2.3.If we assume that θ is smooth in the sense that it is continuous and piecewise continuously differentiable, then it admits a Fourier series representation In this case the function γ appearing in Lemma 2.2 is given by the modified Fourier series which can be seen as follows.By Theorem 3.1 in [12] the coefficients of this series appear in a representation of the log-characteristic function and the relation γ(x) = e αx ψ(e −x ) = −e αx ψ(ie −x ) easily shows (2.4).
A natural question which arises is if for two admissable functions θ 1 , θ 2 with respect to the parameters α 1 ∈ (0, 1) and c 1 > 1, respectively α 2 ∈ (0, 1) and c 2 > 1, the composition of the corresponding semi-fractional derivatives given by (1.10) can again be a semi-fractional derivative of order α := α 1 + α 2 .We concentrate on the easiest case when α ∈ (0, 1) and θ 1 , θ 2 have the same periodicity, i.e. c . In view of (1.11) for the corresponding log-characteristic functions we need to show that (2.6) where ψ is the log-characteristic function of a semistable distribution corresponding to an admissable function θ with respect to the parameters α ∈ (0, 1) and c := for the corresponding log-Laplace exponents and we have self-similarity in view of Lemma 2.2 this is equivalent to require that ψ is a Bernstein function.
Corollary 2.4.ψ 1 • ψ 2 is a Bernstein function iff there exists an admissable function θ with respect to the parameters α = α 1 + α 2 ∈ (0, 1) and c = c Remark 2.5.In case θ 1 , θ 2 are smooth admissable functions with Fourier series representations as in (2.3) , then by (2.5) and the Cauchy product rule we easily get where Uniqueness of the Fourier coefficients then gives us the representation (2.9) for the admissable function θ from Corollary 2.4, provided that (2.8) is a Bernstein function.
Finally, we want to show that the semi-fractional derivative of order α ∈ (0, 1) can be seen as a special case of a generalized fractional derivative introduced in [16].Starting with (1.10) for a smooth admissable function θ, integration by parts shows that as laid out in [12].Introducing the kernel function k(y) := y −α θ(log y) and restricting considerations to functions with support on the positiv real line, this can be interpreted as a semi-fractional derivative of Caputo type On the other hand, interchanging the order of integration and differentiation gives a semi-fractional derivative of Riemann-Liouville type which is also called of convolution type in [35].The relationship between these forms is given by the formula which can be derived as (2.33) in [21] and for more general kernel functions D (k) f is called a generalized fractional derivative in [16].For further approaches into this direction see [3,17,19,24,28,35].Of particular interest are non-negative locally integrable kernel functions k such that the operator D (k) possesses a right inverse such that D (k) I (k) f = f .Using the theory of complete Bernstein functions and the relationship to the Stieltjes class, it is shown in [16] that this is possible with for locally bounded measurable functions f if (k, k * ) forms a Sonine pair of kernels, i.e. k * k * ≡ 1; cf. also [32,26,36].In this case I (k) f is called a generalized fractional integral of order α and it also holds that for absolutely continuous functions f .As shown in [8], necessarily the kernel function k must have an integrable singularity at 0. We will now show that a Sonine kernel k * may exist for our specific kernel function k(y) = y −α θ(log y) working in the more general framework of Bernstein functions and their relation to completely monotone functions.Hence we aim to extend the list of specific kernel functions given in section 6 of [19] by a new example.Therefore we have to relax the definition of a Sonine pair in the following sense.
By virtue of Lemma 2.7 we may interpret as a semi-fractional integral of order α for locally bounded and measurable functions f .Since I (k) f (0) = 0, we get and for absolutely continuous functions f with density f ′ we get The semi-fractional integral is also determined by a self-similar Bernstein function.Proof.From Lemma 2.2 we get the scaling relation In particular it follows that ρ({0}) = 0 and by the Fubini-Tonelli theorem we get where the Borel measure µ on (0, ∞) is given by dµ(t) := 1 t dρ(t) and integrates min{1, t} as shown in the proof of Theorem 3.2 in [30]; cf.Proposition 3.5 in [30].
Thus the primitive G * I (x) is a Bernstein function and the scaling relation gives us I is a self-similar Bernstein function with respect to 1 − α ∈ (0, 1) and d > 1.
Remark 2.9.By Lemma 2.2 the Lévy measure µ corresponding to G * I is given by µ(t, ∞) = t α−1 σ(log t) for an admissable function σ with respect to the parameters 1 − α ∈ (0, 1) and d > 1.If this admissable function σ is smooth, we get a Sonine pair (k, k * ) in the original sense as follows.Integration by parts yields and hence we get as the derivative where the kernel k * is non-negative by Lemma A.1.To show that (k, k * ) is a Sonine pair simply calculate the Laplace transform as in the proof of Lemma 2.7.Note that in general we cannot expect k * to be completely monotone as in the approach of [16] with complete Bernstein functions.Nor can we expect that the admissible function σ is smooth in general.

Discrete approximation of the generator
Recall that by (1.10) the semi-fractional derivative operator of order α ∈ (0, 1) is given by the negative generator of the continuous convolution semigroup (ν * t ) t≥0 , where ν is the semistable distribution with log-characteristic function (1.9).If the tail of the Lévy measure is given by φ(t, ∞) = Γ(1 − α) t −α , i.e. the admissable function θ ≡ Γ(1 − α) is constant, it is well known that the semi-fractional derivative coincides with the ordinary Riemann-Liouvile fractional derivative of order α ∈ (0, 1) and can be approximated by means of the Grünwald-Letnikov formula for functions f in the Sobolev space W 2,1 (R); see section 2.1 in [21] for details.The coefficients in the Grünwald-Letnikov approximation formula appear in a discrete distribution on the positive integers called Sibuya distribution which first appeared in [31].A discrete random variable X α on N is Sibuya distributed with parameter α ∈ (0, 1) if For further details and extensions of the Sibuya distribution we refer to [5,6,18] and the literature mentioned therein.Using (3.2) we may rewrite (3.1) as which shows that the Grünwald-Letnikov formula is in fact a discrete approximation of the generator.For further relations of the Sibuya distribution to fractional diffusion equations see [22,23].
Our aim is to generalize this formula for semi-fractional derivatives by means of the corresponding self-similar Bernstein functions.Note that for the Bernstein function ψ(x) = x α the nominator on the right-hand side of (3.2) is given by ψ (j) (1) and hence we may define a semi-fractional Sibuya distribution in the following way.
Definition 3.1.Given an admissable function θ with respect to α ∈ (0, 1) and c > 1, a semi-fractional Sibuya distributed random variable X θ on N is given by ψ( 1) where ψ is the corresponding self-similar Bernstein function.
Clearly, the expression in (3.3) is non-negative by (1.1).Moreover, by monotone convergence and a Taylor series approach justified by Proposition 3.6 in [30] we get (−1) j−1 j! ψ (j) (1) This shows that indeed (3.3) defines a proper distribution on N with pgf for |z| ≤ 1.Note that in the above arguments self-similarity of the Bernstein function is not needed.Hence (3.3) defines a proper distribution on N for every Bernstein function ψ with ψ(0) = 0. Now let θ be a smooth admissable function having Fourier series representation Then by Theorem 3.1 in [12] the corresponding self-similar Bernstein function can be expressed in terms of the log-characteristic function ψ as and hece for j ∈ N 0 we have Note that these coefficients appear in a Grünwald-Letnikov type approximation of the semi-fractional derivative given in Theorem 4.1 of [12] which enables us to prove the following approximation formula.
Theorem 3.2.Let θ be a smooth admissable function with respect to α ∈ (0, 1) and c > 1 with corresponding self-similar Bernstein function ψ and semi-fractional Sibuya distributed random variable X θ given by (3.3).Then for f ∈ W 2,1 (R) the semi-fractional derivative can be approximated along the subsequence Proof.First note that h ikc m = c −imkc/α = e −2πimk = 1 and by self-similarity we have for all m ∈ N. Hence we get where the last convergence follows from Theorem 4.1 in [12].

Space-time duality for semi-fractional diffusions
We first give a sufficient condition for an inverse function to be a Bernstein function.
Bernstein function and f (n) (x) = 0 for all x > 0 and n ∈ N. Then its inverse f −1 is a Bernstein function with (f −1 ) (n) (x) = 0 for all x > 0 and n ∈ N. log-characteristic function ψ from (1.13), possesses C ∞ (R)-densities x → p(x, t) for every t > 0 with P {X t ∈ A} = A p(x, t) dx and for t = 0 we may write p(x, 0) = δ(x) corresponding to X 0 = 0 almost surely.It is shown in [12] that these densities are the point source solution to the semi-fractional diffusion equation with the negative semi-fractional derivative of order α from (1.14) acting on the space variable.Since the semi-fractional derivative is a non-local operator, this equation is hard to interpret from a physical point of view, whereas non-locality in time may correspond to long memory effects [9].As a generalization of a space-time duality result for fractional diffusions [4,10] based on a corresponding result for stable densities by Zolotarev [38,39] it was shown in [11] that space-time duality may also hold for semi-fractional diffusions.Theorem 3.3 in [11] states that for x > 0 and t > 0 we have p(x, t) = α −1 h(x, t), where h(x, t) is the point source solution to the semi-fractional differential equation with a semi-fractional derivative of order 1/α acting on the time variable, provided that τ and ̺ are admissable functions with respect to 1/α ∈ ( 1 2 , 1) and d = c 1/α > 1 which remains an open problem in [11].We will now show that indeed τ is admissable and ̺(x) = −ατ ′ (x), provided that τ is smooth.Note that in general ̺ will not be admissable as conjectured in [11] but we will justify the inhomogenity in (4.3) by different arguments.In [11] the function τ appears in the following way.The above inverse ζ −1 is called ξ in [11] and its existence is shown in Lemma 4.1 of [11].It was further shown in Lemma 4.2 of [11] that ξ(t) = t 1/α g(log t) for a continuously differentiable and log(c)-periodic function g.Since we now know that ξ = ζ −1 is a Bernstein function, in fact g is a C ∞ (R)-function.Since g is a smooth log(c)-periodic function, it is representable by its Fourier series Proof.As shown above ξ = ζ −1 is a Bernstein function and with d = c 1/α and ξ(t) = t 1/α g(log t) for a log(c)-periodic function g we get showing that ξ is a self-similar Bernstein function with respect to 1/α ∈ ( 1 2 , 1) and d = c 1/α .By Lemma 2.2 we have where µ is a semistable Lévy measure with µ(t, ∞) = t −1/α τ (log t) for an admissable function τ with respect to 1/α ∈ ( 1 2 , 1) and d = c 1/α .It remains to show that τ is indeed the function we are looking for.Let us assume for a moment that τ is smooth and thus admits the Fourier series In this case the function γ appearing in Lemma 2.2 fulfills and by (2.4) has the Fourier series representation A comparison with (4.4) and uniqueness of the Fourier coefficients shows that a n coincides with d n /Γ(in d − 1 α + 1) and thus τ is indeed the function in (4.6).Finally, we have to show that the series in (4.6) converges.Using the asymptotic behavior of the gamma function in Corollary 1.4.4 of [1], the Fourier coefficients fulfill for a constant K > 0 and all n ∈ Z \ {0}.According to our assumption (4.5) we for some ε > 0 and all n ∈ Z \ {0} showing that the series in (4.6) converges and the resulting function is continuously differentiable by Theorem 2.6 in [7].
Admissability of τ in combination with Theorem 3.3 in [11] finally enables us to completely solve space-time duality for semi-fractional diffusions.The equation (4.3) is derived in [11] by Laplace inversion of the equation where f is defined in the proof of Theorem 3.1 in [11] as (4.9) and m is a log(c 1/α )-periodic function given by (4.10) ζ(s) = x α m(log x).

4 Lemma 4 . 2 . 2 , 1 )
d = c 1/α ,where by Lemma 1 in §12 of[2] we have|d n | ≤ C • e − π2 |n| d for some C > 0 and all n ∈ Z.If we require a little more quality, namely that the Fourier coefficients even decay as (4.5) |d n | ≤ C • e − π for some ε > 0 and all n ∈ Z \ {0}, then we can define τ by the Fourier series (If (4.5) holds, then the function τ in (4.6) is well-defined and a smooth admissable function with respect to 1/α ∈ ( 1 and d = c 1/α .Moreover, ζ −1 is a self-similar Bernstein function with respect to the same parameters.