A Note on the Generalized Relativistic Diffusion Equation

We study here a generalization of the time-fractional relativistic diffusion equation based on the application of Caputo fractional derivatives of a function with respect to another function. We find the Fourier transform of the fundamental solution and discuss the probabilistic meaning of the results obtained in relation to the time-scaled fractional relativistic stable process. We briefly consider also the application of fractional derivatives of a function with respect to another function in order to generalize fractional Riesz-Bessel equations, suggesting their stochastic meaning.


Introduction
In recent papers, relativistic diffusion equations have been investigated both from the physical [1] and mathematical [2,3] points of view. It is well known that these kinds of space-fractional equations are strictly related to relativistic stable processes (see, e.g., [2]).
In [2], a fractional relativistic stable process was considered, connected with a time-fractional relativistic diffusion equation involving a derivative in the sense of Caputo. In [3], the Fourier transform of the fundamental solution and the probabilistic interpretation of the solution for the Cauchy problem of a time-fractional relativistic equation was discussed. On the other hand, the fractional derivative of a function with respect to another function is a useful mathematical tool that is recently gaining more interest in relation to models involving time-varying coefficients, see, e.g., [4][5][6] and the references therein.
The aim of this short paper is to consider the generalization of the time-fractional relativistic diffusion equation by means of a Caputo fractional derivative of a function with respect to another function.
We are able to find the Fourier transform of the fundamental solution for this new class of generalized relativistic diffusion equations. Moreover, we discuss its probabilistic meaning, showing the role of the time-scaling involved in the application of time-fractional operators. A time-scaled fractional tempered stable process is considered in connection to this class of equations. The time-scaled fractional relativistic stable process is also analyzed by evaluating the covariance and correlation coefficient and the connections with fractional-type equations.
We finally apply fractional derivatives with respect to another function in order to generalize the fractional Bessel-Riesz motion recently considered in [7].
The density of the fractional TS process T β µ,α (t) satisfies the fractional equation (see [2] Theorem 6) under initial-boundary conditions We here denote by D β t the Caputo fractional derivative of order β ∈ (0, 1), i.e., Definition 2. The fractional relativistic stable process is defined as the time-changed Brownian motion The density of the fractional relativistic stable process X β µ,α (t) coincides with the solution of the time-fractional relativistic diffusion equation ( [2], Theorem 16) under initial and boundary conditions We will here present some generalizations of Equations (7) and (11) obtained by using fractional derivatives of a function with respect to another function.

Fractional Derivatives of a Function with Respect to Another Function
Fractional derivatives of a function with respect to another function have been considered in the classical monograph by Samko et al. [10] and Kilbas et al. [11] (Section 2.5). Recently they were reconsidered by Almeida in [4] where the Caputo-type regularization of the existing definition and some interesting properties are provided. We recall the basic definitions and properties for the reader's convenience. Let an increasing function such that f (t) = 0 in [a, t], the fractional integral of a function g(t) with respect to another function f (t) is given by Observe that for f (t) = t β we recover the definition of Erdélyi-Kober fractional integral recently applied, for example, in connection with the Generalized Grey Brownian Motion [12].
The corresponding Caputo-type evolution operator (according to our notation) is given by where n = [ν] + 1. Here we have used the symbol O ν, f t (·) in order to underline the generic integro-differential nature of the time-evolution operator, beyond the fractional calculus theory.
A relevant property of Equation (14) is As a consequence, the composite Mittag-Leffler function (see [6,13] for Mittag-Leffler functions) is an eigenfunction of the operator O ν, f t . In the next section we will consider a = 0. Hereafter we denote by f (k) the Fourier transform of a given function f (x). T], such that f (0) = 0 and f = 0 ∀t ∈ (0, T], then the Fourier transform of the fundamental solution of the generalized time-fractional relativistic diffusion equation

On the Generalized Relativistic Diffusion Equation
is given by where Moreover, in the case ν = 1, the fundamental solution of the equation is given by Proof. By taking the spatial Fourier transform of Equation (17), we have that under the initial condition u(k, 0) = 1. Then, by using the fact that we obtain the claimed result. The case ν = 1 can be checked by simple calculations.

Remark 1.
Observe that, for m = 0, we obtain the Fourier transform of the fundamental solution of the generalized space-time-fractional diffusion equation while, for α = 2, m = 0, we have the Fourier transform of the generalized time-fractional heat equation In the latter, some interesting cases have been considered in the literature. In particular • if f (t) = t, we recover the time-fractional diffusion equation, which is widely studied in the literature, (we refer, for example, to [14,15]). • if f (t) = (1 − e −βt )/β, we have the fractional Dodson diffusion equation studied in [6]. • if f (t) = ln t, we have the time-fractional diffusion involving a regularized Hadamard fractional derivative.

Remark 2.
In Theorem 1, we take for simplicity the condition that a = 0 in the definition of the fractional operator O ν, f t and we consider the renstriction to functions f such that f (0) = 0. In the more general case a > 0, f (a) = 0, we have that the Fourier transform of the fundamental solution for Equation (17) is given by Much care should be given, for example, to the case of logarithmic functions, where "a" must be greater than zero. (20) with α = 1 can be euristically interpreted as a relativistic Schrödinger-type equation with time-dependent mass.

Time-Scaled Fractional Tempered Stable Process
Let us consider the following equation under the initial-boundary conditions Let us denote by f the Laplace transform of a function f , i.e., f (η) := +∞ 0 e −ηx f (x)dx. We now evaluate the Laplace transform, with respect to the space argument, of Equation (27) with initial condition h(η, 0) = 1 and where ψ α,µ (η) := (η + µ) α − µ α (see [2] for details). Thus, we can write its solution as By comparing Equation (29) with Equation (20) in [2], we can derive the following relationship holding for the corresponding process T β, f µ,α (t), for any t > 0 and, by the unicity of the Laplace transform, we can write the following equality of the one-dimensional distributions T which holds for any t > 0. Thus the process T β, f µ,α can be called time-scaled fractional tempered stable process. Therefore, we have the following: From the finiteness of the moments of the standard tempered stable process and by taking into account Equation (6), we can draw the conclusion that the moment generating function of T β, f µ,α exists, for any f and α, β ∈ (0, 1), and is equal to Equation (30), for κ = −η. By differentiating we then obtain the first and second moments of T β, f µ,α as follows which, for β = 1 and f (t) = t, for any t, coincide with those of the standard tempered stable process (see [8]). Since, in this case, the process is not Lévy, we also loose the stationarity of increments.
We study the tails' behavior of the distribution of the time-scaled fractional TS process T β, f µ,α (t), i.e., we obtain an estimate for P(T β, f µ,α (t) > x), for x → +∞, by means of the asymptotic result holding for the usual TS subordinator T µ,α (t): It is well-known that, for x → +∞, (see [16]). Let f (0) = 0, for simplicity, and let l β (x, t) := P{L β (t) ∈ dx} be the transition density of the inverse β-stable subordinator, then we can write, by considering Equations (6) and (29) together, that Taking the derivative out of the integral, in the previous lines, is allowed by the standard arguments for the moment generating function of the inverse stable subordinator +∞ 0 e kz l β (z, f (t))dz = E β (k f (t) β ) (see also [17]).
Going back to the problem considered in the previous section, we are now able to provide a stochastic interpretation for the fundamental solution of the generalized fractional relativistic diffusion equation. (17) coincides with the density of the time-scaled fractional relativistic stable process X ν

Theorem 3. The fundamental solution of the generalized time-fractional relativistic diffusion Equation
We omit the complete proof, since it can be easily derived from [2], Corollary 13. Indeed, we here underline that, by using a time-fractional derivative with respect to a function, we obtain a deterministic time-change; therefore the related process is the time-scaled counterpart of the original process.
We can evaluate, by conditioning the expected value and variance of the time-scaled fractional relativistic stable process X ν m,α/2 . Indeed, we can write

An Application to Generalized Fractional Bessel-Riesz Motion
In the recent paper [7], the authors give a stochastic representation for the solution of the following fractional Cauchy problem where (−∆) α/2 and (I − ∆) γ/2 are the inverses of the Riesz and the Bessel potential respectively. The Fourier transform of the Cauchy Equation (34), has been obtained in the form In [7], it was proved that the fundamental solution for Equation (34) coincides with the density of the stochastic process where W is the n-dimensional Brownian motion, L β (t) is the inverse stable subordinator (according to our notation) and L α,γ is the Lévy subordinator with Laplace exponent All these three processes are jointly independent. Let us consider the generalized fractional counterpart of the Cauchy Equation (34) involving fractional derivatives with respect to another function, i.e., Assuming that f (t) is a suitable smooth function such that f (0) = 0, we have that the fundamental solution of Equation (39) given by and therefore it coincides with the density of the time-scaled stochastic processes depending on f X BR (t) = W(L α,γ (L β ( f (t)))), t ≥ 0 (41)

Conclusions and Open Problems
We have analyzed here some applications of the time-fractional derivative with respect to a function, in the context of the so-called fractional relativistic diffusion equation. It is well-known that the most interesting case from the physical point of view corresponds to α = 1, where the operator −H 1,m appearing in Equation (5) represents the free energy of the relativistic Schrödinger equation. A physical discussion about the utility and meaning of the time-fractional generalizations of the relativistic diffusion is still missing. In view of the wide exhisting literature on the time-fractional Schrödinger equation, it is worth investigating this topic for future studies, also for the stochastic interpretation discussed here.
From the mathematical point of view, another interesting topic is the connection between higher order equations and time-fractional equations (see, e.g., [20]). In this case the relation is not trivial and it probably works only for specific values of the real order of the derivative.
Finally, a probabilistic interpretation of the fractional integrals with respect to a function can be developed, following the discussion presented in [21].