Time-Changed Dirac–Fokker–Planck Equations on the Lattice

Abstract

A time-changed discretization for the Dirac equation is proposed. More precisely, we consider a Dirac equation with discrete space and continuous time perturbed by a time-dependent diffusion term $\sigma ^2Ht^{2H-1}$ that seamlessly describes a latticizing version of the time-changed Fokker–Planck equation carrying the Hurst parameter $0<H<1$. Our model problem formulated on the space-time lattice ${{\mathbb {R}}}_{h,\alpha }^n\times [0,\infty )$ ($h>0$ and $0<\alpha <\frac{1}{2}$) preserves the main features of the Dirac–Kähler type discretization over the space-time lattice $h{{\mathbb {Z}}}^n\times [0,\infty )$ in case of $\alpha ,H \rightarrow 0$, and encompasses a regularization of Wilson’s approach (Phys Rev 10(8):2445, 1974] for values of H in the range $0<H\le \frac{1}{2}$ (limit condition $\alpha \rightarrow \frac{1}{2}$). The main focus here is the representation of the solutions by means of discrete convolution formulae involving a kernel function encoded by (unnormalized) Hartman–Watson distributions—ubiquitous on stochastic processes of Bessel type—and the solutions of a semi-discrete equation of Klein–Gordon type. Namely, on our main construction the ansatz function $\widehat{\varPsi }_H(y,t)$ appearing on the discrete convolution representation may be rewritten as a Mellin convolution type integral involving the solutions $\varPsi (x,t|p)$ of a semi-discrete equation of Klein–Gordon type and a Lévy one-sided distribution $L_H(u)$ in disguise. Interesting enough, by employing Mellin-Barnes integral representations it turns out that the underlying solutions of Klein–Gordon type may be represented through generalized Wright functions of type ${~}_1\Psi _1$, that converge uniformly in case that the quantity $\alpha +\frac{1}{2}$ may be regarded as an lower estimate for the Hurst parameter in the superdiffusive case (that is, if $\alpha +\frac{1}{2}\le H<1$).

Appendix A: Fractional Calculus Background

We aim at presenting in this appendix a systematic account of basic properties and characteristics of generalized Wright functions (also known as Fox-Wright functions (cf. [22])) in interplay with the Mellin transform.

1.1 The Mellin Transform

The well-known Mellin transform ${\mathcal {M}}$ (cf. [3]) is defined for a locally integrable function f on $]0,\infty [$ by the integral

$$\begin{aligned} {\mathcal {M}}\{f(t)\}(s)=\int _{0}^{\infty } f(t)t^{s-1}dt,&\text{ with }&s\in {{\mathbb {C}}}. \end{aligned}$$

(A.1)

In order to provide the existence of the inverse ${\mathcal {M}}^{-1}$ of (A.1) through the inversion formula

$$ \begin{aligned} f(t)=\frac{1}{2\pi i}\int _{c-i\infty }^{c+i\infty }{\mathcal {M}}\{f(t)\}(s)~t^{-s}~ds,&\text{ with }&t>0 ~~~ \& ~~ c=\mathfrak {R}(s) \end{aligned}$$

(A.2)

in such way that the contour integral is independent of the choice of the parameter c, one needs to restrict the domain of analyticity of the complex-valued function ${\mathcal {M}}\{f(t)\}(s)$ to the fundamental strip $-a<\text{ Re }(s)<-b$ paralell to the imaginary axis $i{{\mathbb {R}}}$, whereby the parameters a and b are determined through the asymptotic constraint

$$\begin{aligned} f(t)=\left\{ \begin{array}{lll} O(t^{-a-1}) &{} \text{ if } &{} t\rightarrow 0^+ \\ O(t^{-b-1}) &{} \text{ if } &{} t\rightarrow \infty \end{array}\right. . \end{aligned}$$

It is straighforward to see after a wise change of variable on the right hand side of (A.1), we infer that

$$\begin{aligned} {\mathcal {M}}\{t^\beta f(t)\}(s)={\mathcal {M}}\{f(t)\}\left( s+\beta \right) ,&\text{ for }&\beta \in {{\mathbb {C}}}\\ {\mathcal {M}}\{f(t^\gamma )\}(s)=\frac{1}{|\gamma |}({\mathcal {M}}f)\left( \frac{s}{\gamma }\right) ,&\text{ for }&\gamma \in {{\mathbb {C}}}\setminus \{0\} \\ {\mathcal {M}}\{f(\kappa t)\}(s)=\kappa ^{-s}({\mathcal {M}}f)(s),&\text{ for }&\kappa >0. \end{aligned}$$

With the above sequence of operational identities, neatly amalgamated through the compact formula

$$\begin{aligned} {\mathcal {M}}\{t^\beta f(\kappa t^\gamma )\}(s)=\frac{1}{|\gamma |}\kappa ^{-\frac{s+\beta }{\gamma }}{\mathcal {M}}\{f\}\left( \frac{s+\beta }{\gamma }\right) \end{aligned}$$

(A.3)

carrying the parameters $\beta \in {{\mathbb {C}}},~\gamma \in {{\mathbb {C}}}\setminus \{0\}$ and $\kappa >0$, there holds the Mellin convolution theorem

$$\begin{aligned} {\mathcal {M}}\{f\star _{{\mathcal {M}}}g\}(s)={\mathcal {M}}\{f\}(s){\mathcal {M}}\{g\}(s) \end{aligned}$$

(A.4)

encoded by the convolution type integral (cf. [3, Theorem 3.])

$$\begin{aligned} (f\star _{{\mathcal {M}}}g)(t):=\int _{0}^{\infty }f\left( \frac{t}{p}\right) g(p)\frac{dp}{p}. \end{aligned}$$

(A.5)

We refer to [3, Section 4.] for additional properties associated to the Mellin convolution (A.5). In particular, the Parseval type property

$$\begin{aligned} {\mathcal {M}}\{f(t)g(t)\}(\omega )=\frac{1}{2\pi i}\int _{c-i\infty }^{c+i\infty } {\mathcal {M}}\{f(t)\}(\omega -s)~ {\mathcal {M}}\{g(t)\}(s)~ds \end{aligned}$$

(A.6)

yields straightforwardy from the combination of the set of identities

$$\begin{aligned} {\mathcal {M}}\{f(t)\}(\omega -s)= & {} {\mathcal {M}}\left\{ t^{-\omega }f\left( \frac{1}{t}\right) \right\} (s), \\ {\mathcal {M}}\{f(t)g(t)\}(\omega )= & {} \left( t^{-\omega }f\left( \frac{1}{t}\right) \star _{{\mathcal {M}}}g\right) (1) \end{aligned}$$

resulting from (A.3) and (A.4), respectively, with the set of properties (A.5) and (A.2).

1.2 Generalized Wright Functions

Generalized Wright functions ${~}_p\Psi _q$ are a rich class of analytic functions that include generalized hypergeometric functions ${~}_pF_q$ and stable distributions (cf. [22] & [24, Chapter 3]). With the aim of amalgamate some the technical work required in Sects. 4.2 and 4.3 we will take into account the definition of ${~}_p\Psi _q$ in terms of series expansion

$$\begin{aligned} {~}_p\Psi _q \left[ \begin{array}{l|} (a_k,\alpha _k)_{1,p} \\ (b_l,\beta _l)_{1,q} \end{array} ~ \lambda \right] =\sum _{m=0}^\infty \dfrac{\prod _{k=1}^p\Gamma (a_k+\alpha _km)}{\prod _{l=1}^q\Gamma (b_l+\beta _l m)}~\dfrac{\lambda ^m}{m!}, \end{aligned}$$

(A.7)

where $\lambda \in {{\mathbb {C}}}$, $a_k,b_l\in {{\mathbb {C}}}$ and $\alpha _k,\beta _l\in {{\mathbb {R}}}\setminus \{0\}$ ($k=1,\ldots ,p$; $l=1,\ldots ,q$).

Here and elsewhere

$$\begin{aligned} \Gamma (s)=\int _{0}^{\infty } e^{-t}t^{s-1}dt \end{aligned}$$

(A.8)

stands for the Eulerian representation for the Gamma function.

We note that in particular, that the trigonometric functions may be seen as particular cases of the Mittag-Leffler and Wright functions

$$\begin{aligned} \displaystyle E_{\rho ,\beta }(\lambda )={~}_1\Psi _1 \left[ \begin{array}{l|} (1,1) \\ (\beta ,\rho ) \end{array}~ \lambda \right] \hbox { resp. }\displaystyle \phi (\rho ,\beta ;\lambda )={~}_0\Psi _1 \left[ \begin{array}{l|} \\ (\beta ,\rho ) \end{array}~ \lambda \right] . \end{aligned}$$

Namely, in view of (A.12) and on the Legendre’s duplication formula

$$\begin{aligned} \Gamma (2s)=\frac{2^{2s-1}}{\sqrt{\pi }}\Gamma (s)\Gamma \left( s+\frac{1}{2}\right) \end{aligned}$$

(A.9)

one readily has

$$\begin{aligned} \cos (\lambda )= & {} {~}_1\Psi _1 \left[ \begin{array}{l|} (1,1) \\ (1,2) \end{array} -\lambda ^2 \right] =\sqrt{\pi }{~}_0\Psi _1 \left[ \begin{array}{l|} \\ \left( \frac{1}{2},1\right) \end{array} -\dfrac{\lambda ^2}{4} \right] \end{aligned}$$

(A.10)

$$\begin{aligned} \dfrac{\sin (\lambda )}{\lambda }= & {} {~}_1\Psi _1 \left[ \begin{array}{l|} (1,1) \\ (2,2) \end{array} -\lambda ^2 \right] =\dfrac{\sqrt{\pi }}{2}{~}_0\Psi _1 \left[ \begin{array}{l|} \\ \left( \frac{3}{2},1\right) \end{array} -\dfrac{\lambda ^2}{4} \right] . \end{aligned}$$

(A.11)

showing that $\cos (\lambda )$ and $\frac{\sin (\lambda )}{\lambda }$ are spherical Bessel functions in disguise.

In the paper [20], Kilbas et al have checked for $\alpha _k,\beta _l>0$ that ${~}_p\Psi _q$ admits the the Mellin-Barnes type integral representation

$$\begin{aligned}&\displaystyle {~}_p\Psi _q \left[ \begin{array}{l|} (a_k,\alpha _k)_{1,p} \\ (b_l,\beta _l)_{1,q} \end{array} ~ \lambda \right] \nonumber \\&\quad =\displaystyle \dfrac{1}{2\pi i} \int _{c-i\infty }^{c+i\infty } \dfrac{\Gamma (s)\prod _{k=1}^p\Gamma (a_k-\alpha _ks)}{\prod _{l=1}^q\Gamma (b_l-\beta _l s)}(-\lambda )^{-s}~ds \end{aligned}$$

(A.12)

in a way that ${~}_p\Psi _q$ and the inverse of the Mellin transform (see Eqs. (A.1) and (A.2) ) are interrelated by the operational formula

$$\begin{aligned} {~}_p\Psi _q \left[ \begin{array}{l|} (a_k,\alpha _k)_{1,p} \\ (b_l,\beta _l)_{1,q} \end{array} ~ \lambda \right]= & {} {\mathcal {M}}^{-1}\left\{ \dfrac{\Gamma (s)\prod _{k=1}^p\Gamma (a_k-\alpha _ks)}{\prod _{l=1}^t\Gamma (b_l-\beta _l s)}\right\} (-\lambda ). \end{aligned}$$

This result may be summarized as follows: if intersection between the simple poles $b_l =-m$ ($m\in {{\mathbb {N}}}_0$) of $\Gamma (s)$ and the simple poles $\frac{a_k+m}{\alpha _k}$ ($k=1,\ldots ,p;m\in {{\mathbb {N}}}_0$) of $\Gamma (a_k-\alpha _k s)$ ($k=1,\ldots ,p$) satisfies the condition $\frac{a_k+m}{\alpha _k}\ne -m$, we have the following characterization:

• (1) In case of $\displaystyle \sum _{l=1}^q\beta _l-\sum _{k=1}^p \alpha _k>-1$, the series expansion (A.7) is absolutely convergent for all $\lambda \in {{\mathbb {C}}}$.

• (2) In case of $\displaystyle \sum _{l=1}^q\beta _l-\sum _{k=1}^p \alpha _k=-1$, the series expansion (A.7) is absolutely convergent for all values of $|\lambda |<\rho$ and of $|\lambda |=\rho$, $\text{ Re }(\kappa )>\frac{1}{2}$, with

  $$\begin{aligned} \displaystyle \rho =\displaystyle \dfrac{\Pi _{l=1}^q |\beta _l|^{\beta _l}}{\Pi _{k=1}^p |\alpha _k|^{\alpha _k}}&\text{ and }&\kappa =\sum _{l=1}^q b_l-\sum _{k=1}^p a_k+\frac{p-q}{2}. \end{aligned}$$

Other important classes of generalized Wright functions are the modified Bessel functions

$$\begin{aligned} I_{\nu }(u)=\left( \dfrac{u}{2}\right) ^\nu {~}_0\Psi _1 \left[ \begin{array}{l|} \\ (\nu +1,1) \end{array} ~ \dfrac{u^2}{4} \right] \end{aligned}$$

of order $\nu$ and the one-sided Lévy distribution $L_\nu$ which is represented through the Laplace identity

$$\begin{aligned} \exp (-s^\nu )=\int _{0}^{\infty }e^{-su}L_\nu (u)~du,&0<\nu <1. \end{aligned}$$

(A.13)

For the later one we would like to emphasize that $L_\nu$ may be seamlessly described in terms of the Wright functions $\displaystyle \phi (\rho ,\beta ;\lambda )={~}_0\Psi _1 \left[ \begin{array}{l|} \\ (\beta ,\rho ) \end{array}~ \lambda \right]$ ($-1<\rho <0$) (cf. [13, 22]). In concrete, the term-by-term integration of the $k-$terms of $\phi (\rho ,\beta ;\lambda )$ provided by (A.8) yields

$$\begin{aligned} e^{-s^\nu }=\int _{0}^\infty e^{-su} {~}_0\Psi _1 \left[ \begin{array}{l|} \\ (0,-\nu ) \end{array}~ \frac{1}{u^\nu } \right] \dfrac{du}{u} \end{aligned}$$

(A.14)

so that (A.13) may be reformulated in terms of the Mellin convolution (A.5). That is, $e^{-s^\nu }=(f\star _{{\mathcal {M}}}g)(1)$, with

$$\begin{aligned} f(t)={~}_0\Psi _1 \left[ \begin{array}{l|} \\ (0,-\nu ) \end{array}~ {t^\nu } \right]&\text{ and }&g(t)=e^{-st}. \end{aligned}$$

Moreover, $L_\nu (u)$ is uniquely determined by

$$\begin{aligned} L_\nu (u)=\dfrac{1}{u}{~}_0\Psi _1 \left[ \begin{array}{l|} \\ (0,-\nu ) \end{array}~ \dfrac{1}{u^\nu } \right] . \end{aligned}$$