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 \(\alpha +\frac{1}{2}\le H<1\)).

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Acknowledgements

The author would like to thank to the anonymous referees for the careful reading of the paper and for the criticism throughout the reports. That allowed to improve the quality of the submitted version in a clever style.

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Communicated by Luis Vega.

Appendix A: Fractional Calculus Background

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.

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).

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. (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. (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

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

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Faustino, N. Time-Changed Dirac–Fokker–Planck Equations on the Lattice. J Fourier Anal Appl 26, 44 (2020). https://doi.org/10.1007/s00041-020-09754-6

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Keywords

  • Discrete Fourier transform
  • Discretized Klein–Gordon equations
  • Lévy one-sided distributions
  • Modified Bessel functions
  • Time-changed Fokker–Planck equations
  • Wright functions

Mathematics Subject Classification

  • Primary 30G35
  • 35Q41
  • 42B05
  • Secondary 33E12
  • 35Q84
  • 39A12
  • 44A20