On Time-Fractional Relativistic Diffusion Equations
Narn-Rueih Shieh
†March 3, 2012
Abstract
The purpose of this paper is to consider the Cauchy problem for the time- fractional (both sub- and super- diffusive) relativistic diffusion equation. Based on the viewpoint of theory of pseudo-differential operators, we regard the equation as a pseudo differential equation, and we act the equation via the space-time transform. The solution is expressed as the convolution of the Green’s function (heat kernel) and the initial data. The Green’s functions are determined by their Fourier transforms, and are written in terms of Mittag-Leffler functions.
1 Introduction
The purpose of this paper is to consider the Cauchy problem for the following time- fractional relativistic diffusion equation (TFRDE, for brevity),
∂β
∂tβu(t, x) = Hα,mu(t, x), u(0, x) = u0(x), t ≥ 0, x ∈ Rn. (1.1) In the above, the fractional temporal derivative ∂t∂ββ is in the Caputo-Djrbashian sense (see the end of this section); while the spatial differential operator
Hα,m := m − (mα2 − ∆)α2
†Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong, PRC. On leave from National Taiwan University. Correspondence to E-mail: shiehnr@ntu.edu.tw
is a relativistic diffusion operator with the spatial-fractional parameter α ∈ (0, 2) and the normalized mass parameter m > 0.
The operator Hα,m appears in vast literature of mathematics and physics. The prominent case is α = 1, which −H1,m represents the free energy of the relativistic Schr¨odinger operator with a relativistic particle of mass m; see the seminal paper of Carmona et al. [2] for mathematical discussions. For study on general α ∈ (0, 2), one may refer to Ryznar [10], Baeumer et al. [1], and the references therein. The operator Hα,m has also played an essential role in the theory of computer vision; see a special volume edited by Kimmel et al. [6], in which Hα,mis employed to connect the PDEs and the computer vision theory. In this paper, we consider (1.1) from the viewpoint of theory of pseudo-differential operators; see, for example, the book of Wong [14]. We regard (1.1) as a pseudo differential equation, and we act the equation via the space-time transform;
see the statement and the proof of Proposition 1. We will also present in Proposition 3 a corresponding stochastic processes viewpoint, when β ≤ 1. We should remark that the viewpoint of pseudo-differential operators is very inspiring to catch the essence of the solution of (1.1), and also works for the whole time-fractional range β ∈ (0, 2). We state our results in Section 2, and all the proofs are given in Section 3. In the final Section 4, we give remarks on the two-scale property of relativistic Green’s functions (heat kernels).
Here, we review briefly the fractional time-derivatives as follow; see, for example, Djrbashian [4] for details.
dβf dtβ (t) :=
( f(m)(t) if β = m ∈ N
1 Γ(m−β)
Rt 0
f(m)(τ )
(t−τ )β+1−mdτ if β ∈ (m − 1, m), (1.2) where f(m)(t) denotes the ordinary (non-fractional) derivative of order m of a causal function f (t) (i.e., f is vanishing for t < 0). In this note, we mainly use 0 < β < 1 and 1 < β < 2; which are referred respectively as the sub-diffusive and the super-diffusive, since the β = 1 is well-known as the diffusive.
For the expression of the Green’s function in the time-fractional index β, we need the
following Mittag-Leffler (ML) functions; see, for example, [4, Chapter 1] for the details.
The ML function is defined by, for k = 1, 2, Eβ,k(z) =
∞
X
i=0
zi
Γ(βi + k), z ∈ C. (1.3)
We remark that a ML function is an entire function on the complex plane and its asymptotic behaviors, when β ∈ (0, 2), β 6= 1, have the inverse power law as follows:
|Eβ,k(z)| ∼ O( 1
|z|), |z| → ∞ with |arg(−z)| < π(1 − β
2), (1.4)
where arg: C → (−π, π) and the notation f (z) ∼ O(g(z)) means that f (z)/g(z) remains bounded as z approaches the indicated limit point; for (1.4), see the classic book by Erd´elyi et al. [5] (pp. 206-212, especially p. 206 (7) and p. 210 (21)).
Acknowledgement: Most of this work is achieved while the author visited De- partment of Mathematics and Statistics, York University, Canada, in Fall 2011. The hospitality of the host is appreciated.
2 Main results
Since (1.1) is of linear-parabolic type, the solution must be expressed as the convolution of the Green’s function ( heat kernel) and the initial condition. Therefore, the form of the solution expressed below is not surprising; however the defining display for the Green’s function is completely new, to our knowledge. For an f ∈ L2(Rn, Leb), we use f to denote the Fourier (-Plancherel) transform of f . In the context henceforth, we skipˆ the indices α, m from the Green’s functions for most time; yet will resume these two indices when it is necessary.
Proposition 1. The solution u(t, x) of (1.1) is expressed:
• β = 1(diffusive): with the given initial u0(x) = f (x), and with the Green’s function denoted by G = Gα,m,
u(t, x) = Z
Rn
G(t, x − y)f (y)dy.
• β : 0 < β < 1(sub-diffusive): with the given initial u0(x) = f (x), and with the associated Green’s function denoted by Gβ,1 = Gβ,1;α,m,
u(t, x) = Z
Rn
Gβ,1(t, x − y)f (y)dy.
• β : 1 < β < 2(super-diffusive): with the two given initials u0(x) = f (x), and
∂tu
∂t|t=0(x) = g(x), and with the two associated Green’s functions denoted by Gβ,i = Gβ,i;α,m, i = 1, 2 ( respectively associated with the two initials f (x), g(x) ),
u(t, x) = Z
Rn
Gβ,1(t, x − y)f (y)dy + Z
Rn
Gβ,2(t, x − y)g(y)dy.
In the above, the initial(s) f (and g in the super-diffusive) are assumed in the fol- lowing subspace DA of L2(Rn, Leb).
DA:= {f ∈ L2(Rn, Leb) : Z
Rn
θ(λ)| ˆf (λ)|2dλ < ∞}, with
θ(λ) := (mα2 + |λ|2)α2 − m > 0, ∀λ 6= 0.
The defining displays for the above Green’s functions are, respectively, via the spatial Fourier transforms as follows: for each t > 0 fixed, for λ ∈ Rn, with < ·, · > denoting the inner product in Rn,
• β = 1,
Z
Rn
ei<λ,x>G(t, x)dx = e−tθ(λ). (2.1)
• β ∈ (0, 2), β 6= 1,
Z
Rn
ei<λ,x>Gβ,1(t, x)dx = Eβ,1(−tβθ(λ)). (2.2)
• β ∈ (1, 2),
Z
Rn
ei<λ,x>
Gβ,2(t, x)dx = t · Eβ,2(−tβθ(λ)). (2.3)
For the prominent case, that is, α = 1 and m > 0, from the explicit expression of the Green’s function G1,m(t, x) appeared in Wong [13, pp. 195-6] (we need to multiply emt to the Kt(x) there to meet our present situation), we see that G1,m(t, x) is positive valued, and its integral over x is 1. See also Carmona et al. [2, pp. 123-4] from the viewpoint of stochastic processes. The following property shows that the corresponding sub-diffusive Green’s function also behaves in the same manner.
Proposition 2. For the sub-diffusive 0 < β < 1 with α = 1 and m > 0; for each t > 0, the associated Green’s function Gβ,1(t, x) satisfies
Gβ,1(t, x) > 0;
Z
Rn
Gβ,1(t, x)dx = Eβ,1(−tβθ(0)) = 1. (2.4) Remarks: 1. For the super-diffusive β ∈ (1, 2) and the two associated Green’s functions Gβ,k, k = 1, 2, we can view Proposition 2 in the following informal way: let us rewrite the latter as Gβ,k+1, k = 0, 1, and let θ(λ) be the θ function defined in Proposition 1. Then, we have
Z
Rn
Gβ,k+1(x, t)dx = Z
Rn
1 (2π)n
Z
Rn
e−i<λ,x>
tkEβ,k+1(−θ(λ)tβ)dλ dx. (2.5) Using the symbolic relation of the δ0-function
δ0(λ) = 1 (2π)n
Z
Rn
e−i<λ,x>
dx, we see that the display (2.5) is symbolically equal to
Z
Rn
δ0(λ)tkEβ,k+1(−θ(λ))dλ = tkEβ,k+1(0) = tk.
The last relation in the above seems to indicate that the super-diffusive does not have the propobilistic interpretation; such an interpretation for the diffusive and the sub-diffusive will be given in the below, Proposition 3.
2. Proposition 2 is stated and proved only for the case α = 1, since we use the explicit expression of G1,m(t, x) to see that it is postive valued; this positivity property seems not readily seen from the Fourier transform expression of the Gα,m(t, x), for general α ∈ (0, 2).
The following property a probabilistic viewpoint for the solution of (1.1) in the dif- fusive and the sub-diffusive cases; it is based on the (Bochner) subordination of the Brownian motion in Rn; this probabilistic approach in, say, [3] allows to consider the sub-diffusive RDE on a bounded domain.
Proposition 3. In the diffusive and the sub-diffusive cases, the solution u(t, x) of (1.1) has a probabilistic interpretation: with the given initial data f (x) = u0(x), for the diffu- sive, u(t, x) = Ex[f (Xt)], and for the sub-diffusive, u(t, x) = Ex[f (Yt)]. The stochastic process Xtis the Brownian motion Bton Rnsubordinated by a relativistic α2-subordinator.
The stochastic process Yt is the Xt further subordinated by a β-subordinator. The nota- tion Ex[·] means the expectation w.r.t. the process starting at x, and the foot-index x is skipped when the process starts at x = 0.
3 Proofs
Proof of Proposition 1. The diffusive case is typical; indeed the display for the Green’s function G(t, x) is a consequence of our viewing that RDE (β = 1) as a pseudo differential equation. The proof of the sub-diffusive case can be imbedded in the super- diffusive case (with g(x) identically to be zero). Therefore we proceed the proof for the latter case, 1 < β < 2. The proof is based on the temporal-spatial transform arguments:
take the Laplace transform L(·) with respect to the temporal variable t, and then take the Fourier transform F (·) with respect to the spatial variable x. For the notational consistency, we use (ˆx, ˆt) to denote the spatial-temporal variable in the transformed space-time. Firstly, by the convolution for Laplace transform and the integration-by- parts on the defining display of the fractional derivative (1.2) for the causal functions, we see that u(x, t) must satisfy
{−∂u
∂tu(x, 0) − ˆtu(x, 0) + ˆt2Lu(x, ˆt)}ˆtβ−2 = Hα,mLu(x, ˆt). (3.1)
Then, we take the spatial Fourier transform F (·) to (3.1); we then have {−F∂u
∂t(ˆx, 0) − ˆtF u(ˆx, 0) + ˆt2F Lu(ˆx, ˆt)}ˆtβ−2 = −{(mα2 + |ˆx|2)α2 − m}F Lu(ˆx, ˆt), which can be rewritten as
F Lu(ˆx, ˆt) = −ˆtβ−2
−{(mα2 + |ˆx|2)α2 − m} − ˆtβ(ˆtF u(ˆx, 0) + F∂u
∂t(ˆx, 0)). (3.2) For θ = θ(ˆx) := (m2α + |ˆx|2)α2 − m > 0, we expand the −θ−ˆ−ˆtβ−1tβ and the −θ−ˆ−ˆtβ−2tβ in powers of tˆ−1by geometric series and represent the inverse power of ˆt by the integral representation for the Γ-function (see, for example, Podlubny [9, (1.80)]), then we can have
−ˆtβ−1
−θ − ˆtβ = L(Eβ,1(−θ β))(ˆt) and
−ˆtβ−2
−θ − ˆtβ = L( Eβ,2(−θ β))(ˆt),
in which wants to mean the variable in t to be under the L. Then, we take the inverse Laplace transform to (3.2) to obtain
F u(ˆx, t) = Eβ,1(−θtβ) + tEβ,2(−θtβ).
As a final step, we take the the inverse Fourier transform to the above, and then use the convolution to obtain the solution; the displays for the Green’s functions are consequently derived.
Proof of Proposition 2. From Wyss and Wyss [15], noting that we are now in the sub-diffusive, we can express the associated Green’s function as
Gβ,1(t, x) = Z ∞
0
fβ(z)G(tβz, x)dz, x ∈ Rn, t > 0; (3.3) in which, G(t, x) is the Green’s function determined by the diffusive case, with α = 1 and m > 0 (see Proposition 1), and fβ(z) is a nonnegative-valued function of z ≥ 0 represented in terms of so-called H-function (see, for example, Schneider [12, p. 284])
fβ(z) = H1110 z| (1 − β, β) (0, 1)
!
, z ≥ 0; (3.4)
indeed fβ is with Laplace transform Z ∞
0
e−szfβ(z)dz = Eβ,1(−s), s ≥ 0. (3.5) By (3.3), and the positivity of G(t, x) as we remark in Section 2, we see that Gβ,1(t, x) >
0. Moreover, Z
Rn
Gβ,1(t, x) dx = Z
Rn
Z ∞ 0
fβ(z)G(tβz, x) dzdx
= Z ∞
0
fβ(z) Z
Rn
G(tβz, x) dxdz
= Z ∞
0
fβ(z) · e−tβzθ(0) dz = Eβ,1(−tβθ(0)) = 1. (3.6) In the above, we use Tonelli Theorem (it is legitimate to use, since the integrand is known to be nonnegative) to change the order of integrations, and we also use the (3.5) and the defining display of G(t, x) in Proposition 1. Note that θ(0) = 0, and that Eβ,1(0) = 1.
Proof of Proposition 3. The proof is adapted from two references, Ryznar [10] and Chen et al. [3]. In [10], the relativistic relativistic (α/2)-subordinator is introduced, as a L´evy process Ttwith increasing sample paths and with the Laplace function determined by
Eh e−uTti
= e−t{(m
α2+u)α2−m}
, u > 0.
Assume that Tt and the Brownian motion Bt in Rn are totally independent, then the subordinated process Xt= BTt is a L´evy process Rn, for which the characteristic function is given by,
Eh
ei<λ,Xt>i
= e−t{(m
α2+|λ|2)α2−m}
, λ ∈ Rn.
This means that the transition density function of the process Xt is exactly the Green’s function of the (1.1). Now, for the sub-diffusive case β < 1, the general theory for sub-diffusive processes mentioned, for example, in [3] and the references therein, asserts the second part of the proposition. Indeed, let St be a subordinator for which Laplace function is determined by
Eh e−uSti
= e−tuβ, u > 0.
Assume that the processes Stis totally independent from a Markov process Xt for which the infinitesimal generator is denoted by A, then, by [3, Section 3], the sub-fractional diffusive equation, regarded as an evolution equation,
∂βu
∂tβ = Au, u(0) = f,
has the solution u(t, x) = Ex[f (Yt)], where the process Yt := XEt, and Et is the right- inverse of St, i.e. Et = inf{s : Ss > t}. The generator of the L´evy process Xt is Hα,m, which can be seen from the characteristic function expression of Xtgiven above and the theory of L´evy processes in, say, the book of Sato [11]. This gives the assertion.
Remark: We should remark that Proposition 3 is not fully equivalent to Proposi- tion 1, since our domain of the action DA in Proposition 1 is not, in general, known to be the same as the domain of the Markov generator A.
4 Remark on Green’s functions
Here, we make the remarks on how the viewpoint of pseudo-differential operators gives us the insight on the RDEs. The Fourier expression bGα,mof the Green’s function Gα,m(t, x) in Proposition 1 (we add foot-index α, m here to clarity their roles in the below) gives us the following two-scale property:
When T → ∞,
Gbα,m(T t, T−12λ) = expn
T t(m − (mα2 + T−1|λ|2)α2)o
→ expn
− tα
2m1−α2|λ|2o . When ε → 0,
Gbα,m(εt, ε−α1λ) = eεtme−εt(m
α2+ε− 2α|λ|2)α2 → e−t|λ|α.
The first one is observed by the limiting on the first two terms in the concerned fractional- binomial expansion, and the second one is observed by the direct limiting.
The above two-scale property asserts that: the large-scale is dominated by the mass index m, and the small-scale is dominated by the spatial-fractional index α. This two- scale property is key to consider the multi-scaling property of RDEs, as shown in [7, 8].
As for the time-fractional case, the Fourier expression bGβ,1 of the first associated Green’s function Gβ,1 in Proposition 1 also exhibits the two-scale via its Fourier trans- form: bGβ,1(T t, T−β2λ), for large T , and bGβ,1(εt, ε−βαλ), for small ε. The displays will then use the ML function Eβ,1 to replace the exponential function.
However, the the Fourier expression bGβ,2 of the second associated Green’s function Gβ,2, appearing only in the super-diffusive, has a aggregating time factor t in its expres- sion, which should be quite different from the above observations.
Finally, we mention that, though the explicit expression for the prominent G1,m(t, x) has appeared in Carmona et al. [2, pp. 123-4] and Wong [13, pp. 195-6], it seems that there is no such explicit expression for Gα,m(t, x) for general α ∈ (0, 2).
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