PHYSICAL REVIEW E, VOLUME 65, 041103
Stochastic solution of space-time fractional diffusion equations
Mark M. Meerschaert*
Department of Mathematics, University of Nevada, Reno, Nevada 89557-0084
David A. Benson
Desert Research Institute, 2215 Raggio Parkway, Reno, Nevada 89506-0220
Hans-Peter Scheffler
Fachbereich Mathematik, University of Dortmund, 44221 Dortmund, Germany
Boris Baeumer
Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand
共Received 16 July 2001; published 28 March 2002兲
Classical and anomalous diffusion equations employ integer derivatives, fractional derivatives, and other
pseudodifferential operators in space. In this paper we show that replacing the integer time derivative by a
fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process
whose probability distributions are Mittag-Leffler type. This leads to explicit solutions for space-time fractional
diffusion equations with multiscaling space-fractional derivatives, and additional insight into the meaning of
these equations.
DOI: 10.1103/PhysRevE.65.041103
PACS number共s兲: 05.40.Fb, 02.50.⫺r, 05.10.Gg, 47.55.Mh
I. INTRODUCTION
Space-fractional diffusion equations 关1– 4兴 have been useful as models of anomalous transport in many diverse disciplines, including finance, semiconductor research, biology,
and hydrogeology 关5–7兴. In the context of the flow in porous
media, fractional space derivatives model large motions
through highly conductive layers or fractures, while fractional time derivatives describe particles that remain motionless for extended periods of time. Dissolved solutes may sorb
to solid material 关8兴 or diffuse into immobile-water zones of
various sizes 关9兴. The scalar space-fractional diffusion equation governs Lévy motion, and the tail parameter ␣ of the
Lévy motion equals the order of the fractional derivative.
Solutions to the vector space-fractional diffusion equation
are operator Lévy motions 关10兴 that may scale at different
rates in different directions. The matrix exponent of the fractional derivative is related to the scaling rates in a similar
manner 关11,12兴. A more general diffusion equation governs
any Lévy process X(t) 关13,14兴. The probability density
p(x,t) of any such process solves a diffusion-type equation
⳵ p 共 x,t 兲
⫽L p 共 x,t 兲 ;
⳵t
p 共 x,0 兲 ⫽ ␦ 共 x兲 ,
共1兲
where L is the generator of the Feller semigroup S t f (x)
⫽ 兰 f (x⫺y)p(y,t)dy 关15–17兴. In this case, we say that X(t)
is the stochastic solution to Eq. 共1兲. The generator L f (x)
⫽limt↓0 t ⫺1 关 S t f (x)⫺ f (x) 兴 . If X(t) is an ␣ -stable Lévy motion without drift, then L is a fractional derivative operator of
order ␣ .
*Electronic address: mcubed@unr.edu
1063-651X/2002/65共4兲/041103共4兲/$20.00
Fractional time derivatives are important in reactive transport, since solutes may interact with the immobile porous
medium in highly nonlinear ways. There is evidence that
solutes may sorb for random amounts of time that have a
power law distribution 关8兴, or move into irregularly sized
blocks of relatively immobile water, producing similar behavior 关9兴. If the first moment of these time delays diverges,
then a fractional time derivative applies 关6兴. The fractional
time derivative ⳵ ␥ g(t)/ ⳵ t ␥ for 0⬍ ␥ ⬍1 is the inverse
Laplace transform of s ␥ g(s), where g(s)⫽L关 g(t) 兴 is the
usual Laplace transform. In this paper, we find the stochastic
solution to the space-time fractional diffusion equation
⳵ ␥ q 共 x,t 兲
⳵t␥
⫽Lq 共 x,t 兲 ⫹ ␦ 共 x兲
t ⫺␥
.
⌫ 共 1⫺ ␥ 兲
共2兲
We show that if X(t) is the stochastic solution to Eq. 共1兲 then
X(V t ) is the corresponding solution to Eq. 共2兲, where V t is
the inverse Lévy process 关18兴 for the stable subordinator
with index ␥ . The fractional time derivative subordinates
X(t) to the inverse stable subordinator V t .
The space-time fractional diffusion equation is also connected with scaling limits of continuous time random walks
共CTRW, see 关6兴兲. The spatial operator L depends on the jump
size distribution 关11,12兴. A fractional time derivative of order
0⬍ ␥ ⬍1 pertains when the random waiting time T between
jumps satisfies P(T⬎t)⬇t ⫺ ␥ so that E(T)⫽⬁. The infinite
mean waiting time CTRW limit is the finite mean waiting
time CTRW limit, subordinated to the inverse stable subordinator V t . The random variable V t has a Mittag-Leffler distribution 关19兴 previously noted in connection with fractional
time derivatives 关4,20兴 and relaxation 关21兴.
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©2002 The American Physical Society
MEERSCHAERT, BENSON, SCHEFFLER, AND BAEUMER
PHYSICAL REVIEW E 65 041103
c ⫺ ␥ N [ct] ⇒V t
II. CTRW SCALING LIMITS
CTRW were introduced 关22,23兴 to study random walks on
a lattice. They are now used in physics to model a wide
variety of phenomena connected with anomalous diffusion
关23–25兴. With finite mean waiting times, the jump process is
asymptotically linear, and the CTRW behaves in a manner
similar to the original random walk for large time 关20,26兴.
For a scalar process, finite variance jumps lead to Brownian
motion in the scaling limit. Infinite variance jumps with
power law tails lead to Lévy motion. Vector jumps with finite
second moments lead to multivariable Brownian motion.
Vector jumps with power law tails lead to multivariable Lévy
motion, or operator Lévy motion if the power law behavior
varies with the direction of motion 关11,12兴. The speed of
convergence to the CTRW scaling limit, and the implications
for fractional diffusion modeling, are discussed in a recent
paper of Barkai 关27兴.
Many physical applications involve infinite mean waiting
times 关21,28兴. Introducing infinite mean waiting times has
the effect of subordinating the CTRW scaling limit to the
inverse process of a stable subordinator whose index ␥ is the
same as the power law tail index of the waiting times. Essentially, this is because the counting process for particle
jumps is inverse to the jump time process. The jump time
process is asymptotically the stable subordinator, so the
counting process for particle jumps is asymptotically the inverse stable subordinator.
A rigorous mathematical proof appears in 关29兴. We recount the basic ideas here to emphasize the physical applications. Given iid positive random variables J i let T n
n
⫽ 兺 i⫽1
J i denote the time of the nth particle jump. The pon
sition of the particle after the nth jump is W(n)⫽ 兺 i⫽1
Yi
where Yi are iid and assumed independent of J i . Then N t
⫽max兵n:Tn⭐t其 counts the number of particle jumps by time
t⬎0 and the CTRW variable W(N t ) gives the position of the
particle at time t⬎0.
If Y has zero mean and finite second moments, the simple
random walk of particle jumps
c ⫺1/2W共关 ct 兴 兲 ⇒X共 t 兲
as c→⬁,
共3兲
where the scaling limit X(t) is a Brownian motion. Shrinking
the spatial coordinates by c 1/2 compensates expanding the
time scale by c according to the central limit theorem. If
P(J⬎t)⬇t ⫺ ␥ for some 0⬍ ␥ ⬍1 then
c ⫺1/␥ T [ct] ⇒B t
as c→⬁
共4兲
according to the extended central limit theorem 关15兴 where
the scaling limit B t is the stable subordinator process 关13兴.
The ␥ -stable random variable B t is totally positively skewed,
hence this Lévy process is strictly increasing. The inverse
process
V ␶ ⫽inf兵 t:B t ⬎ ␶ 其
is also called the hitting time or first passage time process.
Using the fact that T n ,N t are inverse, so that 兵 N t ⭓x 其
⫽ 兵 T dxe ⭐t 其 , along with Eq. 共4兲 yields
as c→⬁.
Hence N [ct] ⬇c ␥ V t , and together with Eq. 共3兲 this yields
c ⫺ ␥ /2W共 N [ct] 兲 ⬇ 共 c ␥ 兲 ⫺1/2W共关 c ␥ V t 兴 兲 ⇒X共 V t 兲
as c→⬁, so that the Brownian motion X(t) is subordinated
to the inverse stable subordinator V t .
The inverse processes have inverse distributional scaling
B ct ⫽c 1/␥ B t and V ct ⫽c ␥ V t , and together with the classical
scaling for Brownian motion X(ct)⫽c 1/2X(t) this shows that
the CTRW limit is subdiffusive
X共 V ct 兲 ⫽X共 c ␥ V t 兲 ⫽c ␥ /2X共 V t 兲
with Hurst index H⫽ ␥ /2⬍1/2. Since P(V ␶ ⭐t)⫽ P(B t ⭓ ␶ )
⫽ P(t 1/␥ B 1 ⭓ ␶ )⫽ P„(B 1 / ␶ ) ⫺ ␥ ⭐t… the random variable V ␶
has the same density function as ( ␶ /B 1 ) ␥ . The density g ␥ of
the stable random variable B 1 has Laplace transform
␥
L关 g ␥ (t) 兴 ⫽e ⫺s . Computing moments of (t/B 1 ) ␥ shows that
V t has a Mittag-Leffler distribution 关19兴. If p(x,t) is the
density of X(t) then a conditioning argument along with a
simple change of variable shows that X(V t ) has density
q 共 x,t 兲 ⫽
冕
⫽
t
␥
⬁
0
p„x, 共 t/s 兲 ␥ …g ␥ 共 s 兲 ds
冕
⬁
0
p 共 x,u 兲 g ␥ 共 tu ⫺1/␥ 兲 u ⫺1/␥ ⫺1 du.
共5兲
Analytical estimates in 关29兴 show that q(k,t)⭓C 储 k储 ⫺b for
large 储 k储 , so X(V t ) does not have a normal density and hence
cannot be a fractional Brownian motion 关30兴.
If P( 储 Y储 ⬎r)⬇r ⫺ ␣ for some 0⬍ ␣ ⬍2 then X(t) is an
␣ -stable Lévy motion and the CTRW limit X(V t ) has Hurst
index H⫽ ␥ / ␣ . If the tail index varies with the spatial coordinate, operator norming applies 关12兴. Then X(ct)⫽c E X(t)
leads to X(V ct )⫽c ␥ E X(t) so that the Hurst index H⫽ ␥ E is
a matrix. For a diagonal exponent E⫽diag(1/␣ 1 , . . . ,1/␣ d )
the ith coordinate X i (t) is an ␣ i -stable Lévy motion and
X i (V t ) is self-similar with Hurst index ␥ / ␣ i . Diagonalizable
matrix exponents introduce a change of coordinates. Repeated eigenvalues thicken probability tails by a logarithmic
factor, and complex exponents introduce rotations, leading to
discrete scale invariance 关31兴. In every case, the scaling limit
X(t) of the simple random walk is subordinated by the inverse stable subordinator V t and the density changes from
p(x,t) to q(x,t) via Eq. 共5兲 when infinite mean waiting times
are introduced. Next, we show that this change corresponds
to a fractional time derivative in the diffusion equation.
III. TIME-FRACTIONAL DIFFUSION EQUATIONS
Wyss 关32兴 and Schneider and Wyss 关33兴 studied a timefractional diffusion equation. Zaslavsky 关34兴 introduced the
space-time fractional kinetic equation 共2兲 for Hamiltonian
chaos. When L⫽⫺ v ⳵ / ⳵ x⫹D ⳵ ␣ / ⳵ 兩 x 兩 ␣ Saichev and
Zaslavsky 关35兴 show that if p(x,t) solves Eq. 共1兲 then the
function q(x,t) given by Eq. 共5兲 solves Eq. 共2兲. When ␣
⫽2 they call the stochastic solution to Eq. 共2兲 a ‘‘fractal
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STOCHASTIC SOLUTION OF SPACE-TIME . . .
PHYSICAL REVIEW E 65 041103
Brownian motion.’’ We prefer the term ‘‘time-fractional diffusion’’ to avoid confusing this process with the well-known
fractional Brownian motion. In fact, if X(t) is a Brownian
motion, the stochastic solution to Eq. 共1兲 in this case, then
the stochastic solution to Eq. 共2兲 is X(V t ) where V t is the
inverse ␥ -stable subordinator. This interesting stochastic process is self-similar with Hurst index ␥ /2 so it is subdiffusive.
However X(V t ) does not have a Gaussian distribution and it
does not have stationary increments 关29兴, so it is not fractional Brownian motion.
Barkai, Metzler, and Klafter 关20兴 introduce a fractional
Fokker-Planck equation equivalent to Eq. 共2兲 with
⳵ V ⬘共 x 兲
⳵
⫹K 1 2 .
⳵x m␩1
⳵x
tional Cauchy problem 共2兲 via the inverse Lévy transform
共5兲. We summarize the essentials here in order to clarify the
argument.
Use s ␥ ⫺1 ⫽L关 t ⫺ ␥ /⌫(1⫺ ␥ ) 兴 and take Laplace-Fourier
transforms (x哫k,t哫s) in Eq. 共2兲 to get s ␥ q(k,s)
⫽ ␺ (k)q(k,s)⫹s ␥ ⫺1 , where ␺ (k) is the Fourier symbol of
L, so that F 关 L f (x) 兴 ⫽ ␺ (k) f (k). Then
q 共 k,s 兲 ⫽
⫽
2
L⫽⫺
Barkai 关4兴 applies Eq. 共5兲, which he calls the inverse Lévy
transform of p(x,t), to the solution of Eq. 共1兲 in order to
solve this fractional Fokker-Planck equation.
Scalar solutions to Eq. 共1兲 with
L⫽⫺ v
冉
␣
␣
⳵
1⫺ ␤
⳵
1⫹ ␤ ⳵
⫹D
⫹
␣
⳵x
2 ⳵ 共 ⫺x 兲
2 ⳵x␣
s ␥ ⫺1
s ␥ ⫺ ␺ 共 k兲
冕
⬁
0
are multivariable stable densities 关11兴, where
erator with Fourier symbol
冕
储 ␪ 储 ⫽1
␥
exp兵 ⫺ 关 s ␥ ⫺ ␺ 共 k兲兴 u 其 du
共6兲
冕
⬁
0
1/␥ ) ␥
⫽
冕
⬁
e ⫺su
1/␥
v
0
g ␥共 v 兲 d v
e ⫺st g ␥ 共 u ⫺1/␥ t 兲 u ⫺1/␥ dt.
共7兲
Then compute
␥
s ␥ ⫺1 e ⫺s u ⫽
⫽
is the op-
⫺1 d
␥ u ds
1
␥u
冕
⬁
0
冉冕
⬁
0
e ⫺st g ␥ 共 u ⫺1/␥ t 兲 u ⫺1/␥ dt
冊
te ⫺st g ␥ 共 u ⫺1/␥ t 兲 u ⫺1/␥ dt
共8兲
and combine with Eq. 共6兲 to write q(k,s) as
冕 冉␥ 冕
冕 冉冕
⬁
共 ik• ␪ 兲 ␣ m 共 ␪ 兲 d ␪ .
0
⫽
1
u
⬁
⬁
0
e ⫺st
0
q 共 k,t 兲 ⫽
where the generalized fractional derivative
冊
te ⫺st g ␥ 共 u ⫺1/␥ t 兲 u ⫺1/␥ dt p 共 k,u 兲 du
⬁
0
冊
t
p 共 k,u 兲 g ␥ 共 u ⫺1/␥ t 兲 u ⫺1/␥ ⫺1 du dt.
␥
Now invert the Laplace transform to obtain
L⫽⫺ v •“⫹ 12 “•A“⫹F,
冕
0
␥
⫽
If ␣ ⫽2, this integral reduces to (ik)A(ik) ⬘ where the matrix
A has i j component 兰 ␪ i ␪ j m( ␪ )d ␪ , and solutions are vector
Brownian motion.
Operator stable densities, where the stable index depends
on the coordinate, solve Eq. 共1兲 with
Ff 共 x 兲 ⫽
⬁
s ␥ ⫺1 e ⫺s u p 共 k,u 兲 du,
e ⫺s u ⫽e ⫺(su
␣
L⫽⫺ v •“⫹D“ m
␣
“m
冕
using 兰 ⬁0 e ⫺au du⫽a ⫺1 and p(k,t)⫽e ␺ (k)t , which follows
␥
␥
from Eq. 共1兲. Use d(e ⫺s u )/ds⫽⫺ ␥ s ␥ ⫺1 ue ⫺s u to get
␥
␥
␥
s ␥ ⫺1 e ⫺s u ⫽⫺( ␥ u) ⫺1 d(e ⫺s u )/ds. Recall that e ⫺s
⫽L关 g ␥ (t) 兴 and write
冊
are ␣ -stable densities 关36,37兴, purely symmetric when the
skewness ␤ ⫽0 关2,38兴 and maximally skewed when ␤ ⫽1
关3,34兴. When ␣ ⫽2 the skewness ␤ is irrelevant, and the
solutions are normal densities. Vector solutions for
⫽s ␥ ⫺1
t
␥
冕
⬁
0
p 共 k,u 兲 g ␥ 共 tu ⫺1/␥ 兲 u ⫺1/␥ ⫺1 du
and invert the Fourier transform to get Eq. 共5兲.
关 f 共 x⫺y 兲 ⫺ f 共 x 兲 ⫹y“ f 共 x 兲兴 d ␾ 共 y 兲
IV. CONCLUSIONS
and d ␾ (r E ␪ )⫽r ⫺2 dr m( ␪ )d ␪ is an operator stable Lévy
measure 关10,12,39兴. These are all abstract Cauchy problems
关16,17兴 whose solution p(x,t) is the family of densities for a
Lévy process, a stationary independent increment process
that includes Brownian motion and 共operator兲 Lévy motion
as special cases. Baeumer and Meerschaert 关40兴 give a rigorous mathematical proof that any solution to the abstract
Cauchy problem 共1兲 is transformed to a solution of the frac-
Infinite mean waiting times subordinate CTRW scaling
limits to an inverse stable subordinator, equivalent to applying an inverse Lévy transform 共5兲 to the solution density.
Since the solutions to time-fractional diffusion equations are
also obtained via the inverse Lévy transform, a ␥ -fractional
time derivative in a diffusion equation has the effect of subordinating the stochastic solution to the inverse process of a
␥ -stable subordinator. When applied to the classical diffusion
equation, this procedure produces the ‘‘fractal Brownian mo-
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MEERSCHAERT, BENSON, SCHEFFLER, AND BAEUMER
PHYSICAL REVIEW E 65 041103
tion’’ of Saichev and Zaslavsky as the solution to the timefractional diffusion equation, a model for subdiffusion. This
interesting stochastic process is not the same as fractional
Brownian motion, but rather a completely different stochastic process. For CTRW models with coupled memory, in
which particle jumps Yi and waiting times J i are dependent,
the effect of infinite mean waiting times is more complicated
关26兴. In a forthcoming paper we will show that infinite mean
waiting times in a coupled CTRW model also induce subor-
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关12兴
关13兴
关14兴
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ACKNOWLEDGMENTS
M.M.M. and B.B. were partially supported by NSF DES
Grant No. 9980484, and D.A.B. was partially supported by
NSF-DES Grant No. 9980489 and DOE-BES Grant No. DEFG03-98ER14885.
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