Reflected stable Lévy motions and their
governing equations
Center for Applied Mathematics Colloquium
Cornell University
Friday, September 25 2015
Mark M. Meerschaert
Department of Statistics and Probability
Michigan State University
mcubed@stt.msu.edu
http://www.stt.msu.edu/users/mcubed
Partially supported by NSF grants DMS-1462156 and EAR-1344280.
Abstract
Stable Lévy motions are now used in many areas of science and
engineering to model anomalous super-diffusion, where a plume
of particles spreads faster than the traditional Brownian motion
model predicts. Their probability densities solve a partial differential equation that involves fractional derivatives. When the
process is reflected to stay in the positive the real line, it becomes a Markov process. In this talk, we explicitly compute the
transition densities of this Markov process, and their governing
equation. This Kolmogorov forward equation involves a fractional boundary condition to enforce the no-flux boundary. This
seems to be the first known derivation of a fractional reflecting
boundary condition. We then apply numerical methods to explicitly compute the transition density of this space-inhomogeneous
Markov process, for any starting point, to any desired degree of
accuracy. Finally, we discuss an application to fractional Cauchy
problems, which involve a positive Caputo fractional derivative
in time.
Acknowledgments
Boris Baeumer, Maths & Stats, University of Otago, New Zealand
David Benson, Geological Engineering, Colorado School of Mines.
Mihály Kovács, Maths & Stats, University of Otago, New Zealand
Hans-Peter Scheffler, Math, Universität Siegen, Germany.
René L. Schilling, Institute of Math. Stoch., TU Dresden
Rina Schumer, Hydrology, Desert Research Institute, Reno, Nevada.
Alla Sikorskii, Statistics and Probability, Michigan State
Peter Straka, Applied Mathematics, U New South Wales
Charles Tadjeran, Mathematics, U Nevada
Stephen W. Wheatcraft, Geological Sciences, U Nevada
Two Recent Books
Stochastic Models for Fractional Calculus
Mark M. Meerschaert and Alla Sikorskii
De Gruyter Studies in Mathematics 43, 2012
Advanced graduate textbook
ISBN 978-3-11-025869-1
Mathematical Modeling, 4th Edition
Mark M. Meerschaert
Academic Press, Elsevier, 2013
Advanced undergraduate / beginning graduate textbook
ISBN 978-0-12-386912-8
Fractional derivatives: An old idea gets new life
Fractional derivatives dαf (x)/dxα for any α > 0 were invented by
Leibniz soon after the more familiar integer derivatives.
Some derivative formulas extended to the fractional case:
dα h λxi
α eλx
e
=
λ
dxα
π
dα
[sin x] = sin x + α
dxα
2
dα
Γ(p + 1)
p−α
p
x
x
=
[
]
dxα
Γ(p − α + 1)
Probability and transforms
If the random variable X has density f (x) so that
P (a ≤ X ≤ b) =
Z b
a
f (x)dx
then f (x) has Fourier transform (FT)
Z ∞ 1
1 − ikx + (ikx)2 + · · · f (x)dx
fˆ(k) =
2!
−∞
1
= 1 − ikµ1 − k2µ2 + · · ·
2
where the pth moment
µp =
Z ∞
−∞
xpf (x)dx
Central limit theorem
If µ1 = 0 and µ2 = 2 then fˆ(k) = 1 − k2 + · · ·
The IID sum Sn = X1 +· · ·+Xn has FT fˆ(k)n and the normalized
√
sum Sn/ n has FT
n
√ 2
√ n
fˆ(k/ n) = 1 − (k/ n) + · · ·
= 1−
2
k2
n
+ ···
→ e−k ≡ ĝ(k)
!n
as n → ∞.
Inverting the Fourier transform reveals a Gaussian density
1 −x2/4
e
g(x) = √
4π
Brownian motion
If Xn represents a particle jump at time n then Sn = X1 +· · ·+Xn
is the location of the particle at time n. Expanding the time
scale by a factor of r > 0 and taking limits as r → ∞ shows that
r −1/2S[rt] ⇒ At since
fˆ(r −1/2k)[rt] = 1 −
k2
r
+ ···
![rt]
2
→ e−k t ≡ ĉ(k, t)
for all t > 0. Inverting the FT shows that the density of the
limiting Brownian motion process At is Gaussian
c(x, t) = √
2
1
e−x /(4t).
4πt
0.5
0.0
S
1.0
Simple random walk simulation
2
4
6
8
10
t
R codes at www.stt.msu.edu/users/mcubed/Rcodes.zip
−7
−5
S
−3
−1
Longer time scale
0
20
40
60
t
80
100
0
5
S
10
15
Scaling limit: Brownian motion
0
200
400
600
800
1000
t
Any finite variance jumps lead to the same limit process.
The graph is a random fractal with dimension 2 − 1/2.
The diffusion equation
Taking Fourier transforms in the classical diffusion equation
∂c(x, t)
∂ 2c(x, t)
=
∂t
∂x2
yields [since f ′(x) has FT (ik)fˆ(k)]
d
ĉ(k, t) = (ik)2ĉ(k, t) = −k2ĉ(k, t)
dt
whose solution
2
ĉ(k, t) = e−k t
inverts to the same limit density for the Brownian motion At.
For a cloud of diffusing particles, c(x, t) is the particle density.
Properties of Brownian motion
Since the density of t1/2A1 has FT
√ 2
2
tk)
−tk
−(
=e
= ĉ(k, t)
e
d
we see that At = t1/2A1 (same density).
A cloud of diffusing particles spreads at the rate t1/2.
2
The plume tails off like e−x .
Real plumes often spread faster, with a heavier tail.
0.20
0.10
0.00
density
Classical diffusion profile
−6
−4
−2
0
2
4
6
x1
Brownian motion Gaussian (Normal) density at time t = 1, 2, 3
showing square root spreading rate and fast tail decay.
Heavy (power law) tails
If P (X > x) ≈ x−α then f (x) ≈ αx−α−1 and some moments
µk =
Z ∞
−∞
xk f (x)dx
do not exist. If 1 < α < 2 and µ1 = 0 then a typical X has FT
fˆ(k) = 1 + (ik)α + · · ·
and n−1/α(X1 + · · · + Xn) has FT
n
−1/α
n
−1/α
α
fˆ(n
k) = 1 + (n
ik) + · · ·
!n
α
(ik)
= 1+
+ ···
n
α
(ik)
→e
≡ ĝ(k)
as n → ∞.
The inverse Fourier transform g(x) is called a stable density.
This random walk is called a Lévy flight.
Stable Lévy motion
If Sn = X1 + · · · + Xn is particle location at time n, then the
scaling limit r −1/αS[rt] ⇒ At, since
fˆ(r −1/αk)[rt] = 1 +
(ik)α
r
+ ···
![rt]
α
t(ik)
→e
≡ ĉ(k, t).
1/α α
α
d
Now At = t1/αA1, since t1/αA1 has FT e(ikt ) = et(ik) .
The limit process At is called a stable Lévy motion.
Lévy motion models super-diffusion, where a plume spreads like
t1/α for α < 2, faster then Brownian motion.
−2.0
−1.0
S
0.0
1.0
Heavy tail random walk simulation
2
4
6
8
10
t
Random walk jumps P (X > x) = x−α with α = 1.5.
−30
−50
S
−10 0
Longer time scale
0
20
40
60
t
Converging to an α-stable Lévy motion.
80
100
−200
−400
S
0
Scaling limit: Stable Lévy motion
0
200
400
600
800
1000
t
Power law jumps persist in the limit process (non-locality).
Graph is a random fractal of dimension 2 − 1/α.
Fractional diffusion equation
To solve the fractional diffusion equation
∂c(x, t)
∂ αc(x, t)
=
∂t
∂xα
take Fourier transforms to get
d
ĉ(k, t) = (ik)αĉ(k, t)
dt
and solve to obtain
ĉ(k, t) = et(ik)
α
so the density of the Lévy motion solves this fractional PDE!
In this case, c(x, t) falls off like x−α−1 as x → ∞.
0.00
0.10
c
0.20
0.30
Fractional diffusion profile
−4
−2
0
2
4
x
Stable α = 1.5 Lévy motion density at time t = 1, 2, 3 showing
super-diffusive spreading rate, skewness, and power law tail.
Application to groundwater hydrology [BSMW01]
0.15
plume data
α-stable
Gaussian
0.10
Normalized Mass
Normalized Mass
0.15
0.05
0.00
-100
0
100
200
Longitudinal Distance (meters)
0.10
0.05
0.00
-100
300
a) Snapshot 1 (day #27)
0
100
200
Longitudinal Distance (meters)
300
b) Snapshot 1 (day #132)
0.15
0.15
plume data
α-stable
Gaussian
0.10
Normalized Mass
Normalized Mass
plume data
α-stable
Gaussian
0.05
0.00
-100
0
100
200
Longitudinal Distance (meters)
c) Snapshot 3 (day #224)
300
plume data
α-stable
Gaussian
0.10
0.05
0.00
-100
0
100
200
Longitudinal Distance (meters)
d) Snapshot 4 (day #328)
300
Power law tail downstream
SOXPH GDWD
B VWDEOH
*DXVVLDQ
1RUPDOL]HG 0DVV
1RUPDOL]HG 0DVV
Fractional diffusion equation with α = 1.1 captures power law
leading tail at the MADE experimental site [BSMW01].
SOXPH GDWD
B VWDEOH
*DXVVLDQ
E 6QDSVKRW GD\ D 6QDSVKRW GD\ /RQJLWXGLQDO 'LVWDQFH PHWHUV
/RQJLWXGLQDO 'LVWDQFH PHWHUV
∂c(x, t)
∂c(x, t)
∂ αc(x, t)
Governing equation:
= −0.12
+ 0.14
∂t
∂x
∂xα
Application to Ecology
Fisher equation for population growth and dispersal [BKM08]
∂c(x, y, t)
=a
∂t
∂ 2c(x, y, t)
∂x2
+b
∂ 2c(x, y, t)
∂y 2
+ rc(x, y, t) 1 −
c(x, y, t)
K
20
y
10
0
−10
−20
−20
0
20
40
60
80
x
Population diffuses slowly through gap in barrier.
100
!
Fractional Fisher equation
Fractional derivatives model power law movements [BKM08].
∂c(x, y, t)
=a
∂t
∂ 1.7c(x, y, t)
∂x1.7
+b
∂ 2c(x, y, t)
∂y 2
c(x, y, t)
+rc(x, y, t) 1 −
K
20
y
10
0
−10
−20
−20
0
20
40
60
80
100
x
Fractional diffusion jumps the barrier (nonlocal operator).
Real invasive species data shows this behavior.
!
Reflected stable process
Take a stable Lévy process Yt with negative jumps
α
−ikY
t(−ik)
t
ĉ(k, t) = E[e
]=e
for some 1 < α ≤ 2, and define the reflected process
Zt = Yt − inf{Ys : 0 ≤ s ≤ t}.
Then Zt ≥ 0 is a Markov process on the real line. The semigroup
Ttf (x) = E[f (Zt+s)|Zs = x]
on the Banach space C∞(R) satisfies the Feller property
kTtf − f k := sup{|Ttf (x) − f (x)| : x ∈ R} → 0
as t → 0.
We show [BKMSS15, Cor 2.4] that Zt+s given Zs = x has a
smooth transition density y 7→ p(x, y, t).
Reflected stable sample path
50
0
−100
Y
150
Stable process Yt (thin line) with α = 1.3 and reflected stable Zt
(thick line). See Appendix for R code.
0
200
400
600
t
800
1000
The forward equation [BKMSS15, Theorem 2.3]
The transition density p(x, y, t) of Zt solves the forward equation
∂p(x, y, t)
∂ αp(x, y, t)
=a
;
α
∂t
∂(−y)
∂ α−1p(x, 0+, t)
= 0.
α−1
∂(−y)
Here dαf (x)/d(−x)α has FT (−ik)αfˆ(k).
When α = 2 (reflected Brownian motion) this is just the diffusion
equation with the reflecting boundary condition
∂
∂
p(x, 0+, t) = − p(x, y, t)
∂(−y)
∂y
= 0.
y=0+
The fractional BC enforces a no-flux condition at the boundary.
This boundary condition keeps all the probability mass in [0, ∞).
Reflected stable transition density
Transition densities y 7→ p(x, y, t) with index α = 1.8 and initial
state x = 0, 1, 2, 4 (left to right) at times t = 0.5 (left), t = 1
(middle), and t = 2 (right). See Appendix for Matlab code.
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
5
0
0
5
0
0
5
Numerical method
For f is bounded, and f (k) ∈ L1(R) for k ≤ 2 [MS12, Prop 2.1]
∞
X
dαf (x)
(−1)j Γ(1 + α)
−α
= lim h
f (x ∓ jh).
α
h→0
d(±x)
j=0 j!Γ(1 + α − j)
Explicit/implicit Euler schemes based on this Grünwald-Letnikov
approximation are unstable [MT04, Props 2.1, 2.3].
Shift: Replace jh by (j − 1)h to get a stable convergent implicit
Euler method [MT04, Th 2.7].
Enforce the fractional no-flux BC at y = 0.
See [BKMSS15, Section 3] for complete details.
The backward equation
The transition density p(x, y, t) of Zt also solves the backward
equation
∂ αp(x, y, t)
∂p(x, y, t)
=a
;
α
∂t
∂x
∂p(0+, y, t)
= 0.
∂x
The negative fractional derivative in the forward equation becomes a positive fractional derivative in the backward equation.
The fractional boundary condition in the forward equation becomes an integer derivative in the backward equation.
Bernyk, Dalang and Peskir (2011) compute the backward generator ∂ α/∂xα
Patie and Simon (2012) compute exact domain of the generator,
and show that the boundary condition ∂p/∂x = 0 holds.
Application: Fractional Cauchy problems
Suppose Xt is a Brownian motion on Rd with generator L = ∆.
Then p(x, t) = E[f (XZt )|X0 = x] solves the fractional Cauchy
problem
∂ β p(x, t)
t−β
= Lp(x, t) + f (x)
β
∂t
Γ(1 − β)
for any f ∈ D(L), when β = 1/α for some α ∈ (1, 2).
Proof: Dt = inf{r > 0; Yr > t} is a β-stable subordinator whose
inverse Et = inf{u > 0; Du > t} is also the supremum process
(inverse of the inverse) of Yt.
The result is known for the inverse stable subordinator Et [BM01,
Theorem 3.1 and MS04, Theorem 5.1].
But Prop VI.3 in Bertoin (1996) implies Et = Zt in distribution.
Ongoing Research: Reflected stable on [a, b]
Plan: Take a stable Lévy process Yt with negative jumps
ĉ(k, t) = E[e−ikYt ] = et(−ik)
α
for some 1 < α ≤ 2, and reflect to stay in [a, b].
Burdzy et al. (2009) solve the Skorokhod Problem:
Zt = Yt + At − Bt ∈ [a, b], At, Bt are non-decreasing, and
Z ∞
0
I(Zt > a)dAt =
Z ∞
0
I(Zt < b)dBt = 0.
Apply the Itô Representation and the compensation formula to
compute the backward generator.
Compute adjoints (integration by parts) to get the forward generator and the BC.
Governing equations on [a, b]
Conjecture: The forward equation is
∂p(x, y, t)
α p(x, y, t);
= D∂[y,b]
∂t
The backward equation is
α−1
∂[y,b]
p(x, a+, t) = 0, ∂y p(x, b−, t) = 0.
∂p(x, y, t)
α−1
α p(x, y, t);
= D∂[a,x]
∂[a,x]
p(b−, y, t) = 0, ∂xp(a+, y, t) = 0.
∂t
Here we use the positive Caputo fractional derivative
x
1
α
f ′′(y)(x − y)1−αdy
∂[a,x]f (x) =
Γ(2 − α) a
and the negative Caputo fractional derivative
Z
b
1
α
∂[x,b]f (x) =
f ′′(y)(y − x)1−αdy
Γ(2 − α) x
on the finite interval.
Z
Interpret: Fractional BC prevents jumps past a, first order BC
prevents drift past b, in the forward equation.
References
1. B. Baeumer, M. Kov acs, M.M. Meerschaert, Numerical solutions for fractional reactiondiffusion equations, Computers and Mathematics with Applications, 55 (2008), 2212–
2226.
2. B. Baeumer, M. Kováks, M.M. Meerschaert, P. Straka, and R. Schilling (2015) Reflected spectrally negative stable processes and their governing equations. Trans. Amer.
Math. Soc., to appear. Preprint at www.stt.msu.edu/users/mcubed/ReflectedStable.pdf
3. D.A. Benson, R. Schumer, M.M. Meerschaert and S.W. Wheatcraft (2001) Fractional
dispersion, Lévy motions, and the MADE tracer tests. Transport in Porous Media 42,
211–240.
4. V. Bernyk, R.C. Dalang and G. Peskir (2011) Predicting the ultimate supremum of a
stable Lévy process with no negative jumps. Ann. Probab., 39, 2385–2423.
5. J. Bertoin (1996) Lévy Processes. Cambridge University Press, Cambridge.
6. K. Burdzy, W. Kang, and K. Ramanan (2009) The Skorokhod problem in a timedependent interval. Stoch. Proc. Appl. 119(2), 428–452.
7. K.J. Falconer, Fractal Geometry–Mathematical Foundations and Applications, Wiley,
Chichester, 1990.
8. M.M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional
advection-dispersion flow equations. J. Comput. Appl. Math. 172 (2004), 65–77.
9. M.M. Meerschaert and Y. Xiao, Dimension results for sample paths of operator stable
Lévy processes, Stochastic Processes and Their Applications Vol. 115 (2005), No. 1,
pp. 55–75.
10. M. M. Meerschaert and A. Sikorskii (2012) Stochastic Models for Fractional Calculus.
De Gruyter Studies in Mathematics 43, De Gruyter, Berlin, 2012, ISBN 978-3-11025869-1.
11. M.M. Meerschaert, Fractional Calculus, Anomalous Diffusion, and Probability, Fractional Dynamics, R. Metzler and J. Klafter, Eds., World Scientific, Singapore, pp.
265–284, 2012.
12. P. Patie and T. Simon (2012) Intertwining certain fractional derivatives. Potential Anal.
36, 569–587.
R code for reflected stable sample path
# Plot stable Y_t with characteristic function exp(t(ik)^a)
# and the reflected stable process Z_t=Y_t-inf{Y_u:0<=u<=t}
#
# You need to install the stabledist package on your R platform.
# Try Packages > Load package to see if stabledist is available.
# If not then use Packages > Install package(s)
library(stabledist)
t=seq(1:1000)
a=1.3
g=(abs(cos(pi*a/2)))^(1/a)
y=rstable(t,alpha=a,beta=-1.0,gamma=g,delta=0.0,pm=1)
Y=cumsum(y)
Z=Y-cummin(Y)
plot(t,Y,type="l",ylim=c(min(Y),max(Z)))
lines(t,Z,lwd=2)
axis=t-t
lines(t,axis)
MATLAB code for reflected stable pdf
%%% Matlab script to compute p(x,y,t)
%% enter variables
alpha=1.2; ymax=12; N=1200; t=[0,.5,1,2]; x=2;
%% initialise parameters
h=ymax/N; y=(h:h:ymax)’;
u0=zeros(N,1);u0(floor(x/h)+1)=1/h; % initial condition
%% Make Grunwald matrix
w=ones(1,N+1);
for k=1:N
w(k+1)=w(k)*(k-alpha-1)/k;
end
w=w/h^alpha;
M=spdiags(repmat(w,N,1),-1:1:N-1,N,N); %enter w’s along diagonals
M(1,:)=-cumsum(w(1:N))’; %change first row for BC
%% Solve ODE system
[~,p]=ode113(@(t,u) M*u,t,u0);