Reflected stable subordinators for fractional Cauchy problems

0.35 MB·English

by  Boris Baeumer

REFLECTED STABLE SUBORDINATORS FOR FRACTIONAL
CAUCHY PROBLEMS
BORIS BAEUMER, MIHA´LY KOVA´CS, MARK M. MEERSCHAERT, RENE´ L. SCHILLING,
AND PETER STRAKA
3
1
Abstract. In a fractional Cauchy problem, the first time derivative is replaced by
0
a Caputo fractional derivative of order less than one. If the original Cauchy problem
2
governsaMarkovprocess,anon-Markoviantimechangeyieldsastochasticsolutionto
n
the fractional Cauchy problem, using the first passage time of a stable subordinator.
a
J This paper proves that a spectrally negative stable process reflected at its infimum
3 hasthe sameonedimensionaldistributionsasthe inversestablesubordinator. There-
2 fore, this Markov process can also be used as a time change, to produce stochastic
solutionstofractionalCauchyproblems. Theproofusesanextensionofthe D.Andr´e

]
reflectionprinciple. Theforwardequationofthereflectedstableprocessisestablished,
R
including the appropriate fractional boundary condition, and its transition densities
P
are explicitly computed.
.
h
t
a
m
[
1. Introduction

1
v A classical paper of Einstein [13] outlines the link between random walks, Brownian
5 motion, and the diffusion equation ∂ u(x,t) = ∂2u(x,t). Sokolov and Klafter [33]
0 t x
review modern extensions of the theory to encompass random walks with heavy tails,
6
5 jump processes, and fractional diffusion. This connection between probability and
.
1 differentialequationshasprofoundconsequences formathematics[1,7,8,25], andforits
0
applications in science and engineering [16, 27, 28, 31], including a probabilistic method
3
1 for solving fractional differential equations, by exploiting the associated Markov process
: [37]. More details on fractional calculus, and its connection to probability theory, may
v
i be found in the recent book of Meerschaert and Sikorskii [26].
X
The time-fractional diffusion equation ∂βu(x,t) = ∂2u(x,t) with 0 < β < 1 governs
r t x
a the long-time limit of a random walk with power law waiting times P(W > t) ∼ Ct−β
between particle jumps [22, Theorem 4.2], see also [26, Sec. 4.5]. The limit process
is a Brownian motion, time-changed by the inverse or hitting time E of a β-stable
t
subordinator. This time change process is not Markovian, since it remains constant
over time intervals whose distribution is not exponential.
The time-fractional diffusion equation is one example of a fractional Cauchy problem.
More generally, for any strongly continuous Markov semigroup T f(x) = Ex[f(X )]
t t
with generator L, it follows from [3, Theorem 3.1] along with [22, Corollary 3.1] that
Keywordsandphrases. Markovprocess,fractionalderivative,Cauchyproblem,stablesubordinator,
hitting time, forward equation, boundary condition.
M.M. Meerschaert was partially supported by NSF grants DMS-1025486 and DMS-0803360, and
NIH grant R01-EB012079.
1
p(x,t) = Ex[f(X )] solves the fractional Cauchy problem
Et
(1.1) ∂βp(x,t) = Lp(x,t); p(x,0) = f(x)
t
of order 0 < β < 1, for any f in the domain of the generator L. The Caputo fractional
derivative in (1.1) is defined by
1 ∞
(1.2) ∂βf(t) = f′(t−r)r−βdr.
t Γ(1−β)
Z0
The stochastic process X provides the basis for particle tracking solutions to the
Et
time-fractional Cauchy problem, in which the underlying process is simulated, and the
resulting histogram for many sample paths approximates the solution [21, 38].
The goal of this paper is to provide an alternative method for particle tracking solu-
tions of fractional Cauchy problems, by establishing an equivalent solution method in
terms of a Markovian time change. This solves an open problem in [5, Remark 5.2]. We
will show that a certain Markov process Z has the same one dimensional distributions
t
as the inverse stable subordinator E , and therefore, one can also compute solutions
t
to fractional Cauchy problems by the probabilistic formula p(x,t) = Ex[f(X )]. The
Zt
proof uses a variation of the D. Andr´e reflection principle, which may also be useful in
other contexts. The Markov process Z is a stable L´evy process with index α = 1/β
t
and no positive jumps, reflected at the origin. At the end of this paper, we establish the
forward (Fokker-Planck) equation for this Markov process, which involves a fractional
boundary condition, and we explicitly compute the transition densities (see Figures 1
and 2).
Notation. Most of our notation is standard or self-explicatory. Since our paper ad-
dresses both probabilists and analysts, some words on function spaces and convergence
seem to be in order. We write C [0,∞) for the continuous functions which vanish
∞
at infinity, i.e., lim u(x) = 0, and this space is always equipped with uniform
x→∞
norm kuk = sup |u(x)|. Its topological dual is the space of (signed) Radon mea-
x∈[0,∞)
sures M [0,∞), and by M [0,∞) we mean the absolutely continuous (with respect
b ac
to Lebesgue measure) elements in M [0,∞). On M [0,∞) we use vague convergence
b b
which is, topologically, the weak-* convergence; i.e., µ → µ vaguely if, and only if,
n
udµ → udµ for all u ∈ C [0,∞). The subscripts c,b,ac,∞ stand for ‘compact
n ∞
support’, ‘bounded’, ‘absolutely continuous’ and ‘vanishing at infinity’.
R R
Fractional integrals and derivatives in the Riemann-Liouville sense are denoted by
Iα and Dα, cf. (4.2)–(4.5) while Caputo derivatives are written as ∂α, cf. (4.11).
2. Reflection principle
In this section, we note a useful extension of the D. Andr´e reflection principle for
Brownian motion. Since this extension is not completely standard, we include a simple
proof.
Theorem 2.1 (reflection principle). Suppose that Y is a L´evy process started at the
t
origin, with no positive jumps, and let S = sup{Y : 0 6 u 6 t}. Assume that
t u
2
P(Y > 0) = P(Y > 0), for all t > 0. Then
t 1
P(Y > x)
(2.1) P(S > x) = P(Y > x|Y > 0) = t
t t t P(Y > 0)
t
for all t,x > 0.
Proof. Let τ := inf{u > 0 : Y > x} denote the first passage time process. Since
x u
(Y ) has stationary independent increments, it follows that (Y −Y ) is a L´evy
t t>0 t+τx τx t>0
process, which is independent of the σ-algebra generated by (Y ) , and it has the
t t6τx
same finite dimensional distributions as (Y ) . Consequently
t t>0
P(τ 6 t,Y < Y ) = P(τ 6 t,Y −Y < 0)
x t τx x t τx
= P(τ 6 t)P(Y < 0)
x 1
P(Y < 0)
(2.2) = 1 P(τ 6 t)P(Y > 0)
P(Y > 0) x 1
1
P(Y < 0)
= 1 P(τ 6 t,Y > Y ).
P(Y > 0) x t τx
1
Observe that we have for any c`adl`ag process
{τ < t} ⊂ {S > x} ⊂ {τ 6 t} ⊂ {S > x}
x t x t
for all t and x > 0. Therefore,
P(S > x) 6 P(τ 6 t) = P(τ 6 t,Y > Y )+P(τ 6 t,Y < Y )
t x x t τx x t τx
P(Y < 0)
= 1+ 1 P(τ 6 t,Y > Y )
P(Y > 0) x t τx
(cid:18) 1 (cid:19)
1
= P(τ 6 t,Y > Y )
P(Y > 0) x t τx
1
1
= P(τ 6 t,Y > Y )
P(Y > 0) x t τx
t
1
= P(τ 6 t,Y > Y ,Y > 0)
P(Y > 0) x t τx t
t
= P(τ 6 t,Y > Y |Y > 0).
x t τx t
3
On the other hand,
P(S > x) > P(τ < t) = P(τ < t,Y > Y )+P(τ < t,Y < Y )
t x x t τx x t τx
P(Y < 0)
= 1+ 1 P(τ < t,Y > Y )
P(Y > 0) x t τx
(cid:18) 1 (cid:19)
1
= P(τ < t,Y > Y )
P(Y > 0) x t τx
1
1
= P(τ < t,Y > Y )
P(Y > 0) x t τx
t
1
= P(τ < t,Y > Y ,Y > 0)
P(Y > 0) x t τx t
t
= P(τ < t,Y > Y |Y > 0)
x t τx t
> P(τ < t,Y > Y |Y > 0).
x t τx t
Therefore,
P(τ < t,Y > Y |Y > 0) 6 P(S > x) 6 P(τ 6 t,Y > Y |Y > 0).
x t τx t t x t τx t
Since Y has no upward jumps, Y = x and {τ < t}∩{Y > x} = {Y > x}. Therefore,
t τx x t t
P(Y > x|Y > 0) 6 P(S > x) 6 P(Y > x|Y > 0).
t t t t t
We can now use standard approximation techniques:
{X > x} = X > x−1/n = X > x−1/n
n n
\(cid:8) (cid:9) \(cid:8) (cid:9)
and
{X > x} = {X > x+1/n}
n
[
to get
P(Y > x|Y > 0) = lim P(Y > x−1/n|Y > 0)
t t t t
n→∞
6 lim P(S > x−1/n) = P(S > x)
t t
n→∞
and
P(S > x) = lim P(S > x−1/n) 6 lim P(Y > x−1/n|Y > 0)
t t t t
n→∞ n→∞
= P(Y > x|Y > 0)
t t
which proves P(S > x) = P(Y > x|Y > 0). (cid:3)
t t t
Remark 2.2. If (Y ) is a Brownian motion, then (2.1) becomes the classical reflection
t t>0
principle: P(Y > 0) = 1/2, so that (2.1) is equivalent to P(S > x) = 2P(Y > x).
t t t
The proof of Theorem 2.1 relies essentially on local symmetry and the strong Markov
property. Let Y be a strong Markov process with c`adl`ag paths and transition function
t
p (z,dy) = Pz(Y ∈ dy), and write τz = inf{u > 0 : Y −z > x} for the first passage
t t x u
time above the level x + z for the process Y started at z; observe that, in general,
t
4
Y > x+ z. We can use the strong Markov property in (2.2) to get for any starting
τz
x
point z
Pz τz 6 t,Y < Y = Pz τz 6 t,Y −Y < 0
x t τxz x t τxz
(cid:0) (cid:1) = (cid:0) PYτxz(ω) Yt−τz(ω) −(cid:1) Y0 < 0 Pz(dω).
x
Z{τxz6t}
(cid:0) (cid:1)
If we assume, in addition, some local ‘symmetry’, i.e., that for some constant c ∈ (0,∞)
we have
Pz(Y −z < 0)
(2.3) t = c for all t > 0, z ∈ R,
Pz(Y −z > 0)
t
then we get PYτxz(ω) Yt−τz(ω)−Y0 < 0 = cPYτxz(ω) Yt−τz(ω)−Y0 > 0 and, with a similar
x x
argument,
(cid:0) (cid:1) (cid:0) (cid:1)
Pz τz 6 t,Y < Y = cPz τz 6 t,Y > Y .
x t τxz x t τxz
This means that we ca(cid:0)n follow the lin(cid:1)es of th(cid:0)e proof of Theor(cid:1)em 2.1 to derive the
following general result.
Theorem 2.3 (Markov reflection principle). Suppose (Y ,Pz) is a strong Markov pro-
t
cess satisfying the local symmetry condition (2.3). Set S = sup{Y −Y : 0 6 u 6 t}.
t u 0
Then we have for all t,x > 0 and z ∈ R
Pz(S > x) 6 Pz(S > x,Y > Y |Y > z)
t t t τz t
(2.4) x
6 Pz(Y −z > x|Y > z).
t t
If Y has only non-positive jumps, then Y = x+z a.s., and we get for all t,x > 0 and
t τz
z ∈ R x
(2.5) Pz(S > x) = Pz(Y −z > x|Y > z).
t t t
3. An equivalent Markov time change
Suppose that D is a standard stable subordinator, a L´evy process with E[e−sDt] =
t
e−tsβ for all t > 0, and define its inverse (hitting time, first passage time) process by
(3.1) E = inf{r > 0 : D > t}
t r
for all t > 0. In this section, we prove that a certain Markov process Z has the
t
same one dimensional distributions as the inverse stable subordinator (3.1) with index
β ∈ [1/2,1). Then we use the Markov process Z as a time change, to develop efficient
t
probabilistic solutions to fractional Cauchy problems.
To construct this Markov process, first consider a stable L´evy process Y with char-
t
acteristic function
(3.2) E[eikYt] = et(ik)α
where α = 1/β. If β = 1/2, then α = 2, and Y is a Brownian motion with variance 2t.
t
Now define
(3.3) Z = Y −inf{Y : 0 6 s 6 t}.
t t s
5
The reflected stable process (3.3) is also the recurrent extension of the process Y killed
t
at zero, which instantaneously and continuously leaves zero, see Patie and Simon [29].
Let C (R) denote the Banach space of continuous functions f : R → R that tend to
∞
zero as |x| → ∞, with the supremum norm. We say that a time-homogeneous Markov
process X is a Feller process if the semigroup T f(x) = E[f(X )|X = x] satisfies
t t t+s s
T f ∈ C (R) and T f → f as t → 0 in the Banach space (supremum) norm, for all
t ∞ t
f ∈ C (R).
∞
Lemma 3.1. Let E be the inverse stable subordinator (3.1), where D is a standard
t t
β-stable subordinator with E[e−sDt] = e−tsβ and β ∈ [1/2,1). Define Z by (3.3), where
t
Y is a stable L´evy process with characteristic function (3.2), with index α = 1/β. Then
t
Z is a Feller process, with the same (one-dimensional) marginal distributions (given
t
Z = 0) as E , for any fixed t > 0.
0 t
Proof. Define the running infimum I = inf{Y : 0 6 s 6 t} and the running supremum
t s
S = sup{Y : 0 6 s 6 t}. Let Yˆ = −Y denote the dual process, and let Iˆ and
t s t t t
Sˆ denote the running infimum and supremum of Yˆ, respectively. Since Yˆ is also a
t t t
L´evy process, it follows from [10, Section VI.1, Proposition 1] that Sˆ − Yˆ is a Feller
t t
ˆ ˆ ˆ
process, and since S = −I , it follows that S − Y = −I + Y = Z . Hence Z is a
t t t t t t t t
Feller process. Since the standard stable subordinator D has characteristic function
t
E[eikDt] = exp(−t(−ik)γ), an application of the Zolotarev duality for stable densities
[5, Theorem 4.1] implies that
(3.4) P(E > x) = P(Y > x|Y > 0) for all t > 0 and all x > 0.
t t t
The stable L´evy process Y has no positive jumps, and since Y is self-similar (e.g., see
t t
[26, p. 105]), it follows that P(Y > 0) = P(t1/αY > 0) = P(Y > 0) for all t > 0.
t 1 1
Since every stable process has a Lebesgue density (e.g., see [26, p. 107]), we also have
P(Y = 0) = 0 for all t > 0. Then it follows from Theorem 2.1 that
t
(3.5) P(Y > x|Y > 0) = P(S > x) for all t > 0 and all x > 0.
t t t
Finally, a fluctuation identity [10, Section VI.1, Prop. 3] implies that
(3.6) P(S > x) = P(Z > x) for all t > 0 and all x > 0.
t t
Then the theorem follows by combining (3.4), (3.5), and (3.6). (cid:3)
Remark 3.2. It follows from the Zolotarev duality formula for stable laws [20, 39] that
in fact P(Y > 0) = 1/α, see [5, Theorem 4.1].
t
The next result shows that fractional Cauchy problems can be solved by a proba-
bilistic method, using a Markovian time change.
Theorem 3.3. For any β ∈ [1/2,1), let Z be given by (3.3), where Y is an α-stable
t t
L´evy process with characteristic function (3.2) for α = 1/β. If X is an independent
t
Markov process such that T f(x) = Ex[f(X )] forms a uniformly bounded, strongly
t t
continuous semigroup on some Banach space B of real valued functions, with generator
L, then p(x,t) = Ex[f(X )] solves the fractional Cauchy problem (1.1) for any f ∈
Zt
D(L), the domain of the generator.
6
Proof. A general result [3, Theorem 3.1] shows that, if L is the generator of a uniformly
bounded, strongly continuous semigroup on some Banach space B, and if u(·,t) ∈ B
solves the Cauchy problem ∂ u(x,t) = Lu(x,t); u(x,0) = f(x) on that space for some
t
f ∈ Dom(L), then the solution to the associated fractional Cauchy problem (1.1) on
that Banach space is given by
∞
(3.7) p(x,t) = u(x,r)h(r,t)dr
Z0
where
t
(3.8) h(r,t) = r−1−1/βg (tr−1/β)
β
β
and g (t) is the function with Laplace transform
β
∞
Lg(s) = e−stg (t)dt = e−sβ
β
Z0
for some 0 < β < 1. Of course g (t) is the probability density function of a standard
β
β-stable subordinator. Let D be the associated L´evy process with E[e−sDt] = e−tsβ
t
and apply [22, Corollary 3.1] to see that (3.8) is also the probability density of the hit-
ting time (3.1). Since T f(x) = Ex[f(X )] is a uniformly bounded, strongly continuous
t t
semigroup, it follows that p(x,t) = Ex[f(X )] solves the fractional Cauchy problem
Et
(1.1). Lemma 3.1 implies that the random variables E and Z have the same distribu-
t t
tion for any t > 0, and we also have Z = E = 0 almost surely. Then it follows that
0 0
Ex[f(X )] = Ex[f(X )], which completes the proof. (cid:3)
Zt Et
Remark 3.4. Theorem 3.3 confirms a conjecture in the paper [5, Remark 5.2]. There
we set Z = Y where σ(t) = inf{u > 0 : H > t} and H = u1 ds. In essence,
t σ(t) u u 0 Ys>0
the negative excursions are cut away, and the positive excursions are joined together
R
without any gaps in time. Since Y has no positive jumps, any up-crossing at the origin
t
is a renewal point, so this process has the same distribution as (3.3).
Remark 3.5. Since the stable L´evy process Y with index α ∈ (1,2] and characteristic
t
function (3.2) has no positive jumps, the first passage time D := inf{r > 0;Y > t}
t r
is a stable subordinator with E[e−sDt] = exp(−ts1/α), and the supremum process E =
t
sup{Y : 0 6 r 6 t} is the inverse stable process (3.1) of D , see Bingham [12]. This
r t
allows us to compare the processes Z and E in terms of their sample paths. It can
t t
also be used to give an alternative proof of Lemma 3.1: Since S = E , we certainly
t t
have P(E > x) = P(S > x), and then the result follows by the fluctuation identity
t t
(3.6). For the case α = 2, Z is the reflected Brownian motion, and E is the supremum
t t
of that same Brownian motion.
4. The reflected stable process
The equivalence of one dimensional distributions established in Lemma 3.1 also has
some interesting implications regarding the reflected stable process Z defined in (3.3).
t
7
Proposition 4.1. Let Y be a stable L´evy process with characteristic function (3.2)
t
and index α ∈ (1,2). Then the reflected process Z in (3.3) has the right-continuous
t
probability density
tβ−1x−1−1/βg (tx−1/β) x > 0
β
(4.1) p(x,t) = t−β/Γ(1−β) x = 0

0 x < 0
for any t > 0, where β = 1/α and gβ(t) is the probability density function with Laplace
transform Lg(s) = exp(−s1/α).
Proof. Lemma 3.1 shows that the probability distribution of Z is also the probability
t
distribution of the random variable E , the hitting time (3.1) of a stable subordinator
t
D such that E[e−sDt] = exp(−tsβ), and β = 1/α. Then [22, Corollary 3.1] implies
t
that E has a Lebesgue density h(x,t) = tβ−1x−1−1/βg (tx−1/β) for x > 0. Since D
t β t
is right-continuous, it follows from (3.1) that E > 0 for t > 0, so that p(x,t) = 0 for
t
x < 0 and t > 0. Finally, since g (x) ∼ βx−β−1/Γ(1−β) as x → ∞, it is easy to check
β
that p(x,t) → t−β/Γ(1−β) as x → 0+ for any t > 0. (cid:3)
The next result shows thatthe density (4.1) of Z startedat Z = 0 solves a fractional
t 0
boundary value problem. Given a real number α > 0 which is not an integer, define
the positive Riemann-Liouville fractional integral
1 ∞ 1 x
(4.2) Iαf(x) = f(x−y)yα−1dy = f(y)(x−y)α−1dy,
x Γ(α) Γ(α)
Z0 Z−∞
the negative Riemann-Liouville fractional integral
1 ∞ 1 ∞
(4.3) Iα f(x) = f(x+y)yα−1dy = f(y)(y−x)α−1dy,
−x Γ(α) Γ(α)
Z0 Zx
the positive Riemann-Liouville fractional derivative
dn 1 dn x
(4.4) Dαf(x) := In−αf(x) = f(y)(x−y)n−α−1dy,
x dxn x Γ(n−α)dxn
Z−∞
and the negative Riemann-Liouville fractional derivative
dn (−1)n dn ∞
(4.5) Dα f(x) := In−αf(x) = f(y)(y−x)n−α−1dy,
−x d(−x)n −x Γ(n−α)dxn
Zx
where n−1 < α < n. If α ∈ (1,2), then n = 2.
Theorem 4.2. Let Y be a stable L´evy process with characteristic function (3.2) and
t
index 1 < α < 2, and let Z be the reflected process (3.3). Then the probability density
t
(4.1) of Z solves the fractional boundary value problem:
t
∂ p(x,t) = Dα p(x,t) for t > 0 and x > 0;
t −x
(4.6)
0 = Dα−1p(x,t) for t > 0 and x = 0.
−x
8
Proof. Let q(x,t) denote the probability density of the stable L´evy process Y . An
t
application of the Zolotarev duality formula for stable densities [5, Theorem 4.1] to-
gether with Lemma 3.1 shows that βp(x,t) = q(x,t) for all x > 0 and t > 0, where
β = P(Y > 0) for any t > 0. It follows from [26, Example 3.29] that the stable density
t
q(x,t) solves
(4.7) ∂ q(x,t) = Dα q(x,t)
t −x
for all t > 0 and all x ∈ R. Hence we also have ∂ p(x,t) = Dα p(x,t) for t > 0 and
t −x
x > 0. When x = 0, it follows from Proposition 4.1 and the continuity of x 7→ q(x,t)
that βp(0,t) = q(0,t) for t > 0. Then ∂ p(x,t) = Dα p(x,t) at x = 0 as well, since
t −x
Dα p(x,t) at x = 0 depends only on the values of p(x,t) for x > 0.
−x
It remains to verify the fractional boundary condition. From the general definition
(4.5) it follows that for 1 < α < 2 we have
1 d2 ∞
Dα f(x) = f(y)(y−x)1−αdy,
−x Γ(2−α)dx2
(4.8) Zx
−1 d ∞
Dα−1f(x) = f(y)(y−x)1−αdy.
−x Γ(2−α)dx
Zx
Since 1 = P(Z > 0) is constant for all t > 0, it follows that
t
∞
0 = ∂ P(Z > 0) = ∂ p(x,t)dx
t t t
Z0
∞
= Dα p(x,t)dx
−x
Z0
(4.9) ∞ d 1 d ∞
= p(y,t)(y−x)1−αdy dx
dx Γ(2−α)dx
Z0 (cid:18) Zx (cid:19)
∞ d
= − Dα−1p(x,t)dx
dx −x
Z0
= Dα−1p(0,t),
−x
since lim Dα−1p(x,t) = αlim Dα−1q(x,t) = 0, by an elementary estimate using
x→∞ −x x→∞ −x
the fact that q(x,t) ∼ Cx−α−1 for some C > 0 as x → ∞. (cid:3)
Let Z be the stochastic process defined in (3.3), where Y is a stable L´evy process
t t
with index 1 < α < 2 and characteristic function (3.2). Since Z > 0 by definition, it
t
follows from Lemma 3.1 that Z is a conservative time-homogeneous Markov process
t
whose (backward) semigroup T f(x) = E[f(Z )|Z = x] is strongly continuous and
t t+s s
contractive (i.e., kT fk 6 kfk) on the Banach space X = C [0,∞). Let A denote the
t ∞
generator of this semigroup, with domain D(A). Next we will show that this semigroup
is also analytic and give a core for the generator. Recall that a core C of a closed linear
A
operator A is a subset of its domain that is dense within the domain if the domain is
equipped with the graph norm; i.e., C is a core of A if for each f ∈ D(A) there exists
A
a sequence {f } ⊂ C such that f → f and Af → Af.
n A n n
9
Write
S := f ∈ C [0,∞) : f′′ ∈ C(0,∞), f′′(x) = O(1) as x → ∞,
b ∞
(4.10)
n
f′′(x) = O(xα−2) as x → 0, f′ ∈ C (0,∞), f′(0+) = 0
b
and denote by ∂α the (positive) Caputo fractional derivative of order α > 0, whicoh can
x
be defined by
1 x
(4.11) ∂αf(x) = In−αf(n)(x) = (x−y)n−1−αf(n)(y)dy
x x Γ(n−α)
Z0
wheren−1 < α < nandf(n) isthenthderivativeoff. TheCaputofractionalderivative
differs from the Riemann-Liouville form (4.4) because the operations of differentiation
and (fractional) integration do not commute in general. For example, when 0 < α < 1
we have ∂αf(x) = Dαf(x)−f(0)x−α/Γ(1−α) for suitably nice bounded functions (e.g.,
x x
see [26, p. 39]).
Theorem 4.3. Let Z denote the Feller process (3.3) where Y is a stable L´evy pro-
t t
cess with index α = 1/β ∈ (1,2) and characteristic function (3.2), and let T f(x) =
t
E[f(Z )|Z = x] be the associated transition semigroup on C [0,∞) with generator
t+s s ∞
A. Then {T } is analytic, C = {f ∈ S : ∂αf ∈ C [0,∞)} is a core of A and
t t>0 A b x ∞
Af = ∂αf for all f ∈ C , where ∂α is the Caputo fractional derivative (4.11).
x A x
Proof. It follows from [9, Proposition 4] that C ⊂ D(A) and Af = ∂αf for all f ∈ C .
A x A
Note that the extension from f′′ bounded to |f′′(x)| = O(xα−2) at x = 0+ is also
mentioned in the proof.
Since A generates a strongly continuous contraction semigroup, the resolvent opera-
tors R(λ,A) := (λ−A)−1 exist for all λ with Reλ > 0 and they are bounded operators.
Let S be the span of {x 7→ exp(−cx) : c > 0}, which is dense in C [0,∞) by a
e ∞
standard Stone-Weierstraß argument. If we can show that R(λ,A)S ⊂ C for some
e A
λ > 0, then C is a core of A.
A
Indeed, pick f ∈ D(A) and g = λf −Af. Since S is dense in C [0,∞), there exists
e ∞
a sequence {g } ⊂ S with g → g. Thus, f = R(λ,A)g → f and Af = λf −g →
n e n n n n n n
λf −g = Af. Since f ∈ R(λ,A)S ⊂ C , we see that C is a core.
n e A A
Itremainstoshowthatf = R(λ,A)gisanelementofC foranyg ∈ S . TheLaplace
λ A e
transform of a bounded and measurable function g is the unique analytic function
defined by Lg(s) = ∞e−sxg(x)dx for Res > 0; in particular, if g(x) = e−cx for some
0
c > 0, its Laplace transform is given by Lg(s) = 1/(s+c) for any s with Res > 0.
R
We will first define the function f via its Laplace transform, and later we will show
λ
that in fact f = R(λ,A)g. Let
λ
1 sα−1λ1/α−1
−
s+c λ1/α +c
(4.12) f (s) := .
λ
λ−sα
Next we show that fλ is the Lbaplace transform of a function fλ ∈ Sb. Note that fλ can
be extended to an analytic function on C \(−∞,0], since the singularity at s = λ1/α
is removable. Furthebrmore, sf (s) is bounded on the complement of any neighbobrhood
λ
10
b