TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 368, Number 1, January 2016, Pages 227–248
http://dx.doi.org/10.1090/tran/6360
Article electronically published on April 20, 2015
REFLECTED SPECTRALLY NEGATIVE STABLE PROCESSES
AND THEIR GOVERNING EQUATIONS
BORIS BAEUMER, MIHÁLY KOVÁCS, MARK M. MEERSCHAERT, RENÉ L. SCHILLING,
AND PETER STRAKA
Abstract. This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its
infimum. First we derive the forward equation using the theory of sun-dual
semigroups. The resulting forward equation is a boundary value problem on
the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fractional reflecting boundary condition at the origin.
Then we 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.
1. Introduction
Consider a spectrally negative (no positive jumps) stable Lévy process Yt with
characteristic function
(1.1)
E[eikYt ] = et(ik)
α
for some 1 < α 2. If α = 2, then Yt is a Brownian motion with variance 2t. Now
define
(1.2)
Zt = Yt − inf{Ys : 0 s t}.
The reflected stable process (1.2) is also the recurrent extension of the process Yt
killed at zero, which instantaneously and continuously leaves zero; see Patie and
Simon [42]. 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 timehomogeneous Markov process Xt is a Feller process if the semigroup Tt f (x) =
E[f (Xt+s )|Xs = x] satisfies Tt f ∈ C∞ (R) and Tt f → f as t → 0 in the Banach
space (supremum) norm, for all f ∈ C∞ (R). It is not hard to show (see Theorem
2.1 in this paper) that Zt is a Feller process, and since Zt 0 by definition, the
space-inhomogeneous Markov process Zt lives on the state space [0, ∞).
If α = 2, then this process is called “reflected Brownian motion” and the governing differential equation (forward Kolmogorov equation) for the transition density
Received by the editors February 15, 2013 and, in revised form, August 6, 2013 and October 29,
2013.
2010 Mathematics Subject Classification. Primary 60G52, 60J50; Secondary 26A33, 60J22.
Key words and phrases. Stable process, reflecting boundary condition, Markov process, fractional derivative, Cauchy problem.
The third author was partially supported by NSF grants DMS-1025486 and DMS-0803360,
and NIH grant R01-EB012079.
c
2015
American Mathematical Society
227
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228
BORIS BAEUMER ET AL.
p(x, y, t) of Zt+s = y given Zs = x is the diffusion equation ∂t p(x, y, t) = ∂y2 p(x, y, t)
together with the reflecting boundary condition
p(x, y + h, t) − p(x, y, t) := lim
= 0 for all t > 0,
(1.3) ∂y p(x, y, t)
h→0+
h
y=0+
y=0
(i.e., the normal derivative vanishes); see for example Itô and McKean [25, Eq. 8].
This paper extends to the case of a reflected stable process. The stable process Yt
without reflection is a space-homogeneous Markov process, so its transition density
p(y, t) is independent of the initial state x. This density solves a fractional difα
p(y, t) that involves a negative Riemann-Liouville
fusion equation, ∂t p(y, t) = D−y
fractional derivative in space; see (2.4) below for the definition. The fractional
derivative reduces to the usual second derivative in the case α = 2. The forward
equation for the reflected stable process turns out to be the fractional diffusion
α
p(x, y, t) with the fractional reflecting boundary condiequation ∂t p(x, y, t) = D−y
tion
(1.4)
∞
1 α−1
α−1
:= lim α−1
wk p(x, y + kh, t)
= 0 for all t > 0,
D−y p(x, y, t)
h→0+ h
y=0+
y=0
k=0
using the (fractional) binomial coefficients
wkα := (−1)k
α
.
k
When α = 2 we have w0α−1 = 1, w1α−1 = −1, and wkα−1 = 0 for k > 1, so that (1.4)
reduces to the classical condition (1.3), i.e., the one-sided first derivative. In either
case (α = 2 or 1 < α < 2), the boundary term enforces a no-flux condition at the
point y = 0 in the state space.
The connection between probability and differential equations has profound consequences for mathematics [2, 8, 9, 36], and for its applications in science and engineering [21, 39, 40, 47], including a probabilistic method called particle tracking
for solving fractional differential equations, by exploiting the associated Markov
process [10, 54, 55]. More details on fractional calculus, and its connection to probability theory, may be found in the recent book of Meerschaert and Sikorskii [37].
Since fractional derivatives are nonlocal operators, the appropriate specification of
boundary conditions requires a new approach [1, 16, 27, 29, 41, 50]. For example,
one can apply the theory of Volterra integral equations [44] or general nonlocal
operators [17]. The results in this paper can help clarify the meaning of reflecting
boundary conditions for fractional diffusion.
We believe that this idea will find many useful applications, both inside and
outside mathematics. As a first application, we show in Theorem 4.1 that a reflected
stable process can be used to solve a fractional Cauchy problem, in which the usual
first time derivative is replaced by a Caputo fractional derivative of order 0 < β < 1.
Notation. We write C∞ [0, ∞) for the Banach space of continuous functions that
vanish at infinity, i.e., limx→∞ u(x) = 0, with the uniform norm
u =
sup |u(x)|.
x∈[0,∞)
Its topological dual is the space of (signed) Radon measures Mb [0, ∞), and by
Mac [0, ∞) we mean the absolutely continuous (with respect to Lebesgue measure)
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REFLECTED SPECTRALLY NEGATIVE STABLE PROCESSES
229
elements in Mb [0,
∞). On Mb [0, ∞) we use vague (weak-∗) convergence; i.e., μn → μ
if, and only if, u dμn → u dμ for all u ∈ C∞ [0, ∞). The subscripts c, b, ac, ∞
stand for ‘compact support’, ‘bounded’, ‘absolutely continuous’ and ‘vanishing at
infinity’. Fractional integrals and derivatives in the Riemann-Liouville sense are
denoted by I α and Dα (see (2.1)–(2.4)) while Caputo derivatives are written as ∂ α ;
see (2.6).
∞
Finally, Lg(s) = L[g(t)](s) = 0 e−st g(t) dt denotes the usual Laplace trans∞
form, and L−∞ g(s) = L−∞ [g(t)](s) = −∞ e−st g(t) dt is the bilateral Laplace
transform.
2. The reflected stable process
Given a real number α > 0 that is not an integer, define the positive RiemannLiouville fractional integral
∞
x
1
1
(2.1)
Ixα f (x) =
f (x − y)y α−1 dy =
f (y)(x − y)α−1 dy,
Γ(α) 0
Γ(α) −∞
the negative Riemann-Liouville fractional integral
∞
∞
1
1
α
α−1
(2.2)
I−x f (x) =
f (x + y)y
dy =
f (y)(y − x)α−1 dy,
Γ(α) 0
Γ(α) x
the positive Riemann-Liouville fractional derivative
(2.3)
Dxα f (x) :=
dn
dn n−α
1
Ix f (x) =
n
dx
Γ(n − α) dxn
x
−∞
f (y)(x − y)n−α−1 dy,
and the negative Riemann-Liouville fractional derivative
∞
dn
(−1)n dn
n−α
α
f (x) :=
I
f
(x)
=
f (y)(y − x)n−α−1 dy,
(2.4) D−x
d(−x)n −x
Γ(n − α) dxn x
where n − 1 < α < n. If α ∈ (1, 2), then n = 2. See [7, 37, 45] for more details.
Let Zt be the stochastic process defined in (1.2), where Yt is a stable Lévy process
with index 1 < α < 2 and characteristic function (1.1). Next we will show that Zt
is a conservative time-homogeneous Markov process whose (backward) semigroup
Tt f (x) = E[f (Zt+s )|Zs = x] is strongly continuous (Tt f − f → 0 as t ↓ 0),
contractive (Tt f f ), and analytic (the mapping t → T (t)f has an analytic
extension to the sectorial region {reiθ ∈ C : r > 0, |θ| < α} for some α > 0) on the
Banach space X = C∞ [0, ∞), and give a core for the generator. Recall that a core
CA of a closed linear operator A is a subset of its domain D(A) that is dense within
the domain in the graph norm; i.e., for each f ∈ D(A) there exists a sequence
{fn } ⊂ CA such that fn → f and Afn → Af .
Write
Sb := f ∈ C∞ [0, ∞) : f ∈ C(0, ∞), f (x) = O(1) as x → ∞,
(2.5)
f (x) = O(xα−2 ) as x → 0, f ∈ Cb (0, ∞), f (0+) = 0
and denote by ∂xα the (positive) Caputo fractional derivative of order α > 0, which
can be defined by
x
1
(2.6)
∂xα f (x) = Ixn−α f (n) (x) =
(x − y)n−1−α f (n) (y) dy
Γ(n − α) 0
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230
BORIS BAEUMER ET AL.
where n − 1 < α < n and f (n) is the nth derivative of f . The Caputo fractional
derivative differs from the Riemann-Liouville form (2.3) because the operations of
differentiation and (fractional) integration do not commute in general. For example,
when 0 < α < 1 we have
x−α
(2.7)
∂xα f (x) = Dxα f (x) − f (0)
Γ(1 − α)
for suitably nice bounded functions (e.g., see [37, p. 39]).
Theorem 2.1. Let Zt denote the reflected process (1.2) where Yt is a stable Lévy
process with index α = 1/β ∈ (1, 2) and characteristic function (1.1). Then Zt is a
Feller process, the transition semigroup Tt f (x) = E[f (Zt+s )|Zs = x] on C∞ [0, ∞)
is analytic, with generator Af = ∂xα f for all f ∈ CA = {f ∈ Sb : ∂xα f ∈ C∞ [0, ∞)},
where ∂xα is the Caputo fractional derivative (2.6), and CA is a core of A.
Proof. Define the running infimum It = inf{Ys : 0 s t} and the running
supremum St = sup{Ys : 0 s t}. Let Ŷt = −Yt denote the dual process, and let
Iˆt and Ŝt denote the running infimum and supremum of Ŷt , respectively. Since Ŷt
is also a Lévy process, it follows from [12, Section VI.1, Proposition 1] that Ŝt − Ŷt
is a Feller process, and since Ŝt = −It , it follows that Ŝt − Ŷt = −It + Yt = Zt .
Hence Zt is a Feller process.
It follows from [11, Proposition 4] that CA ⊂ D(A) and Af = ∂xα f for all f ∈ CA .
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 operators R(λ, A) := (λ − A)−1 exist for all Re λ > 0 and they are bounded operators.
We will now show that
Lg(λ1/α )
(2.8)
R(λ, A)g(x) = −αxα−1 Eα (λxα ) g(x) + 1−1/α Eα (λxα )
λ
is the convolution
for all g ∈ C∞ [0, ∞), where Lg is the Laplace transform,
∞
operator, and the Mittag-Leffler function Eα (x) = n=0 xn /Γ(1 + αn).
Let Rλ,g denote the right-hand side of (2.8). Since
sα−1
sα − λ
(see e.g. [22, 32]), it follows that
L[Eα (λxα )](s) =
and
L[αxα−1 Eα (λxα )](s) =
1
sα − λ
Lg(s)
sα−1 λ1/α Lg(λ1/α )
−
.
λ − sα
λ − sα
λ
Using the fact that [f g] (x) = [f g](x) + f (0)g(x) it follows that Rλ,g is twice
differentiable for any g ∈ C∞ [0, ∞) ∩ C 2 [0, ∞). Equation (2.6) implies that for any
f ∈ C 2 [0, ∞) we have
L[∂xα f ](s) = sα−2 s2 Lf (s) − sf (0) − f (0) = sα Lf (s) − sα−1 f (0).
(2.9)
LRλ,g (s) =
Taking the Laplace transform of λRλ,g − ∂xα Rλ,g , we therefore obtain that
L λRλ,g − ∂xα Rλ,g (s) = λLRλ,g (s) − sα LRλ,g (s) + sα−1 Rλ,g (0)
(2.10)
= Lg(s) −
= Lg(s)
sα−1
Lg(λ1/α ) + sα−1 Rλ,g (0)
λ1−1/α
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REFLECTED SPECTRALLY NEGATIVE STABLE PROCESSES
231
for any g ∈ C∞ [0, ∞) ∩ C 2 [0, ∞), and hence R(λ, A)g = Rλ,g . By continuous
extension, (2.8) holds for all g ∈ C∞ [0, ∞).
2
To show that CA is a core, note that for g ∈ C∞
[0, ∞) := C 2 [0, ∞) ∩ C∞ [0, ∞),
∂xα R(λ, A)g = λR(λ, A)g − g ∈ C∞ [0, ∞)
2
[0, ∞) ⊂ CA . Pick f ∈ D(A) and g = λf − Af . Since
and hence R(λ, A)C∞
2
2
C∞ [0, ∞) is dense in C∞ [0, ∞), there exists a sequence {gn } ⊂ C∞
[0, ∞) with
gn → g. Thus, fn = R(λ, A)gn → f and Afn = λfn − gn → λf − g = Af . Since
2
[0, ∞) ⊂ CA , we see that CA is a core.
fn ∈ R(λ, A)C∞
Finally we show that {Tt }t0 is an analytic semigroup. Since R(λ, A) is a
bounded operator for all Re λ > 0, a general result from the theory of semigroups
[3, Corollary 3.7.12] shows that {Tt } is analytic if for some some M > 0 we have
(2.11)
λR(λ, A)g M g
for all Re λ > 0 and all g ∈ C∞ [0, ∞). Then the result follows from Lemma 5.2,
and this completes the proof.
Remark 2.2. Patie and Simon [42] show that the reflected stable process Zt in
Theorem 2.1 has the backward generator
x
x1−α
y 1−α
+
dy.
f (x − y)
(2.12)
Af (x) = f (0)
Γ(2 − α)
Γ(2 − α)
0
They also give the exact domain of the generator [42, Proposition 2.2]. If f ∈ Sb ,
then f (0) = 0, and Af reduces to the Caputo fractional derivative (2.6).
In view of Theorem 2.1, Tt f (x) = E[f (Zt+s )|Zs = x] is a strongly continuous,
analytic semigroup on the Banach space X := C∞ [0, ∞) with the supremum norm,
with generator Af (x) = ∂xα f (x) for f ∈ Sb such that ∂xα f ∈ X. The dual (or
adjoint) semigroup Tt∗ is defined on the dual space X ∗ = Mb [0, ∞) of finite signed
Radon measures on [0, ∞) equipped with the total variation norm: Given a measure
μ ∈ Mb [0, ∞), use the Jordan decomposition to write μ = μ+ − μ− uniquely as a
difference of two positive measures, and define μ = μ+ [0, ∞) + μ− [0, ∞). The
dual semigroup satisfies
(2.13)
Tt f (x)μ(dx) = f (x)[Tt∗ μ](dx)
for all f ∈ C∞ [0, ∞) and all μ ∈ Mb [0, ∞). See [19, Section 2.5] for more details. In
probabilistic terms, since Tt f (x) = E[f (Zt+s )|Zs = x] for this time-homogeneous
Markov process, equation (2.13) implies that
Tt f (x)μ(dx) = E[f (Zt+s )|Zs = x]μ(dx)
= f (y)Pt (dy, μ) = f (y)[Tt∗ μ](dy)
where Pt (y, μ) = P (y, x, t)μ(dx) and P (y, x, t) = P[Zt+s y|Zs = x] is the
transition probability distribution of the Markov process Zt . Hence, if μ is the
probability distribution of Zs , then Tt∗ μ(dy) = Pt (dy, μ) is the probability distribution of Zt+s . The dual semigroup is also called the forward semigroup associated
with the Markov process Z, since it maps the probability distribution forward in
time.
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232
BORIS BAEUMER ET AL.
Next we will compute the generator A∗ of the forward semigroup. This is the
adjoint of the generator A of the backward semigroup, in the sense that
Af (x)μ(dx) = f (x)[A∗ μ](dx)
for all f ∈ D(A) and μ ∈ D(A∗ ). Theorem 2.3 will show that every measure
μ ∈ D(A∗ ) has a Lebesgue density g ∈ L1 [0, ∞), so that μ(dy) = g(y) dy, and that
the adjoint A∗ g := A∗ μ of the positive fractional Caputo derivative Af (x) = ∂xα f (x)
in our setting is the negative Riemann-Liouville fractional derivative A∗ g(y) =
α
g(y) using (2.4).
D−y
The forward semigroup Tt∗ of a Markov process is not, in general, strongly continuous on Mb [0, ∞). That is, there exist measures μ such that Tt∗ μ → μ in the
total variation norm as t ↓ 0. For example, if Tt∗ is the forward semigroup associated with the diffusion equation ∂t p = ∂x2 p, and μ = δ0 is a point mass at the
origin, then Tt∗ μ is a Gaussian probability measure with mean 0 and variance 2t for
all t > 0, and since μ{0} = 1 and Tt μ{0} = 0 for all t > 0, we have Tt μ − μ = 1
for all t > 0 in the total variation norm.
To handle this situation, we define the sun-dual space of X := C∞ [0, ∞) as
X := {μ ∈ X ∗ : lim Tt∗ μ − μ = 0},
t↓0
a closed subspace of X ∗ = Mb [0, ∞) on which the forward semigroup is strongly
continuous. It follows from basic semigroup theory [19, Section 2.6] that for μ ∈
X , Tt∗ μ ∈ X for all t 0, and X = D(A∗ ). The restriction of {Tt∗ }t0 to
X is called the sun-dual semigroup {Tt }t0 with generator A μ = A∗ μ for all
μ ∈ D(A ), where
(2.14)
D(A ) = {μ ∈ D(A∗ ) : A∗ μ ∈ X }.
For the reflected stable process, we will show in Theorem 2.3 that C∞
[0, ∞) is the
space of absolutely continuous elements of Mb [0, ∞),
Mac [0, ∞) = μ ∈ Mb [0, ∞) : μ(dy) = g(y) dy for some g ∈ L1 [0, ∞) ,
and we will derive the forward equation of the reflected stable process on the sundual space. For a general bounded measure μ ∈ Mb [0, ∞), we will then prove in
Corollary 2.5 that Tt∗ μ can be computed as the vague limit of Tt μn , where μn → μ
[0, ∞) for all n.
vaguely, and μn ∈ C∞
Theorem 2.3. Let Zt denote the Feller process (1.2), where Yt is a stable Lévy process with index α = 1/β ∈ (1, 2) and characteristic function (1.1), with (backward)
[0, ∞) = Mac [0, ∞)
semigroup Tt f (x) = E[f (Zt+s )|Zs = x] on C∞ [0, ∞). Then C∞
and the generator A g := A μ of the sun-dual semigroup {T (t)}t0 is given by
(2.15)
α
A g(y) = D−y
g(y)
α−1
α
with domain D(A ) = {g ∈ L1 [0, ∞) : D−y
g(y) ∈ L1 [0, ∞), D−y
g(0) = 0}.
Proof.
Suppose A∗ μ = ν ∈ Mb [0, ∞) for some μ ∈ Mb [0, ∞), so that Af (x)μ(dx)
= f (x)ν(dx) for all f ∈ D(A). Set v(x) := ν[0, x] for x 0 and v(x) = 0 for x < 0.
If f ∈ Sb with ∂xα f ∈ C∞ [0, ∞), then it follows from Theorem 2.1 that f ∈ D(A)
and Af = ∂xα f . It is obvious from the definition (2.6) that ∂xα−1 f (x) = ∂xα f (x).
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REFLECTED SPECTRALLY NEGATIVE STABLE PROCESSES
Let
233
(x − y)α−1
f (y) dy
Γ(α)
0
denote the positive Riemann-Liouville fractional integral (2.1) of order α > 0 for a
function f ∈ C∞ [0, ∞), and apply the general formula [7, Eq. (1.21)] Ixα−1 ∂xα−1 f (x)
= f (x) − f (0) to see that f (x) = Ixα−1 ∂xα−1 f (x) = Ixα−1 ∂xα f (x) = Ixα−1 Af (x).
Since f (x) → 0 as x → ∞, and v(0) = 0 for x < 0, we can apply the integration by
parts formula [23, Theorem 19.3.13]
b
b
f (x)ν(dx) = f (b)v(b) − f (a)v(a) −
v(x)f (x)dx
Ixα f (x)
x
=
a
a
with a < 0, and then let b → ∞, to see that
∞
f (x)ν(dx) = −
0
∞
f (x)v(x) dx.
0
Thus, for all f ∈ Sb with ∂xα f ∈ C∞ [0, ∞), a Fubini argument yields
∞ x
∞
(x − y)α−2
Af (y) dy v(x) dx
f (x)ν(dx) = −
Γ(α − 1)
0
0
0
∞ ∞
(x − y)α−2
(2.16)
v(x) dx Af (y) dy
=−
Γ(α − 1)
0
y
∞
=
Af (y) μ(dy).
0
Next we will show that S := {Af : f ∈ Sb with ∂xα f ∈ C∞ [0, ∞)} is dense in
C∞ [0, ∞), and then it will follow that any measure μ ∈ D(A∗ ) has a Lebesgue
density
∞
(x − y)α−2
α−1
v(y)
(2.17)
g(y) = −
v(x) dx = −I−y
Γ(α − 1)
y
where A∗ μ = ν and v(x) = ν[0, x]. Let Cc∞ [0, ∞) denote the space of smooth
functions with compact support, i.e., such that h(x) = 0 for all x > M , for some
M > 0. It is not hard to check that the space Q = {h ∈ Cc∞ [0, ∞) : h = 0} is
dense in C∞ [0, ∞). Then we certainly have
M
(x − s)α−1 − xα−1
lim Ixα h(x) = lim
h(s) ds = 0,
x→∞
x→∞ 0
Γ(α)
and therefore, the function f (x) = Ixα h(x) is an element of C∞ [0, ∞) for any h ∈ Q.
Elementary estimates suffice to check that f ∈ Sb as well. Since the positive Caputo
derivative is a left inverse of the positive Riemann-Liouville integral [7, Eq. (1.21)],
we also have ∂xα f (x) = ∂xα Ixα h(x) = h(x) ∈ C∞ [0, ∞). Hence h = Af ∈ S for all
h ∈ Q, and thus Q ⊆ S. Now for any f ∈ C∞ [0, ∞) there exists a sequence fn → f
in the supremum norm,
with fn ∈ S for all n. Then some simple estimates can be
used to verify that f (y)μ(dy) = f (y)g(y) dy, and it follows that the measure
μ has the Lebesgue density g. Hence we can identify D(A∗ ) with a subspace of
L1 [0, ∞).
Next we will show that this subspace is dense in L1 [0, ∞). Define
1
α−1
1
φn (x) := Γ(α)
[0,1/n) (x),
n −x
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234
BORIS BAEUMER ET AL.
α−1
and note that D−x
φn (x) ≡ 1 for x ∈ (0, 1/n) by a straightforward computation.
α−1
Note that the set U := {g(x) − [D−x
g(0)]φn : g ∈ Cc∞ [0, ∞), n ∈ N} is dense in
1
α
1
L [0, ∞), and that D−x g(x) ∈ L [0, ∞) for all g ∈ U . Furthermore, U is a subset
of
α−1
α
g ∈ L1 [0, ∞), D−x
g(0) = 0 .
G = g ∈ L1 [0, ∞) : D−x
α−1
For any g ∈ G and f ∈ Sb with ∂xα f ∈ C∞ [0, ∞), we have D−x
g(0) = 0 and
f (0) = 0. Then Theorem 2.1, a Fubini argument, and integration by parts (twice)
using equation (2.4) yields
∞
∞ y
(y − x)1−α f (x) dx g(y) dy
Af (y) g(y) dy =
Γ(2 − α)
0
0
0
∞ ∞
(y − x)1−α
=
g(y) dy f (x) dx
Γ(2 − α)
0
x
∞
2−α
(2.18)
I−x
g(x)f (x) dx
=
0
∞
α−1
=
D−x
g(x)f (x) dx
0
∞
α
=
f (x) D−x
g(x) dx.
0
Furthermore, as {f ∈ Sb : Af ∈ C∞ [0, ∞)} is a core, for all f ∈ D(A) there exists
a sequence fn in the core such that fn → f and Afn → Af . Hence (2.18) holds
α
for all f ∈ D(A) and therefore g ∈ D(A∗ ) and A∗ g = D−x
g for any g ∈ G. Since
1
∗
U is dense in L [0, ∞), and U ⊆ G ⊆ D(A ), it follows that D(A∗ ) is dense in
[0, ∞) is the smallest closed set containing D(A∗ ) by definition,
L1 [0, ∞). Since C∞
1
and since L [0, ∞) is a closed subspace of Mb [0, ∞), we have shown that L1 [0, ∞) =
[0, ∞).
C∞
[0, ∞) for any μ ∈
Equation (2.14) implies that ν = A∗ μ is an element of C∞
D(A ), and therefore, we have ν(dx) = h(x) dx for some h ∈ L1 [0, ∞), as well
as μ(dy) = g(y) dy for some g ∈ L1 [0, ∞). Since v(x) = ν[0, x], it follows that
α−1
α−1
is a left inverse of I−y
in general, it follows from (2.17)
h(x) = v (x). Since D−y
α−1
that v(y) = −D−y g(y). Then we have
(2.19)
h(x) = v (x) =
d α−1
α
−D−x
g(x) = D−x
g(x) = A∗ g(x)
dx
for all g ∈ D(A ), which proves the generator formula (2.15). Equation (2.19)
α
g(x) ∈ L1 [0, ∞) for all g ∈ D(A ), and since v is continuous
also shows that D−x
α−1
with v(0) = 0, it follows that D−x
g(0) = v(0) = 0 for all g ∈ D(A ). This
proves that D(A ) ⊆ G. Since D(A ) is defined as the set of g ∈ D(A∗ ) such
that A∗ g ∈ L1 [0, ∞), it follows from (2.18) that g ∈ D(A ) for all g ∈ G, so that
G ⊆ D(A ) as well, which completes the proof.
Theorem 2.3 establishes the forward equation of the reflected stable process Zt ,
for certain initial conditions. It shows that, for any initial condition μ0 (dx) =
g(x)dx where g ∈ L1 [0, ∞), μt := T μ solves the Cauchy problem
∂t μ(t) = A μ(t);
μ(0) = μ0
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REFLECTED SPECTRALLY NEGATIVE STABLE PROCESSES
235
on the sun-dual space. This implies that, for any probability density p0 (x) such
that p0 (x) = 0 for x < 0, the function p(x, t) = T p0 (x) solves the forward equation
α−1
α
p(x, t); p(x, 0) = p0 (x), D−x
p(x, t)
≡ 0.
(2.20)
∂t p(x, t) = D−x
x=0
The next two results will allow us to compute the transition probability density
y → p(x, y, t) of the time-homogeneous Markov process y = Zt+s for any initial
state x = Zs , by applying Theorem 2.3 to a sequence of initial conditions μn ∈ X such that μn → δx . The first result shows that the transition density exists.
Corollary 2.4. The semigroup Tt is a strongly continuous bounded analytic semigroup on L1 (R+ ) and the transition probability distributions Tt∗ δx have smooth densities y → p(x, y, t) for all t > 0 and all x 0.
Proof. It is well known that the spectra of A and A∗ coincide, and R(λ, A∗ ) =
R(λ, A)∗ for all λ in the resolvent set of A (and A∗ ). Therefore, since A is a
sectorial operator, being the generator of a bounded analytic semigroup, it follows
that A∗ is a sectorial operator as well, and hence A∗ generates a bounded analytic
∗
[0, ∞) = Mb [0, ∞) by
semigroup (not necessarily strongly continuous at 0) on C∞
∗
[3, Theorem 3.7.1] which coincides with Tt for all t > 0 by the uniqueness of the
Laplace transform. Therefore, the restriction of A∗ to D(A∗ ) = L1 [0, ∞) (i.e., the
operator A ) generates a strongly continuous bounded analytic semigroup Tt on
L1 [0, ∞) by [3, Remark 3.7.13]. Furthermore, [3, Remark 3.7.20] shows that for
all n ∈ N and t > 0 we have that Tt∗ μ ∈ D((A∗ )n ) for all μ ∈ Mb [0, ∞) and that
Tt f ∈ D((A )n ) for all f ∈ L1 [0, ∞). Since D(A∗ ) ⊂ L1 [0, ∞), it follows that
Ts∗ μ ∈ L1 [0, ∞) for all s > 0 and μ ∈ Mb [0, ∞). Therefore for all t > 0 and
μ ∈ Mb [0, ∞) we have
∗
n
Tt∗ μ = T ∗t T ∗t μ = T t T t μ ∈ D((A ) )
2
2
2
2
for all n ∈ N. Thus by taking μ = δx it follows that Tt∗ δx ∈ D((A )n ) for
α
g(y) ∈
all n. Using Theorem 2.3, we have that D(A ) = {g ∈ L1 [0, ∞) : D−y
n
α−1
1
α
∗
1
L [0, ∞), D−y g(0) = 0}, and then it follows that D−y Tt δx ∈ L [0, ∞) for all
n. In particular, the transition probability distribution Tt∗ δx has a density function
y → p(x, y, t) for all t > 0 and all x 0. To see that this function is smooth
note that any positive integer m can be written in the form m = nα − β for some
integer n 1 and some positive real number β < α. A straightforward calculaβ
α n
nα
nα
f = D−y
f and I−y
D−y
f =
tion shows that for f ∈ D((A )n ) we have D−y
nα−β
1
m
m
m
D−y f ∈ L [0, ∞) for all β < α and n 1. Since D−y f = (−1) (d/dy) f , we
have (d/dy)m p(x, y, t) ∈ L1 [0, ∞) for all m.
[0, ∞) = Mac [0, ∞) such that μn → μ vaguely as
Corollary 2.5. Let {μn } ⊂ C∞
n → ∞ for some μ ∈ Mb [0, ∞); then Tt μn → Tt∗ μ vaguely as n → ∞.
Proof. Since Tt μn = Tt∗ μn for all n 1, for all φ ∈ C∞ [0, ∞) we have
φ(x)[Tt μn ](dx) = φ(x)[Tt∗ μn ](dx) = [Tt φ](x)μn (x)dx
(2.21)
n→∞
−−−−→ [Tt φ](x)μ(dx) = φ(x)[Tt∗ μ](dx),
and the result follows.
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236
BORIS BAEUMER ET AL.
In Section 3, we will apply Corollary 2.5 to compute the transition densities of
the reflected stable process to any desired degree of accuracy, by solving the forward
equation numerically. For any initial state Zs = x, we will approximate the initial
condition μ0 = δx in the numerical method by a sequence of measures μn with L1 densities, and then Corollary 2.5 guarantees that the resulting solutions converge
to the transition density in the supremum norm as n → ∞.
Remark 2.6. Using integration by parts, one can write the backward generator in
the form of an integro-differential operator (e.g., see Jacob [26])
(2.22)
Af (x) = b(x)f (x) + [f (x + y) − f (x) − yf (x)] φ(x, dy),
with coefficients
x1−α
and
Γ(2 − α)
α(α − 1) −1−α
α − 1 −α
φ(x, dy) =
|y|
x ε−x (dy).
dy(−x,0) (y) +
Γ(2 − α)
Γ(2 − α)
b(x) =
(2.23)
The jump intensity φ(x, dy) describes the behavior of the process Zt , which truncates jumps of the stable process Yt starting at the point x > 0 in the state space,
so that a jump (they are all negative) of size |y| > x is changed to a jump of size x.
This keeps the sample paths of Zt inside the half-line [0, ∞). Since the drift b(x)
is unbounded, the existence of a Markov process Zt with generator (2.22) would
not follow from general theory (e.g., see [20, Section 4.5] or [48, Section 3]). Hence
the reflected stable process Zt is an interesting example of a Markov process with
unbounded drift coefficient.
3. Transition density of the reflected stable process
To the best of our knowledge, there is no known analytical formula for the
transition density y → p(x; t, y) of the reflected stable process y = Zt+s started at
x = Zs > 0. In this section, we compute and plot this transition density, by numerically solving the associated forward equation (2.20). The existence and smoothness
of these transition densities is guaranteed by Corollary 2.4. Corollary 2.5 shows
that Tt gn (y)dy → Tt∗ δx (dy) in the supremum norm as n → ∞ for any sequence
of functions gn ∈ L1 [0, ∞) such that gn (y)dy → δx (y)dy (vague convergence). We
take gn (y) = n [x,x+1/n] (y). Then the solutions Tt gn provide estimates of the
transition density to any desired degree of accuracy. Theorem 2.3 shows that Tt gn
solves the forward equation (2.20). Hence, we can compute the transition densities
of the stable process by solving the forward equation numerically, with this initial
condition.
In order to compute the probability density p(x, y, t) numerically, we consider
the fractional boundary value problem
(3.1)
α
∂t u(y, t) = D−y
u(y, t);
u(y, 0) = δx (y);
α−1
D−y
u(0, t) = 0.
We develop forward-stepping numerical solutions uh (yi , t) that estimate u(yi , t) at
locations yi = ih for i = 0, 1, . . . , N over an interval [0, ymax ] in the state space,
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REFLECTED SPECTRALLY NEGATIVE STABLE PROCESSES
237
where ymax is chosen large enough and h is chosen small enough so that enlarging
the domain further, or making the step size smaller, has no appreciable effect on the
computed solutions (e.g., the resulting graph does not visibly change). We approximate the delta function initial condition u(y, 0) = δx (y) by setting uh (yi , 0) = 1/h
for yi = x and uh (yi , 0) = 0 otherwise, a numerical representation of the initial
condition gn (y) = n [x,x+1/n] (y) with h = 1/n.
Numerical methods for fractional differential equations are an active area of
research. One important finding [34, 35] is that a shifted version of the Grünwald
finite difference formula (1.4) for the fractional derivative is required to obtain a
stable, convergent method. Hence we approximate
N +1−i
α
1 α
(3.2)
D−y
u(yi , t) ≈ α
wkα uh (yi+k−1 , t) where wkα := (−1)k
k
h
k=0
α
for i 2. Note that this approximation of D−y
u(yi , t) does not depend on the
value of uh at the boundary y0 = 0. We enforce the boundary condition at y0 = 0
at each step; i.e., we set
uh (y0 , t) = −
N
wkα−1 uh (yk , t)
k=1
so that
α−1
u(y0 , t) ≈
D−y
N
wkα−1 uh (yk , t) = 0
k=0
since
w0α−1
= 1. Finally, for i = 1 we approximate
N
1 α
wk uh (y1+k−1 , t)
hα
k=0
N
N
1 α
α−1
wk uh (yk , t) −
wk uh (yk , t)
= α
h
α
D−y
u(y1 , t) ≈
(3.3)
k=1
k=1
N
1 α−1
=− α
wk−1 uh (yk , t)
h
k=1
using an elementary identity for fractional binomial coefficients wkα − wkα−1 =
α−1
.
−wk−1
This leads to the following linear system of ordinary differential equations:
⎛
⎞
⎞
⎛
⎛
α−1 ⎞
uh (y1 , t)
uh (y1 , t)
−1 α − 1 . . . . . . −wN
−1
⎜
⎟
⎟
..
..
α
⎜1
⎟⎜
−α w2α . . . wN
⎜
⎟
⎟
−1 ⎟ ⎜
.
.
⎜
⎜
⎟
⎟
⎜
⎟⎜
..
..
d ⎜
1 ⎜
⎟
⎟
.
.
α
⎜
⎟
.
. wN −2 ⎟ ⎜
..
..
(3.4)
1
⎜
⎟= α⎜ 0
⎟
⎟ h ⎜ .
⎟
⎜
dt ⎜
⎟
.
.
.
.
⎜
⎟
⎟
⎜
.
.
..
..
..
.. ⎠ ⎝
⎝ ..
..
..
⎝
⎠
⎠
0
...
0
1
−α
uh (yN , t)
uh (yN , t)
whose solution can be approximated using any numerical ODE solver (indeed, this
is a linear system of the form u = Au, so it has a solution u(t) = etA u(0) for any
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238
BORIS BAEUMER ET AL.
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0
0
5
0
0
5
0
0
5
Figure 1. The transition densities of p(x, y, t) with α = 1.2, x =
0, 1, 2, 4 (left to right) at times t = 0.5 (left panel), t = 1 (middle
panel), and t = 2 (right panel).
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
Figure 2. The transition densities of p(x, y, t) with α = 1.8, x =
0, 1, 2, 4 (left to right) at times t = 0.5 (left panel), t = 1 (middle
panel), and t = 2 (right panel).
initial condition). The resulting numerical solutions with ymax = 12, h = 0.01,
starting points x = 0, 1, 2, 4, and α = 1.2, 1.8 are depicted in Figures 1 and 2,
where each frame represents a snapshot at times t = 0.5, 1 and 2 respectively. The
L1 -error of the numerical solutions for x = 0 decays linearly with h. For h = 0.01
the L1 -error is less than 0.05 for α = 1.2, and less than 0.004 for α = 1.8, for every
case plotted. A short MATLAB code to compute the numerical solution is included
in the Appendix.
Remark 3.1. It is interesting to note that the matrix in (3.4) is essentially the rate
matrix of a discrete state Markov process in continuous time. Extending the state
space to N = ∞, we obtain a Markov process Zth on the state space {ih : i > 0}
that approximates the reflected stable process, with uh (xi , t) = P(Zth = ih). The
transition rate from state ih to state jh for i > j > 1 is
α
≈ α(α − 1)(i − j)−α−1 /Γ(2 − α),
wi−j+1
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REFLECTED SPECTRALLY NEGATIVE STABLE PROCESSES
239
the jump intensity of the stable process Yt , in view of [37, Eq. (2.5)]. The transition
rate from state ih for i > 1 to state jh for j = 1, in the first row of the rate
α−1
≈ i−α /Γ(1 − α), the rate at which the process Yt would jump
matrix, is wi−1
into the negative half-line. This can be computed as φ(−∞, −ih) where φ(dy) =
α(α − 1)|y|−1−α dy/Γ(2 − α) is the Lévy measure of the process Yt , e.g., see [37,
Proposition 3.12].
Remark 3.2. As noted in the introduction, the fractional boundary condition in
(3.1) is a natural extension of the boundary condition (1.3) for Brownian motion
on the half-line. The fractional boundary condition in (3.1) can be written in
Grünwald finite difference form using [37, Proposition 2.1] to arrive at (1.4).
Remark 3.3. Bernyk, Dalang and Peskir [11, Appendix] computed the backward
generator of a general reflected stable Lévy process. Caballero and Chaumont
[14, Theorem 3] compute the backward generator of a killed stable Lévy process. It
may be possible to develop the forward equation and compute the transition density
for those processes, using the methods of this paper. This would be interesting for
applications to fractional diffusion, since it could elucidate the relevant fractional
boundary conditions.
4. Fractional Cauchy problems
In this section, we show that the reflected stable process (1.2) with index 1 <
α 2 can be used as a time change to solve the fractional Cauchy problem
(4.1)
∂tβ p(x, t) = Lp(x, t);
p(x, 0) = f (x)
of order β = 1/α, when L generates a Feller process. The time-fractional Caputo
derivative in (4.1) is defined by
∞
1
f (t − r)r −β dr,
(4.2)
∂tβ f (t) =
Γ(1 − β) 0
a special case of (2.6). Fractional Cauchy problems are useful in a wide variety of
practical applications [21,39,40,47], and the next result allows a Markovian particle
tracking solution for such problems [10, 54, 55].
Theorem 4.1. For any β ∈ [1/2, 1), let Zt be given by (1.2), where Yt is an α-stable
Lévy process with characteristic function (1.1) for α = 1/β. If Xt is an independent
Markov process such that Tt f (x) = Ex [f (Xt )] forms a uniformly bounded, strongly
continuous semigroup with generator L on some Banach space B of real valued
functions, then p(x, t) = Ex [f (XZt )] solves the fractional Cauchy problem (4.1) for
any f ∈ D(L), the domain of the generator.
Proof. Theorem 3.1 in [4] states that if u(x, t) solves the Cauchy problem ∂t u(x, t) =
Lu(x, t); u(x, 0) = f (x) on B for some f ∈ Dom(L), then the solution to the
associated fractional Cauchy problem (4.1) on B is given by
∞
u(x, r)h(r, t) dr
(4.3)
p(x, t) =
0
where
(4.4)
h(r, t) =
t −1−1/β
r
gβ (tr −1/β )
β
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240
BORIS BAEUMER ET AL.
and gβ (t) is the stable probability density function with Laplace transform
∞
β
Lg(s) =
e−st gβ (t) dt = e−s
0
for some 0 < β < 1. Since Yt has no positive jumps, the first passage time Dt :=
inf{r > 0 : Yr > t} is a stable subordinator with E[e−sDt ] = exp(−ts1/α ), and
the supremum process St = sup{Yr : 0 r t} is also the first passage time
Et = inf{u > 0 : Du > t} of Dt ; see Bingham [13]. Note that P(DEt > t) = 1
[12, Theorem III.4]. Apply [33, Corollary 3.1] (or Exercises 29.7 and 29.18 in
Sato [46]) to see that (4.4) is also the probability density of Et . It follows from
[12, Section VI.1, Prop. 3] that
(4.5)
P(St x) = P(Zt x) for all t > 0 and all x > 0.
Hence (4.4) is also the probability density of Zt , and the theorem follows.
Remark 4.2. An alternate proof of Theorem 4.1 uses the reflection principle for
spectrally negative stable Lévy processes. Extending the usual argument for reflected Brownian motion, let τx = inf{u > 0 : Yu > x}. Since Yt is self-similar,
we have P(Yt > 0) = 1/α for every t > 0 (e.g., see [6, Theorem 4.1 (i)]). Defining
Ytx := Yt−τx − x for t τx , we have
P(St x) = P(St x, Yt > x) + P(St x, Yt x)
= P(Yt > x) + P(τx t, Ytx 0)
and since P(τx t, Ytx 0) = (1 − α−1 )P(St x) it follows that
(4.6)
P(St x) = αP(Yt > x) = P(Yt > x|Yt 0) for all t > 0 and all x > 0.
An application of Zolotarev duality for stable densities [6, Theorem 4.1 (ii)] implies
(4.7)
P(Et > x) = P(Yt > x|Yt 0)
for all t > 0 and all x > 0.
Then the theorem follows using (4.5), [4, Theorem 3.1] and [33, Corollary 3.1].
Remark 4.3. Theorem 4.1 confirms a conjecture in the paper [6, Remark
u 5.2]. There
we set Zt = Yσ(t) where σ(t) = inf{u > 0 : Hu > t} and Hu = 0 Ys >0 ds. In
essence, the negative excursions are cut away, and the positive excursions are joined
together without any gaps in time. Since Yt has no positive jumps, any up-crossing
at the origin is a renewal point, so this process has the same distribution as (1.2).
5. Resolvent estimates
In this section, we develop bounds on the norm of the resolvent used in Theorem
2.1. Note that both terms in the formula (2.8) for the resolvent R(λ, A) diverge
as x → ∞. For example, it follows directly from [22, Eq. (6.4)] that Eα (λxα ) ∼
1/α
α−1 eλ x as x → ∞. Hence it is useful to begin by establishing an alternative
representation. Recall that Yt is a negatively skewed stable process with index
1 < α 2 and characteristic function (1.1). Let gα (x) denote the probability
density function of −Y1 , a totally positively skewed stable law taking values on the
entire real line.
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REFLECTED SPECTRALLY NEGATIVE STABLE PROCESSES
241
Lemma 5.1. For any g ∈ C∞ [0, ∞) and any Re λ > 0 we have
∞ ∞
1
x−ξ
e−λt 1/α gα
R(λ, A)g(x) =
g(ξ) dt dξ
t
t1/α
0
0
(5.1)
x x
Lg(λ1/α ) ∞ −λt
e
gα 1/α dt,
+ 1−1/α
1+1/α
λ
αt
t
0
where Lg is the usual Laplace transform.
Proof. First we will show that the right-hand side of (5.1) vanishes for any x < 0.
Recall that gα (x) is a standard positively skewed stable density with index 1 <
α 2 and characteristic function exp((−ik)α ), and hence gα (−x) is a standard
negatively skewed stable density with index 1 < α 2 and characteristic function
exp((ik)α ), i.e., the density of Y1 in (1.1). For 0 < β < 1, let gβ (x) denote the standard positively skewed stable density with characteristic function exp(−(−ik)β ).
Since this density is supported on the positive real line, we can also write
(5.2)
Lgβ (s) = exp(−sβ )
for all Re s > 0
(e.g., see [56, Lemma 2.2.1]). Apply the Zolotarev Duality Theorem for stable
densities [56, Theorem 2.3.1] to see that when β = 1/α we have
(5.3)
gα (−x) = x−1−α gβ (x−α )
for all x > 0.
Then for x < 0, the first term in (5.1) is
x ∞
u 1
e−λt 1/α gα 1/α g(x − u) dt du
t
t
−∞ 0
x ∞
u −1−α
u −α
−λt 1
− 1/α
=
− 1/α
e
g1/α
g(x − u) dt du
t1/α
t
t
−∞ 0
x ∞
t
d −λt 1
e
=−
g
dt g(x − u) du
dλ
(−u)1+α 1/α (−u)α
−∞ 0
x
∞
t
1 d
1
=
e−λt
g
dt g(x − u) du.
1/α
(−u)α
(−u)α
−∞ u dλ 0
Using (5.2) we have
∞
t
1
−λt
(5.4)
e
g1/α
dt = exp(−[λ(−u)α ]β ) = exp(uλ1/α ),
(−u)α
(−u)α
0
and then the first term in (5.1) equals
∞
x
1/α
1
1 d uλ1/α g(x − u) du = λ1/α−1
e
e(x−y)λ g(y) dy
α
−∞ u dλ
0
1 1/α−1 xλ1/α
= λ
e
Lg(λ1/α )
α
t
Lg(λ1/α ) ∞ −λt 1
=
e
g
dt
1/α
(−x)α
(−x)α
αλ1−1/α 0
x Lg(λ1/α ) ∞ −λt (−x)
dt
e
g
= 1−1/α
α
λ
αt1+1/α
t1/α
0
where we have used (5.4) again in the next-to-last line. Since the last line above is
the negative of the second term in (5.1), it follows that the sum of these two terms
vanishes for all x < 0.
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242
BORIS BAEUMER ET AL.
Since the positively skewed stable density gα (x) tends to zero at a superexponential rate as x → −∞ (e.g., see [56, Theorem 2.5.2]), its bilateral Laplace
transform L−∞ [gα ](s) is well defined for all s > 0, and in fact we can write
∞
(5.5)
e−sx
−∞
1
t
g
1/α α
x α
dx = ets
1/α
t
for all t > 0 and all s > 0.
Then it follows that
(5.6)
∞
∞
x 1
1
e−λt
e−sx 1/α gα 1/α dx dt =
λ − sα
t
t
0
−∞
for all s > 0 and all Re λ > 0.
Then for any g ∈ C∞ [0, ∞) the convolution property of the bilateral Laplace transform implies that the first term in (5.1) satisfies
∞
(5.7)
−∞
e−sx
∞
−∞
∞
e−λt
0
1
t
g
1/α α
x−ξ
t1/α
g(ξ) dt dξ dx =
g̃(s)
λ − sα
for all s > 0 and all Re λ > 0.
As for the second term, in view of the fact that the complex contour integral
∞
λ
1
d
αsα−1
du
=
ds u − sα
λ − sα
(integrate along {λ + r : r > 0}) it follows using (5.6) that
∞
∞
x sα−1
1 ∞ d
−ut
−sx 1
dx
dt
du
=
e
e
g
α
λ − sα
α λ ds 0
t1/α
t1/α
−∞
∞
∞
x ∞
−x
1
−ut
=
e−sx
g
e
du
dt dx
α
α t1/α
t1/α
−∞
0
λ
∞
∞
x 1
x =−
e−sx
e−λt
gα 1/α dt dx.
1+1/α
α t
t
−∞
0
Then it follows immediately that the bilateral Laplace transform of the second term
in (5.1) equals the second term in (2.9). Since the right-hand side of (5.1) vanishes
for x < 0, its bilateral Laplace transform equals its ordinary Laplace transform.
(Note however that neither term on the right-hand side of (5.1) vanishes for x < 0,
only their sum.) But then (5.1) has the same Laplace transform as R(λ, A)g(x), and
since both are continuous in view of (2.8), the result follows using the uniqueness
of the Laplace transform.
Lemma 5.2. Under the assumptions of Theorem 2.1, we have for every 1 < α 2
that (2.11) holds for all Re λ > 0 and all g ∈ C∞ [0, ∞), with M = Mα + 1 +
sec(π/(2α)) for some Mα depending only on α.
Proof. Denote by BUC(R) the Banach space of bounded, uniformly continuous
functions with the supremum norm. Given g ∈ C∞ [0, ∞), define ḡ ∈ BUC(R)
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REFLECTED SPECTRALLY NEGATIVE STABLE PROCESSES
243
by setting ḡ(x) = g(x) for x > 0 and ḡ(x) = g(0) for x 0. Use (5.1) to write
R(λ, A)g(x) = I1 − I2 + I3 where
∞ ∞
x−ξ
1
e−λt 1/α gα
I1 =
ḡ(ξ) dξ dt,
t
t1/α
0
−∞
∞ 0
x−ξ
−λt 1
(5.8)
e
gα
ḡ(ξ) dξ dt,
I2 =
t1/α
t1/α
0
−∞
x x
Lg(λ1/α ) ∞ −λt
dt.
I3 = 1−1/α
e
g
α
λ
αt1+1/α
t1/α
0
The formula
∞
x−ξ
1
Ttα f (x) =
f (ξ) dξ
g
α
1/α
t1/α
−∞ t
∞
∞
ξ
1
=
g
gt,α (ξ)f (x − ξ) dξ,
f
(x
−
ξ)
dξ
:=
1/α α
t1/α
−∞ t
−∞
f ∈ BUC(R),
defines a strongly continuous convolution semigroup on BUC(R). Indeed, T̄tα f =
f ∗ gt,α , f ∈ L1 (R), defines a strongly continuous semigroup on L1 (R); see, e.g.,
α
[24, Theorem 21.4.3], noting that the Fourier transform of gt,α is et(ik) . The latter
follows from (5.5) which also holds for s = ik, as gα is absolutely integrable. Then,
Ttα is a subordinate semigroup (where the right-translation group on BUC(R),
which is strongly continuous [19, Chapter I, Section 4.15], is subordinated against
T̄α,t ) which is strongly continuous by [5, Theorem 4.1].
Next we show that Ttα is a bounded analytic semigroup on BUC(R), by showing
d α
Tt f = Aα Ttα f M t−1 f for some M > 0; see [3, Theorem 3.7.19].
that dt
Here, the operator Aα denotes the generator of Ttα . We have that
∞
d α d
T f dt t dt gt,α (x) dx f .
−∞
Recall Carlson’s inequality (see [15, 18])
∞
−∞
|h(x)| dx C
∞
−∞
1/4 2
|F[h](k)| dk
∞
−∞
2 1/4
d
F[h](k) dk
dk
where F[h](k) = e−ikx h(x) dx denotes the Fourier transform of h. Note that
α
d
F[ dt
gt,α ](k) = (ik)α et(ik) . It is easy to check, using the formula for the gamma
probability density, that
∞ ∞
α 2
α
1
|k|2α e−2tcα |k| dk Cα t−2− α
(ik)α et(ik) dk −∞
−∞
for some cα , Cα > 0, and that
∞ ∞ 2
d
α
α t(−ik)α |k|2(α−1) + t2 |k|2(2α−1) e−2cα t|k| dk
(−ik)
e
dk
2
dk
−∞
−∞
Cα t−2+ α .
1
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244
BORIS BAEUMER ET AL.
Hence Aα Ttα f Kα t−1 f , and so Ttα is a bounded analytic semigroup on
BUC(R). Therefore, by [3, Corollary 3.7.12],
∞
−λt α
e Tt ḡ dt
λI1 = λ
= λR(λ, Aα )ḡ Mα ḡ = Mα g, Re λ > 0.
0
Next write
(5.9)
∞ 0
x−ξ
−λt 1
|λI2 | |λg(0)| e
gα
dξ dt
1/α
1/α
t
t
−∞
0
∞
∞
|λ|g e−λt
gα (y) dy dt
x
0
t1/α
e−λt ∞
∞ 1 ∞
x x
−λt
=|λ|g −
dt .
gα (y) dy +
e
g
1+1/α α t1/α
x
λ
λ
αt
0
1/α
0
t
α
A substitution t = (x/u) leads to
∞
∞
x α
x
−λt
(5.10)
e
gα 1/α dt =
e−λ(x/u) gα (u) du,
1+1/α
αt
t
0
0
and then it follows that
|λI2 | g 0
∞
x α
e−λ( y ) gα (y) dy g
for all Re λ > 0 and all g ∈ C∞ [0, ∞).
Another application of (5.10) shows that |λI3 | |λ1/α Lg(λ1/α )|. If λ = reiθ for
some r > 0 and |θ| < π/2, then λ1/α = r 1/α eiθ/α has real part r 1/α cos(θ/α) r 1/α cos(π/(2α)). Hence |λ1/α | = r 1/α Re[λ1/α ]/ cos(π/(2α)) for all Re λ > 0. It
is not hard to check that Re[λ|Lg(λ)|] g for all Re λ > 0 and all g ∈ C∞ [0, ∞).
Then we have
|λ1/α |
1
1/α
1/α |Lg(λ
)|]
|λI3 | |λ1/α Lg(λ1/α )| = ·
Re[λ
cos(π/(2α)) g
1/α
Re[λ ]
for all Re λ > 0 and all g ∈ C∞ [0, ∞), and the result follows.
Remark 5.3. Following a slightly different path using Fourier transforms, it is possible to show that
λR(λ, A)g(x) = λR(λ, Dxα )ḡ(x) − λR(λ, Dxα )g(0)(−∞,0) (x)
+ cα (λR(λ, Dxα ) − I)(−∞,0) (x)
where ḡ(x) = g(x) for x > 0, ḡ(x) = g(0) for x 0, cα = λ1/α Lg(λ1/α ), I is
the identity operator, and (−∞,0) (x) is the indicator function. This form clarifies
that the resolvent of the Caputo fractional derivative A = ∂xα is a modification of
the resolvent of the Riemann-Liouville fractional derivative Dxα to account for the
boundary term, which is natural in view of (2.7). The same resolvent bound in
Lemma 5.2 can also be obtained using this form.
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REFLECTED SPECTRALLY NEGATIVE STABLE PROCESSES
245
Appendix
The following Matlab code computes the transition density p(x, y, t) in the case
x = 2 for the reflected stable process Zt defined by (1.2), where Yt is a stable Lévy
process with characteristic function (1.1) and index 1 < α < 2. This code was used
to generate the plots in Figures 1 and 2.
%%% 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);
Acknowledgments
The authors would like to thank Zhen-Qing Chen, University of Washington,
Pierre Patie, Université Libre de Bruxelles, and Victor Rivero, Centro de Investigación en Matemáticas for helpful discussions. We would also like to thank an
anonymous referee for many helpful comments that significantly improved the presentation.
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Department of Mathematics and Statistics, University of Otago, Dunedin, New
Zealand
E-mail address: bbaeumer@maths.otago.ac.nz
Department of Mathematics and Statistics, University of Otago, Dunedin, New
Zealand
E-mail address: mkovacs@maths.otago.ac.nz
Department of Probability and Statistics, Michigan State University, East Lansing,
Michigan 48824
E-mail address: mcubed@stt.msu.edu
URL: http://www.stt.msu.edu/users/mcubed/
Institut für Mathematische Stochastik, Technische Universität Dresden, 01069 Dresden, Germany
E-mail address: rene.schilling@tu-dresden.de
URL: http://www.math.tu-dresden.de/sto/schilling/
School of Mathematics and Statistics, The University of New South Wales, Kensington NSW 2052, Australia
E-mail address: p.straka@unsw.edu.au
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