On the solution of two-sided fractional ordinary differential equations of Caputo type

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by  Ma. Elena Hernández-Hernández

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ON THE SOLUTION OF TWO-SIDED FRACTIONAL
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8 ORDINARY DIFFERENTIAL EQUATIONS OF
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CAPUTO TYPE

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Ma. Elena Hern´andez-Hern´andez 1, Vassili N. Kolokoltsov 2
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a Abstract
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This paper provides well-posedness results and stochastic representations for the

1 solutions to equations involving both the right- and the left-sided generalized operators
v of Caputo type. As a special case, these results show the interplay between two-sided
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fractional differential equations and two-sided exit problemsfor certain L´evy processes.
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MSC 2010: Primary 34A08; Secondary 35S15, 26A33, 60H30
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0 Key Words and Phrases: two-sided fractional equations, generalized Caputo type
. derivatives, boundary point, stopping time, Feller process, L´evy process
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1 1. Introduction
:
v The successful use of classical fractional derivatives to describe, for example, re-
i
X laxation phenomena, processes of oscillation, viscoelastic systems and diffusions in
r disordered media (anomalous diffusions) among others, have promoted an increasing
a
research on the field of fractional differential equations. For an account of historical
notes, applications and different methods to solve fractional equations we refer, e.g., to
[7]-[10], [12], [14], [21]-[22], [25], [29]-[32], [34], [39], and references cited therein.
Apartfromthedifferentnotionsoffractionalderivativesfoundintheliterature(e.g.,
the Caputo, the Riemann-Liouville, the Grunwald-Letnikov, the Riesz, the Weyl, the
Marchaud, and the Miller and Ross fractional derivatives), numerous generalizations
(mostly from an analytical point of view) have been proposed by many authors, we
refer, e.g., to [2], [18]-[19], [23]-[24], [33] for details. As for the generalized fractional
operators of Caputo type considered in this work, they were introducedin [27] by oneof
theauthorsasgeneralizations(fromaprobabilisticpointofview)oftheclassicalCaputo
derivatives of order β ∈ (0,1) when applied to regular enough functions. These Caputo
type operators can be thought of as the generators of Feller processes interrupted on
the first attempt to cross certain boundary point (see precise definition later).
2 M.E. Hern´andez-Herna´ndez, V.N. Kolokoltsov
As a continuation of our previous works, which show a new link between stochastic
analysis and fractional equations (see [16]-[17], [27]), this paper appeals to a probabilis-
tic approach to study equations involving both left-sided and right-sided generalized
operators of Caputo type. We address the boundary value problem for the two-sided
generalized linear equation with Caputo type derivatives −D(ν+) and −D(ν−):
a+∗ b−∗
−D(ν+)u(x)−D(ν−)u(x)−Au(x) = λu(x)−g(x), x ∈ (a,b),
a+∗ b−∗
u(a) = u , u(b) = u , (1.1)
a b
whereλ ≥ 0, u ,u ∈Randg isaprescribedfunctionon[a,b]. Notation −A≡ −A(γ,α)
a b
refers to the second order differential operator
d d2
−A(γ,α) := γ(·) +α(·) . (1.2)
dx dx2
Equation (1.1) includes, as special cases, the fractional equations
Dβ1 u(x)+Dβ2 u(x) = g(x), x ∈ (a,b), β ,β ∈(0,1), (1.3)
a+∗ b−∗ 1 2
u(a) = u , u(b) = u ,
a b
where Dβ1 and Dβ2 are the left- and the right-sided Caputo derivatives of order β
a+∗ b−∗ 1
andβ ,respectively. Therearerelativelyscarceresultsdealingwithtwo-sidedfractional
2
ordinary equations. For example, to the best of our knowledge, the Riemann-Liouville
version of (1.3) was analyzed (in the space of distributions) in [35]-[36], whereas the
explicit solution to the two-sided fractional equation in (1.3) was justrecently provided
in [27].
Another special case of equation (1.1) is the two-sided equation:
c Dβ1 u(x)+c Dβ2 u(x)+γ(x)u′(x)+λu(x) = g(x), x ∈ (a,b), (1.4)
1 a+∗ 2 b−∗
u(a) = u , u(b) = u .
a b
Ifc > 0,c = 0,β = 1 andλ = 1,thenthe(one-sided)equationisknownastheBasset
1 2 1 2
equation, well-studied in the literature (see, e.g. [29] and references therein). The one-
sided case with β ∈ (0,1) (known as the composite fractional relaxation equation) was
1
treated via the Laplace transform method in [15, Section 4], whereas the left-sided case
with Caputo type and RL type operators was studied by the authors in [17].
Some other examples showing the relevance of left- and right-sided derivatives in
mathematical modeling appear in the study of FPDE’s on bounded domains, as well
as in fractional calculus of variations, see, e.g.,[1], [3], [21], [31], [37].
In this paper we study the well-posedness of (1.1) by considering two types of
solutions: solutions in the domain of the generator and generalized solutions. The first
typeis understoodas a solution uthat belongs to the domain of thetwo-sided operator
seen as the generator of a Feller process. Since the existence of such a solution is quite
restrictive once one imposes boundary conditions, the notion of generalized solution
is introduced via the limit of approximating solutions taken from the domain of the
generator.
ON THE SOLUTION OF TWO-SIDED FRACTIONAL ... 3
Further, appealing to the relationship between two-sided equations and exit prob-
lems for Feller processes (already mentioned in [27]), we provide some explicit solutions
to two-sided equations in the context of classical fractional derivatives. Even though
exit problems for L´evy processes have been widely studied (see, e.g., [5]-[6], [28], [38]),
to our knowledge fractional equations of the type in (1.3) and their connection with
exit problems seem to be novel in the literature. We believe that the probabilistic so-
lutions presented in this work can be used, for example, to obtain numerical solutions
to classical fractional equations for which explicit solutions are unknown.
The paper is organized as follows. The next Section 2 sets standard notation and
definitions. Section 3 gives a quick review about generalized Caputo type operators.
Section 4 provides preliminary results concerning two-sided generalized operators and
their connections with the generators of Feller processes. Then, Section 5 addresses the
well-posedness for the RL type version of (1.1). Thestudy of the Caputotype equation
(1.1) is given in Section 6. Some examples are presented in Section 7. Finally, Section
8 contains the proofs of some key results established in Section 4.
2. Preliminaries
2.1. Notation. Let N and R be the set of positive integers and the real line, respec-
tively. For any open set A ⊂ R, notation B(A), C(A) and C (A) denote the set
∞
of bounded Borel measurable functions, bounded continuous functions and continuous
functions vanishing at infinity defined on A, respectively, equipped with the sup-norm
||h|| = sup |h(x)|. The space of continuous functions on A with continuous deriva-
x∈A
tives up to and including order k is denoted by Ck(A). This space is equipped with the
norm ||h|| := ||h||+ ||h(k)||. For functions defined on the closure A¯ of A, nota-
Ck k=1
tion Ck(A¯) means the space of k times continuously differentiable functions up to the
P
boundary. Further, spaces C [a,b] and Ck[a,b] stand for the space of continuos func-
0 0
tions on [a,b] vanishing at the boundary and the space of functions C [a,b]∩Ck[a,b],
0
respectively.
Letters PandEare reservedfor theprobability andthemathematical expectation,
respectively. ForastochasticprocessX = (X (t)) withstatespaceA,thesubscript
x x t≥0
x in X (t) means that the process starts at x ∈ A, so that notation E[f(X (t))] is
x x
understood as E[f (X(t))|X(0) =x]. All the processes considered in this paper are
assumed to be defined on a fixed complete probability space (Ω,F,P).
2.2. Feller processes: basic definitions. Let{T } beastrongly continuous semi-
t t≥0
group of linear bounded operators on a Banach space (B,||·|| ), i.e. lim ||T f −
B t→0 t
f|| = 0forallf ∈ B. Its(infinitesimal)generator LwithdomainD ,shortly(L,D ),
B L L
is defined as the (possibly unbounded) operator L : D ⊂ B → B given by the strong
L
limit
T f −f
Lf := lim t , f ∈ D , (2.1)
L
t↓0 t
where the domain of the generator D consists of those f ∈ B for which the limit in
L
(2.1) exists in the norm sense. We also recall that, if L is a closed operator, then a
linear subspace C ⊂ D is called a core for the generator L if the operator L is the
L L
4 M.E. Hern´andez-Herna´ndez, V.N. Kolokoltsov
closure of the restriction L [13, Chapter 1, Section 3]. If additionally T C ⊂ C for
CL t L L
all t ≥ 0, then C is said to be an invariant core. The resolvent operator R of the
L (cid:12) λ
semigroup {T } is defin(cid:12)ed (for any λ > 0) as the Bochner integral (see, e.g., [11,
t t≥0
Chapter 1], [13, Chapter 1])
∞
R g := e−λtT gdt, g ∈ B. (2.2)
λ t
Z0
By taking λ = 0 in (2.2), one obtains the potential operator denoted by R g (whenever
0
it exists).
We say that a (time-homogeneous) Markov process X = (X(t)) taking values on
t≥0
A⊂ Rd is a Feller process (see, e.g., [25, Section 3.6]) if its semigroup {T } , defined
t t≥0
by
T f(x):= E[f (X(t))|X(0) = x], t ≥0, x ∈ A, f ∈ B(A),
t
gives rise to a Feller semigroup when reduced to C (A), i.e. it is a strongly continuous
∞
semigroup on C (A) and it is formed by positive linear contractions (0 ≤ T f ≤ 1
∞ t
whenever 0 ≤ f ≤ 1).
3. Generalized fractional operators of Caputo type and RL type
The generalized Caputo type operators introduced in [27] are defined in terms of a
function ν :R×(R\{0}) → R+ satisfying the condition:
(H0) The function ν(x,y) is continuous as a function of two variables and continu-
ously differentiable in the first variable. Furthermore,
∂
sup min{1,|y|}ν(x,y)dy < ∞, sup min{1,|y|} ν(x,y) dy < ∞,
∂x
x x
Z Z (cid:12) (cid:12)
and (cid:12) (cid:12)
(cid:12) (cid:12)
limsup |y|ν(x,y)dy = 0.
δ→0 x Z|y|≤δ
Definition 3.1. Let a,b ∈ R with a < b. For any function ν satisfying the
(ν) (ν)
condition (H0), the operators −D and −D defined by
a+∗ b−∗
x−a
(ν)
−D h (x) = (h(x−y)−h(x))ν(x,y)dy +
a+∗
(cid:16) (cid:17) Z0 ∞
+(h(a)−h(x)) ν(x,y)dy, (3.1)
Zx−a
for functions h :[a,∞) → R, and by
b−x
(ν)
−D h (x) = (h(x+y)−h(x))ν(x,y)dy +
b−∗
(cid:16) (cid:17) Z0 ∞
+(h(b)−h(x)) ν(x,y)dy, (3.2)
Zb−x
ON THE SOLUTION OF TWO-SIDED FRACTIONAL ... 5
for functions h: (−∞,b] → R, are called the generalized left-sided Caputo type operator
and the generalized right-sided Caputo type operator, respectively. The values a and b
will be referred to as the terminals of the corresponding operators.
Remark 3.1. The sign − appearing in the previous notation is introduced to
comply with the standard notation of fractional derivatives.
Due to assumption (H0), the operators (3.1)-(3.2) are well defined at least on the
space of continuously differentiable functions (with bounded derivative).
Remark 3.2. Theleft-sided (resp. right-sided)generalized Riemann-Liouville type
(ν) (ν)
operator −D (resp. −D ) is defined by setting h(a) = 0 (resp. h(b) = 0) in (3.1)
a+ b−
(resp. (3.2)). Hence,
(ν) (ν) (ν) (ν)
−D h(x) = −D [h−h(a)](x) and −D h(x) = −D [h−h(b)](x).
a+∗ a+ b−∗ b−
3.0.1. Particular cases. For smooth enough functions h, the standard analytical def-
β
initions of theleft-sided Caputoderivative D andthe right-sided Caputoderivatives
a+∗
β
D of order β ∈ (0,1) (see, e.g., [10, Definition 2.2, Definition 3.1]) can be rewritten
b−∗
as (see, e.g., [27, Appendix])
β x−a h(x−y)−h(x) h(x)−h(a)
β
D h (x)= dy− , (3.3)
a+∗ Γ(1−β) y1+β Γ(1−β)(x−a)β
(cid:16) (cid:17) Z0
and
β b−x h(x+y)−h(x) h(x)−h(b)
β
D h (x) = dy− . (3.4)
b−∗ Γ(1−β) y1+β Γ(1−β)(b−x)β
(cid:16) (cid:17) Z0
β β (ν)
Hence, for h regular enough, D h (resp. −D h) is a particular case of −D h
a+∗ b−∗ a+∗
β
(resp. −D h) obtained by taking the function
b−∗
β
ν(x,y) ≡ ν(y)= − , β ∈ (0,1). (3.5)
Γ(1−β)y1+β
Remark 3.3. Other examples of generalized operators −D(ν) include the frac-
a+∗
tional derivatives of variable order, aswellasthegeneralized distributed order fractional
derivatives (see [16], [27] for precise definitions).
4. Two-sided operators of RL type and Caputo type
Given two functions ν and ν satisfying condition (H0), define the function ν :
+ −
R×R\{0} → R+ associated with ν and ν by setting
+ −
ν(x,y) := ν (x,y), y > 0, ν(x,y) := ν (x,−y), y < 0. (4.1)
+ −
6 M.E. Hern´andez-Herna´ndez, V.N. Kolokoltsov
Define the two-sided operator of RL type −L and the two-sided operator of Caputo
[a,b]
type −L by
[a,b]∗
−L f (x):= −D(ν+)f (x)+ −D(ν−)f (x)+ −A(γ,α)f (x), (4.2)
[a,b] a+ b−
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
and (cid:0) (cid:1)
−L f (x) := −D(ν+)f (x)+ −D(ν−)f (x)+ −A(γ,α)f (x). (4.3)
[a,b]∗ a+∗ b−∗
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Notati(cid:0)on −A(γ,α(cid:1)) stands for the differential operator given in (1.2). We will see that
the operator −L can be thought of as the generator of a Feller process on [a,b],
[a,b]∗
whereas −L is related to the generator of a killed process. For that purpose, let
[a,b]
us introduce an additional definition for the regularity of the boundary (see, e.g., [26,
Chapter 6]).
Definition 4.1. For a domain D ⊂ R with boundary ∂D, a point x ∈ ∂D
0
is said to be regular in expectation for a Markov process X (or for its generator ) if
E[τ (x)] → 0, as x → x , x ∈ D, where τ (x) := inf{t ≥ 0 : X (t) ∈/ D}, with the
D 0 D x
usual convention that inf{∅} = ∞.
Theorem 4.1. Let ν be a function satisfying assumption (H0). Suppose that
γ ∈ C3[a,b], α ∈ C3[a,b] with derivative α′ ∈ C [a,b] and α being a positive function.
0 0
Then,
(i) the operator (−L , Dˆ ) generates a Feller process Xˆ on [a,b] with a domain
[a,b]∗ ∗
Dˆ such that
∗
f ∈ C2[a,b] : f′ ∈ C [a,b] ⊂ Dˆ . (4.4)
0 ∗
(ii) The points {a,b} a(cid:8)re regular in expectation fo(cid:9)r (−L[a,b]∗,Dˆ∗). Further, the first
exit time τˆ (x) from the interval (a,b) of Xˆ , x ∈ (a,b), has a finite expectation.
(a,b) x
P r o o f. See proof in Section 8. ✷
Stopped and killed processes. To introduce the notion of solutions to the equation
(1.1) we are interested in, we need the stopped version of Xˆ.
Theorem 4.2. Supposethat the assumptions of Theorem 4.1 hold. Let Xˆ be the
x
process started at x ∈ (a,b) generated by (−L , Dˆ ).
[a,b]∗ ∗
(i) The process X[a,b]∗ defined by X[a,b]∗(s) := Xˆ (s∧τˆ (x)), s ≥ 0, is a Feller
x x x (a,b)
processon[a,b]. Iftheoperator(−L ,Dstop )denotesthegeneratorofX[a,b]∗,
stop [a,b]∗
then for any f ∈ Dˆ satisfying −L f (x) = 0 for x ∈ {a,b}, it holds that
∗ [a,b]∗
f ∈ Dstop and −L f = −L f.
[a,b]∗ stop [a,(cid:0)b]∗ (cid:1)
[a,b] [a,b] [a,b]∗
(ii) The process X defined by X (s) := X (s) for s < τˆ (x) is a Feller
x x x (a,b)
(sub-Markov)processon(a,b). If(−L ,Dkill)denotesthegeneratorofX[a,b],
kill [a,b]
ON THE SOLUTION OF TWO-SIDED FRACTIONAL ... 7
thenforanyf ∈ Dstop satisfyingf(x) = 0forx ∈ {a,b},itholdsthatf ∈ Dkill
[a,b]∗ [a,b]
and −L f = −L f.
kill [a,b]
P r o o f. See proof in Section 8. ✷
Remark 4.1. The operator −L can be obtained from the generator (L,D )
[a,b]∗ L
of a Feller process, say X , given by
x
∞
(Lf)(x) = (f(x+y)−f(x))ν(x,y)dy+γ(x)f′(x)+α(x)f′′(x), (4.5)
Z−∞
by modifying it in such a way that it forces the jumps aimed to be out of the interval
(a,b) to land at the nearest (boundary) point (see also [27]). If, instead, the process
is killed upon leaving (a,b), then the corresponding process has a generator related to
the operator −L . Thus, when starting at the same state x ∈ (a,b), it holds that the
[a,b]
paths of the processes X , Xˆ , X[a,b]∗ and X[a,b] coincide before their first exit time
x x x x
from the interval (a,b). Hence, the first exit time in all cases will always be denoted by
τ (x). We refer to the processes X , Xˆ , X[a,b]∗ and X[a,b] as the underlying process,
(a,b) x x x x
the interrupted process, the stopped process and the killed process, respectively.
5. Two-sided equations involving RL type operators
Let us now study the equation (1.1) for which we will also use the short nota-
tion −L ,λ,g,u ,u . We shall start with the boundary value problem with zero
[a,b]∗ a b
boundary conditions: u = 0= u . Thus, due to the relationship between Caputo and
a b
(cid:0) (cid:1)
RL type operators (see Remark 3.2), the two-sided Caputo type operator −L can
[a,b]∗
be replaced with the RL type operator −L , so that the equation −L ,λ,g,0,0
[a,b] [a,b]
will be called the two-sided RL type equation.
(cid:0) (cid:1)
Definition 5.1. (Solutions to RL type equations) Let g ∈ B[a,b] and λ ≥ 0.
A function u ∈C [a,b] is said to solve the linear equation of RL type (−L ,λ,g,0,0)
0 [a,b]
as (i) a solution in the domain of the generator if uis a solution belonging to Dkill; (ii)
[a,b]
ageneralized solution ifforallsequenceoffunctionsg ∈ C [a,b]suchthatsup ||g ||<
n 0 n n
∞ and lim g → g a.e., it holds that u(x) = lim w (x) for all x ∈ [a,b], where
n→∞ n n→∞ n
w is the unique solution (in the domain of the generator) to the RL type problem
n
(−L ,λ,g ,0,0).
[a,b] n
Definition5.2. Forg ∈ B[a,b]andλ ≥ 0,wesaythattheequation(−L ,λ,g,0,0)
[a,b]
is well-posed in the generalized sense if it has a unique generalized solution according
to Definition 5.1.
8 M.E. Hern´andez-Herna´ndez, V.N. Kolokoltsov
Theorem 5.1. (Well-posedness) Let ν be a function defined in terms of two
functions ν and ν via the equalities in (4.1). Let λ ≥ 0 and assume that the as-
+ −
sumptions of Theorem 4.1 hold. Let Rˆ denote the resolvent operator (or the potential
λ
operator if λ = 0) of the process Xˆ .
x
(i) If g ∈ C [a,b] and Rˆ g (x) = 0 for x ∈ {a,b}, then there exists a unique
0 λ
solution in the doma(cid:16)in of(cid:17)the generator, u∈ C0[a,b], to the two-sided RL type
[a,b] [a,b]
equation (−L ,λ,g,0,0) given by u(x) = R g(x), where R denotes the
[a,b] λ λ
[a,b]
resolvent operator (or potential operator if λ = 0) of the process X .
x
(ii) For any g ∈ B[a,b], the equation (−L ,λ,g,0,0) has a unique generalized
[a,b]
solution u∈ C [a,b] given by
0
τ(a,b)(x)
u(x) = E e−λtg(X (t))dt , (5.1)
x
"Z0 #
where τ (x) denotes the first exit time from the interval (a,b) of the under-
(a,b)
lying process X generated by the operator (4.5).
x
(iii) The solution in (5.1) depends continuously on the function g.
P r o o f. (i) Theorem 4.1 implies that (−L ,Dˆ ) generates a Feller process
[a,b]∗ ∗
Xˆ and a strongly continuous semigroup on C[a,b]. Then, the resolvent equation
−L u = λu − g has a unique solution u ∈ Dˆ given by the resolvent operator
[a,b]∗ ∗
Rˆ g for λ > 0 and for any g ∈ C[a,b] [11, Theorem 1.1]. In particular, the latter state-
λ
ment holds for g ∈ C [a,b] such that Rˆ g (x) = 0 for x ∈ {a,b}. Further, Theorem
0 λ
4.2 implies that Rˆ g = R[a,b]g, so tha(cid:16)t −L(cid:17) u = −L u. Hence, u is a solution to
λ λ [a,b]∗ [a,b]
(−L ,g,λ,0,0) belonging to Dkill, as required.
[a,b] [a,b]
[a,b] [a,b]
Since τ (x) := inf{t ≥ 0 : X (t) ∈/ (a,b)} is the lifetime of the process X ,
(a,b) x x
[a,b]
the definition of R and Fubini’s theorem imply
λ
R[a,b]g(x) = E τ(a,b)(x)e−λtg X[a,b](t) dt , (5.2)
λ x
"Z0 (cid:16) (cid:17) #
[a,b]
yielding (5.1) as the paths of X and X coincide before the time τ (x). If λ = 0,
x x (a,b)
then setting λ = 0 in (5.2) implies (as τ (x) has a finite expectation) that
(a,b)
[a,b]
||R g|| ≤ sup E τ (x) < +∞.
0 (a,b)
x∈[a,b]
(cid:2) (cid:3)
[a,b]
Therefore, the potential operator R g provides the unique solution for λ = 0 belong-
0
ing to the domain Dkill, [11, Theorem 1.1’].
[a,b]
(ii) Take g ∈ B[a,b] and any sequence {g } satisfying Definition 5.1. Fubini’s
n
theoremandthedominatedconvergencetheoremappliedto(5.2)implytheconvergence
ON THE SOLUTION OF TWO-SIDED FRACTIONAL ... 9
[a,b]
of lim R g (x) =: u(x), which in turn implies that u is the unique generalized
n→∞ λ n
solution to (−L ,λ,g,0,0).
[a,b]
(iii) Follows from the fact that, for any λ ≥ 0, the equality (5.1) implies
||u−u || ≤ ||g−g || sup E τ (x) , (5.3)
n n (a,b)
x∈[a,b]
(cid:2) (cid:3)
for the solutions u and u to equations (−L ,λ,g,0,0) and (−L ,λ,g , 0,0), re-
n [a,b] [a,b] n
spectively. ✷
6. Two-sided equations involving Caputo type operators
We now turn our attention to the well-posedness for the Caputo type equation
with general boundaryconditions. We will use that both operators −L and −L
[a,b]∗ [a,b]
coincide on functions h vanishing on {a,b}.
Suppose that u solves (1.1). Take any function φ ∈ Dstop satisfying φ(a) = u
[a,b]∗ a
and φ(b) = u . By Theorem 4.2 we can take, for example, φ ∈ C2[a,b] such that
b
φ′ ∈ C [a,b] with −L φ (x) = 0 for x ∈ {a,b} and φ(a) = u and φ(b) = u .
0 [a,b]∗ a b
Define w(x) := u(x)−φ(x), x ∈[a,b], then
(cid:0) (cid:1)
−L w(x) = −L w(x) = −L u(x)+L φ(x),
[a,b] [a,b]∗ [a,b]∗ [a,b]∗
as w vanishes at the boundary. Hence,
−L w(x) = λu(x)−g(x)+L φ(x),
[a,b] [a,b]∗
= λw(x)+λφ(x)−g(x)+L φ(x), (6.1)
[a,b]∗
yielding the RL type equation (−L , λ, g −L φ−λφ, 0, 0) for the function w.
[a,b] [a,b]∗
Therefore, if w is the (possibly generalized) solution to (6.1), then u = w+φ can be
considered as a generalized solution to the Caputo type equation (1.1). This motivates
the definition below.
Definition 6.1. (Solutions to Caputo type equations) Let g ∈ B[a,b] and
λ ≥ 0. A function u ∈C[a,b] is said to solve the linear equation (1.1) as (i) a solution
in the domain of the generator if u is a solution belonging to Dstop ; (ii) a generalized
[a,b]∗
solution if u can bewritten as u= φ+w, wherew is the (possibly generalized) solution
to the RL type problem
(−L , λ, g−L φ−λφ, 0, 0)
[a,b] [a,b]∗
with φ ∈ C2[a,b] satisfying that φ′ ∈ C [a,b], −L φ (x) = 0 in {a,b}, φ(a) = u
0 [a,b]∗ a
and φ(b) = u .
b
(cid:0) (cid:1)
Definition 6.2. For g ∈ B[a,b]andλ ≥ 0. We say thatthetwo-sided linearequa-
tion (1.1) is well-posed in the generalized sense if it has a unique generalized solution
according to Definition 6.1.
10 M.E. Hern´andez-Herna´ndez, V.N. Kolokoltsov
Theorem 6.1. If a generalized solution u = w + φ exists for the Caputo type
linear equation (1.1) with w and φ as in Definition 6.1, then the solution u is unique
and independent of φ.
P r o o f. Supposethattherearetwodifferentsolutionsu forj ∈{1,2}toequation
j
(1.1). Then, u = w +φ , where w is the unique solution (possibly generalized) to
j j j j
the RL type equation (−L , λ, g −L φ −λφ , 0, 0) for some φ satisfying the
[a,b] [a,b]∗ j j j
conditions stated in Definition 6.1. Define u(x) := u (x)−u (x) for x ∈ [a,b], then
1 2
−L u(x) = −L u(x) = −L u (x)+L u (x) = λu(x).
[a,b] [a,b]∗ [a,b]∗ 1 [a,b]∗ 2
Hence, u solves the RL type equation (−L ,λ, g = 0,0,0) whose unique solution (by
[a,b]
Theorem 5.1) is u ≡ 0, which implies the uniqueness and so the independence of φ. ✷
Theorem 6.2. (Well-posedness) Let λ ≥ 0. Suppose that the assumptions of
Theorem 5.1 hold.
(i) For any g ∈ B[a,b], thetwo-sided equation (1.1)is well-posed in thegeneralized
sense. The solution admits the stochastic representation
u(x) = u E e−λτ(a,b)(x)1
a {Xx(τ(a,b)(x))≤a}
h i
τ(a,b)(x)
+ubE e−λτ(a,b)(x)1{Xx(τ(a,b)(x))≥b} +E e−λtg(Xx(t))dt , (6.2)
h i "Z0 #
where τ (x) and X are as in Theorem 5.1.
(a,b) x
(ii) If g ∈ C[a,b] satisfying that g(a) = λu , g(b) = λu and λRˆ g(x) = g(x) for
a b λ
x ∈ {a,b}, then the solution (6.2) belongs to Dstop .
[a,b]∗
(iii) Thesolution to (1.1) dependscontinuously on thefunction g andon thebound-
ary conditions {u ,u }.
a b
P r o o f. (i) Theorem 4.1 implies that the operator (−L ,Dˆ ) generates a
[a,b]∗ ∗
Feller process Xˆ on [a,b] and also ensures that τ (x) has a finite expectation. Let
(a,b)
us take any function φ ∈ C2[a,b] satisfying the conditions from Definition 6.1. Then
(by Theorem 5.1) the generalized solution w to the RL type equation (−L ,g−λφ−
[a,b]
L φ,λ,0,0) is given by w =I −II, where
[a,b]∗
τ(a,b)(x)
I := E e−λtg X[a,b](t) dt
x
"Z0 (cid:16) (cid:17) #
τ(a,b)(x)
II := E e−λt(λ+L )φ X[a,b](t) dt .
[a,b]∗ x
"Z0 (cid:16) (cid:17) #
Thus,u= w+φis(bydefinition)thegeneralizedsolutionto(1.1). Usingthemartingale
r
Y(r) := e−λrφ X[a,b]∗(r) + e−λs(λ+L )φ X[a,b]∗(s) ds
x [a,b]∗ x
(cid:16) (cid:17) Z0 (cid:16) (cid:17)