Abstract We deepen the study of the elliptic differential operator Au = αu′′+βu′ on(weighted) spaces of continuous functions on a real interval. We establish several suffi-cient conditions implying, at the same time, the generation of positive C0-semigroupssatisfying the Feller property and the existence of suitable cores. Some criteria on theregularity of the derivative, for functions satisfying Wentzell-type boundaries condi-tions, are also presented.
Keywords Weighted continuous function space · Degenerate differential operator ·Feller semigroup
1 Introduction and preliminaries
Starting from the classical papers of Feller [17,18] and motivated by the connectionswith the theory of stochastic processes, a lot of attention has been devoted to the studyof the one-dimensional operator Au = αu′′ + βu′ as generator of a positive stronglycontinuous semigroup and, then, as generator of a Markov process.
In [20], Martini highlighted that such a differential operator plays a fundamentalrole also in approximation theory. Actually, if (Ln)n is an approximation process, then,under suitable assumptions, the asymptotic formula n(Lnu − u) → Au holds true.
As it was shown by Altomare in [4], this formula and Trotter’s theorem [23] are thebasic tools to obtain a representation/approximation of the semigroup generated by Aas limit of powers of positive linear operators Ln .
Communicated by Markus Haase.
S. Milella (B)Dipartimento di Matematica, Università degli Studi di Bari “A. Moro”,Via E. Orabona, 4, 70125 Bari, Italye-mail: [email protected]
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S. Milella
This paper has originated a wide literature on the study of evolution problems asso-ciated with A, also in a multidimensional setting, by using the methods of semigrouptheory coupled with the methods of constructive approximation theory (see, e.g., [7,8]and the references quoted therein). To this end, one of the crucial points is to find acore for A, consisting of sufficiently smooth functions, namely, twice differentiablefunctions with uniformly continuous and bounded second derivative.
Beside the application of Trotter formula in constructive approximation of semi-groups, the existence of suitable cores is also essential for others problems aboutdifferential operators. For example, the description of the generator of a semigroup,the existence of stationary distributions of transition functions or the L p-uniqueness ofa diffusion operator. Recently, for multidimensional second order elliptic differentialoperators, these aspects have been investigated by Eberle [16], Albanese and Mangino[1–3] who have provided several sufficient conditions in order that the space of testfunctions is a core.
In this note we extend the theory given in [7] for the operator Au = αu′′ to the moregeneral case of one-dimensional differential operators with an, eventually unbounded,drift term. Our main aim is to present sufficient conditions, easy to check, under which,at the same time,
– A, equipped with boundary conditions of Wentzell-type, generates a positive C0-semigroup satisfying the Feller property, on a (weighted) space of continuousfunctions,
– there exists a core for A consisting of twice differentiable functions with uniformlycontinuous and bounded second derivative.
Moreover, for functions satisfying Wentzell conditions, we give some results aboutthe regularity of the first derivative on the boundary, useful to describe the maximaldomain of A in C0(J ).
Let J be a real interval, r1 = inf J and r2 = sup J . As usual, C(J ) stands for thespace of all real valued continuous functions on J , C2(J ) for its subspace of twicedifferentiable functions with continuous second derivative and Cb(J ) for the space ofall real valued functions on J which are continuous and bounded.
C0(J ) and C∗(J ) denote the spaces of functions f ∈ Cb(J ) such thatlimx→ri f (x) = 0 and, respectively, limx→ri f (x) ∈ R if ri /∈ J .
If w is a weight on J , i.e., w ∈ Cb(J ) and it is strictly positive on J , Cw0 (J ) denotes
the Banach space of functions f ∈ C(J ) such that w f ∈ C0(J ), naturally endowedwith the weighted norm ‖ f ‖w = ‖w f ‖∞. It is obvious that, whenever w ∈ C0(J ),one has Cb(J ) ⊂ Cw
0 (J ).As in [11], we define the following.
Definition 1.1 If E is the space C0(J )or, providedw ∈ C0(J ), E is an arbitrary closedsublattice of Cb(J ) which is dense in Cw
0 (J ), we say that a positive C0-semigroup(T (t))t≥0 on Cw
0 (J ) satisfies the Feller property with respect to E if:
(F1) T (t)(E) ⊂ E for every t ≥ 0,(F2)
(T (t)|E
)t≥0 is strongly continuous on (E, ‖ · ‖∞),
(F3) each T (t) is positive and contractive on (E, ‖ · ‖∞).
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Cores for generators
If a positive C0-semigroup (T (t))t≥0 on Cw0 (J ) satisfies the Feller property with
respect to C0(J ) or C∗(J ), then it is a transition semigroup. Actually, if (F1) − (F3)
hold true for E = C0(J ) or E = C∗(J ), then, there exists a right-continuous Markovprocess (�,�, (Px )x∈ J̃ , (Zt )0≤t≤+∞) with state space J̃ and whose paths have left-hand limits almost surely, such that
T (t)( f )(x) =∫
J
f d PxZt
for every f ∈ Cw0 (J ), t ≥ 0 and x ∈ J . Here J̃ denotes the one-point or the two
points compactification of J .
2 Generation of semigroup satisfying the Feller property
Given α, β ∈ C(oJ ), α strictly positive, let us consider the differential operator
Au(x):=(αu′′ + βu′)(x),
(u ∈ C2(oJ ) and x ∈ o
J ) and the subspaces of Cw0 (J )
D0(A):={
u ∈ C0(J ) ∩ C2(oJ ) | lim
x→r(αu′′ + βu′)(x) = 0, for r = 0,+∞
},
D∗(A):={
u ∈ C∗(J ) ∩ C2(oJ ) | lim
x→r(αu′′ + βu′)(x) = 0 for r = 0,+∞
},
Dw0 (A):=
{u ∈ Cw
0 (J ) ∩ C2(oJ ) | lim
x→rw(x)(αu′′ + βu′)(x) = 0, for r = 0,+∞
}.
where w ∈ C2(◦J ) is a weight function on J such that
ω:= sup◦J
wA1
w< +∞. (2.1)
If ri ∈ J , for every u ∈ Dw0 (A), the function Au can be continuously extended in ri .
The three realizations of A
A : D0(A) ⊂ C0(J ) → C0(J ),
A : D∗(A) ⊂ C∗(J ) → C∗(J ),
A : Dw0 (A) ⊂ Cw
0 (J ) → Cw0 (J ),
are well defined and closed (see [5, Sect. 2], [15]).In the following, we consider the case where r1 = 0 and r2 = +∞, although all the
results may be easily extended to the general case where J is an arbitrary real interval.
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S. Milella
Let us assume that
(H0) C := sup]0,a]
xβ(x)
α(x)< 1 for some a > 0;
and one of the following conditions hold true
(H1∞) (i) α(x) = O(x2 log x) as x → +∞,
(ii) the function x → ∫ x1
β(s)α(s) ds is locally bounded at +∞;
(H2∞) (i) α(x) = O(x2 log x) as x → +∞,
(ii) inf[b,+∞[ β ≥ 0 for some b > 0;
(H3∞) (i)1
α/∈ L1([1,+∞[),
(ii) sup[b,+∞[
β < 0 for some b > 0,
(iii) there exists a differentiable function � : [b,+∞[→ R such that
inf[b,+∞[ � > 0,1
�/∈ L1([b,+∞[) and lim
x→+∞�2
�′α − �β> 0.
(H4∞) (i) sup[b,+∞[
β ≤ 0 for some b > 0,
(ii) there exists a differentiable function � : [b,+∞[→ R such that
limx→+∞ �(x) = +∞,
1
�/∈ L1([b,+∞[) and lim
x→+∞�2
�′α − �β> 0.
Theorem 2.1 The operator (A, D∗(A)) generates a Feller semigroup (T∗(t))t≥0 onC∗(J ).
Proof By using the well known generation theorem of Clement and Timmermans [15,Theorem 2] we shall prove that
W ∈ L1(0, 1) and
+∞∫
1
W (x)
+∞∫
x
1
α(s)W (s)dsdx = +∞.
where
W (x):= exp
⎛
⎝−x∫
1
β(s)
α(s)ds
⎞
⎠ (x > 0).
Indeed, condition (H0) implies that W (x) = O(x−C ) if x → 0 and hence W ∈L1(0, 1).
If (H1∞) holds true, then there exist m, M, K > 0 such that m ≤ W (x) ≤ M andα(x) ≤ K x2 log x for large x . Hence,
W (x)
+∞∫
x
ds
α(s)W (t)≥ m
M K
+∞∫
x
ds
s2 log s≥ m
2M K
1
x log x
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Cores for generators
so that +∞∫
1
W (x)
+∞∫
x
1
α(s)W (s)dsdx = +∞. (2.2)
If (H2∞) holds true, then W is decreasing at infinity. Accordingly,
W (x)
+∞∫
x
ds
α(s)W (t)≥ 1
K
+∞∫
x
ds
s2 log s
and we get (2.2) again.Finally, suppose that (H3∞) or (H4∞) is fulfilled. Then,
limx→+∞ �(x)W (x) = +∞.
When1
αW/∈ L1(b,+∞), equality (2.2) clearly follows. Otherwise, if
1
αW∈
L1(b,+∞), then
limx→+∞
+∞∫
x
ds
α(s)W (t)= 0,
and, by applying De L’Hôpital rule,
limx→+∞ �(x)W (x)
+∞∫
x
ds
α(s)W (t)= lim
x→+∞�(x)
(�′α − �β)(x)= L > 0.
Therefore, we get
W (x)
+∞∫
x
ds
α(s)W (t)≥ L
2�(x)for large x
and (2.2) holds true. �Theorem 2.2 The operator (A, D0(A)) generates a Feller semigroup (T0(t))t≥0 onC0(J ). Moreover T∗(t)|C0(J ) = T0(t), for every t ≥ 0.
Proof The assertion follows by generation property of (A, D∗(A)) and [13, Chap. 0,pp. 386–388].Actually, D0(A) is dense in C0(J ), since it contains the subalgebra H of all the
functions u ∈ C0(J ) ∩ C2(◦J ) which vanish at +∞, and H is dense in C0(J ).
The operator (A, D0(A)) satisfies the positive maximum principle as well as(A, D∗(A)).
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S. Milella
Finally, for every λ > 0 and f ∈ C0(J ) ⊂ C∗(J ), there exists u ∈ D(A) such that
λu− Au = f. The function u = 1
λ( f + Au) ∈ C0(J ) and so we get (λI − A)D0(A) =
C0(J ). �
We come now to consider the weighted case. The following result, in the moregeneral setting of a locally compact noncompact Hausdorff space X, is proved in [11,Corollary 1].
Theorem 2.3 Let E be the space C0(X) or, provided w ∈ C0(X), an arbitrary closedsublattice of Cb(X) which is dense in Cw
0 (X). Moreover, let A : D(A) ⊂ Cw0 (X) −→
Cw0 (X) be a linear operator and ω ∈ R. Assume that
(i) (A, D(A)) satisfies the generalized positive maximum principle with respect toω, i.e.,
Au(x0) ≤ ωu(x0),
for every u ∈ D(A) and x0 ∈ X such that supx∈X
(wu)(x) = (wu)(x0) > 0;
(ii) there exists a subspace D0 of D(A) ∩ E such that A(D0) ⊂ E,(
A|D0 , D0)
isclosable in E and its closure generates a Feller semigroup on E.
Then, the operator (A, D(A)) is closable in Cw0 (X) and its closure generates a positive
C0-semigroup (T (t))t≥0 on Cw0 (X) such that ‖T (t)‖ ≤ eωt , for every t ≥ 0.
Moreover
(1) (T (t))t≥0 satisfies the Feller property with respect to E;(2)
(T (t)|E
)t≥0 is generated by the closure of
(A|D0 , D0
).
Accordingly, we obtain the following result.
Theorem 2.4 The operator (A, Dw0 (A)) generates a positive C0-semigroup
(Tw(t))t≥0 on Cw0 (J ) such that
‖Tw(t)‖ ≤ eωt for every t ≥ 0,
where ω is defined in (2.1). Moreover
(1) (Tw(t))t≥0 satisfies the Feller property with respect to C0(J ) and Tw(t)|C0(J ) =T0(t), for every t ≥ 0.
(2) If w ∈ C0(J ), then (Tw(t))t≥0 satisfies the Feller property with respect to C∗(J )
and Tw(t)|C∗(J ) = T∗(t), for every t ≥ 0.
Proof Thanks to condition (2.1), the operator (A, Dw0 (A)) satisfies the generalized
positive maximum principle with respect to ω.Indeed, for every u ∈ D(A) and x0 ∈ J such that sup(wu)(x) = (wu)(x0) > 0,
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Cores for generators
we clearly have Au(x0) = 0, if x0 ∈ J\ ◦J . On the other hand, if x0 ∈ ◦
J , then(wu)′(x0) = 0 and (wu)′′(x0) ≤ 0, whence
Au(x0) = α(x0)
(1
w
)′′(x0)(wu)(x0) + α(x0)
(wu)′′(x0)
w(x0)
+ β(x0)
(1
w
)′(x0)(wu)(x0)
≤ A
(1
w
)(x0)(wu)(x0) ≤ ωu(x0).
By taking D0 = C0(J ) or D0 = C∗(J ), we get the assertion. �Remark 2.5 If r1 ∈ R or r2 ∈ R, by assuming
(Hr1) C1 = sup]r1,a]
(x − r1)β(x)
α(x)< 1 or (Hr2) C2 = inf[a,r2[
(r2 − x)β(x)
α(x)> −1,
Theorems 2.1, 2.2 and 2.4 still hold.Indeed, (Hi ) implies that W (x) = O
(|ri − x |Ci)
and so W ∈ L1 on a neighborhoodof ri .
Remark 2.6 Conditions (Hi∞) are easily adaptable to r1 = −∞. Hence Theorems2.1, 2.2 and 2.4 still hold for J = R and we improve the generation results establishedin [9].
Remark 2.7 In the following cases, condition (2.1) holds true.
(1) α(x) = O(x2), β(x) = O(x) as x → +∞ and w(x):=(1 + xm)−1 (x ≥ 0), forevery m > 0.
(2) α(x) = O(x2 log x), β(x) = O(x log x) as x → +∞ and w(x):=(log(x +2))−1
(x ≥ 0).(3) sup(mα − β) < +∞ and w(x):=e−mx (x ≥ 0), for every m > 0.(4) α(x) = O(x2(1− x)2), β(x) = O(x(1− x)) as x → 0, 1 and w(x):=x p(1− x)q
(x ∈]0, 1[), for every p, q > 0.
As a direct consequence of Theorem 2.4 and Corollary 3 of [11], we obtain the nextresult on additive perturbations of A.
Theorem 2.8 Let γ ∈ Cb(J ), γ ≤ 0. The operator (A + γ I, Dw0 (A)) generates a
positive C0-semigroup (Tw(t))t≥0 on Cw0 (J ) such that
‖T (t)‖ ≤ e(ω+sup γ )t for every t ≥ 0.
Moreover
(1) (Tw(t))t≥0 satisfies the Feller property with respect to C0(J ).(2) If w ∈ C0(J ) and γ ∈ C∗(J ), then (Tw(t))t≥0 satisfies the Feller property with
respect to C∗(J ).
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S. Milella
3 Cores
As the authors have already pointed out in [7], under generation assumptions, know-ing a core for (A, D0(A)) or (A, D∗(A)) suffices to get a core for the extension(A, Dw
0 (A)), regardless of the weight w.
For the proof of the following proposition, we need to recall that, if (B, D(B)) is aclosed operator on a Banach space E and if its resolvent set ρ(B) is non empty, thena space D is a core for (B, D(B)) if and only if (λI − B)D is dense in E for one/allλ ∈ ρ(B) (here I stands for the identity operator).
Proposition 3.1 Assume that (A, Dw0 (A)) and (A, D∗(A)) generate strongly contin-
uous semigroups in Cw0 (J ) and C∗(J ), respectively. Let D be a core for the operator
(A, D0(A)) and denote by D̃ the linear subspace generated by D ∪ S, where S is the
space of all functions f ∈ C2(◦J ) which are constant at infinity. Then,
(1) D is a core for the operator (A, Dw0 (A)).
Moreover, if J = [r1,+∞[ or J =] − ∞, r2],(2) D̃ is a core for (A, D∗(A)).
Proof Since the two realizations (A, Dw0 (A)) and (A, D0(A)) generate strongly con-
tinuous semigroups on Cw0 (J ) and on C0(J ), there exists λ ∈ R such that λI − A is
invertible from Dw0 (A) into Cw
0 (J ) and from D0(A) into C0(J ), respectively.Then (λI − A) (D) is dense in C0(J ) and, as a consequence, it is dense in Cw
0 (J ),that is, D is a core for the operator (A, Dw
0 (A)).Now let f ∈ D∗(A) and ε > 0. Set f (+∞):= lim
x→+∞ f (x), the function f̃ := f −f (+∞) belongs to D0(A) and so, there exists h ∈ D such that ‖ f̃ − h‖A ≤ ε.
Whence h̃:= f (+∞) + h ∈ D̃ and ‖ f − h̃‖A = ‖ f̃ − h‖A ≤ ε. �Proposition 3.2 Assume that (A, Dw
0 (A)) and (A, D∗(A)) generate strongly contin-uous semigroups in Cw
0 (J ) and C∗(J ), respectively. Let D be a core for the operator(A, D∗(A)) and w ∈ C0(J )
Then, D is a core for the operator (A, Dw0 (A)).
Below, we are interested in proving that the linear subspaces
D1:=K 2(J ) ∩ D0(A),
D2:={
u ∈ K 2(J ) | u is constant near 0}
are cores for (A, D0(A)), where K 2(J ) denotes the space of all the functions f ∈C2(J ) with compact support.
Theorem 3.3 Let J = [0,+∞[. Assume that there exist two positive and increasingfunctions p and q defined on ]0, κ] such that
(i) xq(x) = O(p(x)) as x → 0+;(ii) x2 ≤ C1 p(x) ≤ α(x) ≤ C2 p(x) as x → 0+, for some C1, C2 > 0;
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Cores for generators
(iii) β(x) = O(q(x)) as x → 0+;(iv) α(x) = O(x2 log x) and β(x) = O(x log x) as x → +∞;(v) lim
x→0+ q(x)u′(x) = 0 and limx→+∞
α(x)u′(x)
x log x= 0, for every u ∈ D0(A).
Then D1 is a core for the operator (A, D0(A)).
Proof First note that, conditions (i), (ii) and (iii) implies that, for every u ∈ D0(A)
limx→0+ p(x)u′′(x) = lim
x→0+ x2u′′(x) = limx→0+ xq(x)u′′(x) = 0,
By limx→0+ x2u′′(x) = 0 we also deduce lim
x→0+ xu′(x) = 0.
Let u ∈ D0(A) and ε > 0. Then, there exist a < κ and b > 3 such that, for everyx ∈]0, a],
Hence, D1 is dense in D0(A) with respect to the graph norm, i.e., D1 is a core forthe operator (A, D0(A)). �Theorem 3.4 Let J = [0,+∞[. Assume that there exists a strictly positive andincreasing function q defined on ]0, κ] such that
(i) q(x) = O(x) as x → 0+;(ii) α(x) = O(q(x)2) and β(x) = O(q(x)) as x → 0+;
(iii) α(x) = O(x2 log x) and β(x) = O(x log x) as x → +∞;(iv) lim
x→0+ q(x)u′(x) = 0 and limx→+∞
α(x)u′(x)
x log x= 0, for every u ∈ D0(A).
Then D2 is a core for the operator (A, D0(A)).
Proof Let u ∈ D0(A) and ε > 0. Without losing generality, we may assume thatu(0) = 0. So, there exists a < κ such that, for every x ∈]0, a]
In a similar way, we obtain the following results.
Theorem 3.5 Let J =]0,+∞[. Assume that there exists a strictly positive andincreasing function q defined on ]0, κ] such that
(i) q(x) = O(x) as x → 0+;(ii) α(x) = O(q(x)2) and β(x) = O(q(x)) as x → 0+;
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S. Milella
(iii) α(x) = O(x2 log x) and β(x) = O(x log x) as x → +∞;(iv) lim
x→0+ q(x)u′(x) = 0 and limx→+∞
α(x)u′(x)
x log x= 0, for every u ∈ D0(A).
Then K 2(J ) is a core for the operator (A, D0(A)).
Theorem 3.6 Let J =]0, 1[. Assume that there exist two positive and increasingfunctions p0 and q0 defined on ]0, κ] and two positive and decreasing functions p1and q1 defined on [ν, 1[ such that
(i) xq0(x) = O(p0(x)) as x → 0+ and (1 − x)q1(x) = O(p1(x)) as x → 1−(ii) x2 ≤ C1 p0(x) ≤ α(x) ≤ C2 p0(x) as x → 0+ and (1 − x)2C3 p1(x) ≤ α(x) ≤
C4 p1(x) as x → 1−, for some C1, C2, C3, C4 > 0;(iii) β(x) = O(q0(x)) as x → 0+ and β(x) = O(q1(x)) as x → 1−,
(iv) limx→0+ q0(x)u′(x) = 0 and lim
x→1− q1(x)u′(x) = 0, for every u ∈ D∗(A).
Then C2([0, 1]) ∩ D∗(A) is a core for the operator (A, D∗(A)).
Theorem 3.7 Let J =]0, 1[. Assume that there exist a strictly positive and increasingfunction q0 defined on ]0, κ] and a strictly positive and decreasing function q1 definedon [ν, 1[ such that
(i) q0(x) = O(x) as x → 0+ and q1(x) = O(1 − x) as x → 1−;(ii) α(x) = O(q0(x)2) and β(x) = O(q0(x)) as x → 0+;
(iii) α(x) = O(q1(x)2) and β(x) = O(q1(x)) as x → 1−;(iv) lim
x→0+ q0(x)u′(x) = 0 and limx→1− q1(x)u′(x) = 0, for every u ∈ D0(A).
Then K 2(]0, 1[) is a core for the operator (A, D0(A)).
The more difficult assumptions to inspect in the previous statements, are those aboutthe control of the first derivative at the endpoints of J, for an arbitrary u in D0(A) orin D∗(A). To this end, by generalizing similar results in [7,9,10], we establish somecriteria useful to obtain information on the derivative u′ on the boundary of J, whenu satisfies Wentzell-type conditions.
Given two functions α, β ∈ C([a, b[), α strictly positive, let again W (x) :=exp
(−∫ x
a
β(s)
α(s)ds
), for every x ∈ [a, b[. Here b is eventually +∞.
Proposition 3.8 Let ϕ : [a, b[→ R be a measurable, strictly positive function and
assume that1
αW∈ L1([a, b[). Then, in each of the following cases
(1) limx→b
ϕ(x)W (x) = 0;(2) (i) lim
x→bϕ(x)W (x) ∈]0,+∞[,
(ii)1
ϕ/∈ L1([a, b[);
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Cores for generators
(3) (i) limx→b
ϕ(x)W (x) = +∞,
(ii)1
ϕ/∈ L1([a, b[),
(iii) ϕ is differentiable and the functionϕ2
ϕβ − ϕ′αis well defined and bounded,
we have that
limx→b
ϕ(x)u′(x) = 0
for every u ∈ Cb([a, b[) ∩ C2([a, b[) such that limx→b
(αu′′ + βu′)(x) = 0.
Proof First, note that, for every x ∈ [a, b[
u′(x)
W (x)= u′(a) +
x∫
a
Au(s)
α(s)W (s)ds
then, if the function1
αW∈ L1([a, b[), there exists
limx→b
u′(x)
W (x)= l ∈ R.
Hence, in case (1), we easily have
limx→b
ϕ(x)u′(x) = limx→b
ϕ(x)W (x)u′(x)
W (x)= 0.
In case (2) or (3), by (ii), W /∈ L1([a, b[) and so l = 0.
Indeed, if l �= 0, then u′ /∈ L1([a, b[) and u diverges in b, while u ∈ Cb([a, b[).Therefore, if lim
x→bϕ(x)W (x) is finite, we get directly lim
x→bϕ(x)u′(x) = 0. Other-
wise, if limx→b
ϕ(x)W (x) = +∞, by De L’Hôpital rule and (iii), we obtain
limx→b
ϕ(x)u′(x)= limx→b
ϕ(x)W (x)
⎛
⎝u′(a)+x∫
a
Au(s)
α(s)W (s)ds
⎞
⎠= limx→b
ϕ2(x)Au(x)
(ϕβ−ϕ′α)(x)=0.
�Proposition 3.9 Let ϕ : [a, b[→ R be a strictly positive function and assume that
(i) limx→b
ϕ(x)W (x) = 0,
(ii) ϕ is differentiable and the functionϕ2
ϕβ − ϕ′αis well defined and bounded.
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S. Milella
Then,
limx→b
ϕ(x)u′(x) = 0
for every u ∈ Cb([a, b[) ∩ C2([a, b[) such that limx→b
(αu′′ + βu′)(x) = 0.
Proof As we saw before, for every x ∈ [a, b[
|ϕ(x)u′(x)| ≤ ϕ(x)W (x)
⎛
⎝u′(a) +x∫
a
|Au(s)|α(s)W (s)
ds
⎞
⎠ .
Then, if|Au|αW
∈ L1(a, b), the assertion easily follows. Whereas, if|Au|αW
/∈ L1(a, b)
limx→b
ϕ(x)W (x)
⎛
⎝u′(a) +x∫
a
|Au(s)|α(s)W (s)
ds
⎞
⎠ = limx→b
ϕ2(x)|Au(x)|(ϕβ − ϕ′α)(x)
= 0.
�Proposition 3.10 Let α ∈ C2([a,+∞[), β ∈ C1([a,+∞[) and ϕ : [a,+∞[→ R
be a positive function. Moreover, assume that
(i)ϕ(x)
α(x)= O
(1
x
)as x → +∞,
(ii)∫ x
a |α′′(s) − β ′(s)|ds = O(x) as x → +∞.
Then,
limx→+∞ ϕ(x)u′(x) = 0
for every u ∈ C0([a,+∞[) ∩ C2([a,+∞[) such that limx→+∞(αu′′ + βu′)(x) = 0.
Proof Note that condition (ii) gives α′ − β = O(x) at infinity.Let ε > 0. Then, there exists δ ≥ a such that, for every x ≥ δ
|Au(x)| ≤ ε and |u(x)| ≤ ε.
Accordingly, since
x∫
δ
Au(s)ds = (αu′ + βu − α′u)(x) +x∫
δ
(α′′ − β ′)(s)u(s)ds − (αu′ + βu)(δ),
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Cores for generators
one has
|ϕ(x)u′(x)| ≤ ϕ(x)
α(x)
⎛
⎝ε(x − δ) + ε|α′(x) − β(x)| + ε
x∫
a
|α′′(s) − β ′(s)|ds + ε
⎞
⎠
which gives |ϕ(x)u′(x)| ≤ ε for large x . �By a slight modification of the arguments used in the previous proof, we get the
following proposition, where b ∈ R.
Proposition 3.11 Let α ∈ C2([a, b[), β ∈ C1([a, b[) and ϕ : [a, b[→ R be a positivefunction. Moreover, assume that
(i)ϕ(x)
α(x)= O (1) as x → b,
(ii) α′′ − β ′ ∈ L1(a, b).
Then,
limx→b
ϕ(x)u′(x) = 0
for every u ∈ C0([a, b[) ∩ C2([a, b[) such that limx→b
(αu′′ + βu′)(x) = 0.
Remark 3.12 Propositions 3.8–3.11 can be easily reformulated by replacing the inter-val [a, b[ with ]a, b] (eventually unbounded).
4 Some good situations
In the following propositions we analyze some cases where, at the same time, sufficientconditions for generation and existence of core hold.
Proposition 4.1 Suppose that, as x → 0+
(i) C1x p ≤ α(x) ≤ C2x p,
(ii) β(x) = O(xq),
for some C1, C2 > 0, 0 ≤ p ≤ 2, q > max{0, p − 1}.Then assumption (H0) and (i–iii) of Theorem 3.3 hold true, moreover
limx→0+ xqu′(x) = 0
for every u ∈ Cb(]0, b]) ∩ C2(]0, b]) such that limx→0+(αu′′ + βu′)(x) = 0.
Proof We clearly havexβ(x)
α(x)= O(xq−p+1), so there exists a > 0 such that
C := sup]0,a]
xβ(x)
α(x)< min{1, q}.
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Accordingly, W (x) = O(x−C ) and limx→0
xq W (x) = 0. Moreover,
x2q
|xqβ(x) − qxq−1α(x)| = xq+1
α(x)
∣∣∣∣xβ(x)
α(x)−q
∣∣∣∣
≤ xq+1
C1x p
(q− xβ(x)
α(x)
) ≤ xq−p+1
C1 (q−C),
so that, by Proposition 3.9, limx→0+ xqu′(x) = 0 and core-conditions in 0 hold true with
p(x) = x p and q(x) = xq . �Proposition 4.2 Suppose that, as x → 0+
(i) C1x p ≤ α(x) ≤ C2x p,
(ii) D1 ≤ β(x) ≤ D2,
for some C1, C2 > 0, D1, D2 < 0 and 0 ≤ p < 1.Then assumption (H0) and (i–iii) of Theorem 3.3 hold true, moreover
limx→0+ u′(x) = 0
for every u ∈ Cb(]0, b]) ∩ C2(]0, b]) such that limx→0+(αu′′ + βu′)(x) = 0.
Proof In this case, we clearly have C = sup]0,a]
xβ(x)
α(x)< 0.
Moreover, for q(x) = 1, we have that
q(x)W (x) = O(x−C ) andq2
|(qβ − q ′α)| = 1
|β| = O(1) as x → 0.
Then, by Proposition 3.9, limx→0
u′(x) = 0 and core-conditions in 0 hold true with
p(x) = x p and q(x) = 1. �Proposition 4.3 Suppose that, as x → +∞(i) C1x p logs x ≤ α(x) ≤ C2x p logs x,
(ii) β(x) = O(xq logr x),
for some C1, C2 > 0, 1 < p < 2, q < p − 1, 0 ≤ r, s ≤ 1.Then assumptions (H1∞), (iv) of Theorem 3.3 and (iii) of Theorem 3.4 hold true,moreover
limx→+∞
α(x)u′(x)
x log x= lim
x→+∞ x p−1u′(x) = 0,
for every u ∈ Cb([a,+∞[) ∩ C2([a,+∞[) such that limx→+∞(αu′′ + βu′)(x) = 0.
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Cores for generators
Proof Indeed,β(x)
α(x)= O(xq−p logr−s x), so condition (H1∞) is satisfied. Moreover
1
α(x)W (x)= O(α(x)−1) and
α(x)
x log x= O(x p−1).
By setting ϕ(x) = x p−1, we have that limx→+∞ ϕ(x)W (x) = +∞ and
ϕ2(x)
|(ϕβ − ϕ′α)(x)| = x2p−2
|x p−1β(x) − (p − 1)x p−2α(x)|= x p
|xβ(x) − (p − 1)α(x)| = O(1),
therefore, by Proposition 3.8, limx→+∞
α(x)u′(x)
x log x= lim
x→+∞ ϕ(x)u′(x) = 0 for every
u ∈ D0(A). �Proposition 4.4 Suppose that, as x → +∞(i) C1x p ≤ α(x) ≤ C2x p,
(ii) β(x) = O(xq log x),
for some C1, C2 > 0, p ≤ 1 and q < p − 1.Then assumptions (H1∞), (iv) of Theorem 3.3 and (iii) of Theorem 3.4 hold true,moreover
limx→+∞
α(x)u′(x)
x log x= lim
x→+∞u′(x)
log x= 0,
for every u ∈ Cb([a,+∞[) ∩ C2([a,+∞[) such that limx→+∞(αu′′ + βu′)(x) = 0.
Proof Indeed,β(x)
α(x)= O(xq−p log x), so condition (H1∞) is satisfied. Moreover
α(x)
x log x= O
(log−1 x
).
By setting ϕ(x) = log−1 x , we have that limx→+∞ ϕ(x)W (x) = 0 and
ϕ2(x)
|(ϕβ − ϕ′α)(x)| = 1
|x log xβ(x) + α(x)| = O(1),
therefore, by Proposition 3.9, limx→+∞
α(x)u′(x)
x log x= lim
x→+∞ ϕ(x)u′(x) = 0 for every
u ∈ D0(A). �
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Proposition 4.5 Suppose that, as x → +∞(i) C1x2 logs x ≤ α(x) ≤ C2x2 logs x,
(ii) limx→+∞ β(x) = b ∈] − ∞, 0[,
for some C1, C2 > 0 and 0 ≤ s ≤ 1.Then assumptions (H4∞), (iv) of Theorem 3.3 and (iii) of Theorem 3.4 hold true,moreover
limx→+∞
α(x)u′(x)
x log x= lim
x→+∞ xu′(x) = 0,
for every u ∈ Cb([a,+∞[) ∩ C2([a,+∞[) such that limx→+∞(αu′′ + βu′)(x) = 0.
Proof Indeed, assumption (H4∞) is simply satisfied with � = 1 . On the other hand,by taking ϕ(x) = x , we obtain lim
x→+∞ xW (x) = +∞ and
ϕ2(x)
|(ϕβ − ϕ′α)(x)| = x2
α(x) − xβ(x)≤ x2
C1x2 logs x − xβ(x)for large x .
So, the last part of the assertion follows from Proposition 3.8-(3). �Proposition 4.6 Suppose that
(i) α ∈ C2([a,+∞[) and β ∈ C1([a,+∞[),(ii) α′′ and β ′are bounded
(iii) inf[a,+∞[ β ≥ 0.
Then assumptions (H2∞), (iv) of Theorem 3.3 and (iii) Theorem 3.4 hold true, moreover
limx→+∞
α(x)u′(x)
x= 0
for every u ∈ C0([a,+∞[) ∩ C2([a,+∞[) such that limx→+∞(αu′′ + βu′)(x) = 0.
Proof (H2∞) is clearly satisfied. The limit on u′ follows from Proposition 3.10. �Note that, by combining the assumptions of the previous propositions, we get
D0(A) ={
u ∈ C0(J ) ∩ C2(oJ ) | αu′′, βu′ ∈ C0(J )
},
D∗(A) ={
u ∈ C∗(J ) ∩ C2(oJ ) | αu′′, βu′ ∈ C0(J )
},
The following examples demonstrate how our results include several others knownresults.
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Cores for generators
Example 4.7 [12] Let J =]0, 1[, a > 0, b ∈ C([0, 1]) and consider the differentialoperator
Au(x) = ax(1 − x)u′(x) + b(x)u′(x) (0 < x < 1).
This operator is, for example, associated to the Wright-Fisher equation, which mod-els evolution problems, occurring in demography or in population genetics, as themigration of alleles in the genome.
If b(0) < a and b(1) > −a, then (H0) and (H1) hold true. So, (A, D0(A)) and(A, D∗((A)) generate Feller semigroups on C0(]0, 1[) and C∗(]0, 1[), respectively.
For the rest, if it happens that
(i) b(0) < 0 or b(x) = O(x) as x → 0+,
(ii) b(1) > 0 or b(x) = O(1 − x) as x → 1−,
then, by Theorem 3.6 it follows that C2([0, 1]) ∩ D∗(A) is a core for (A, D∗(A)).Actually, when (i) holds, by applying Proposition 3.9 with ϕ(x) = 1 or ϕ(x) = x , weget
limx→0+ u′(x) = 0 or lim
x→0+ xu′(x) = 0, respectively.
In the same manner, if (ii) holds, we obtain
limx→1− u′(x) = 0 or lim
x→1−(1 − x)u′(x) = 0, respectively.
Example 4.8 [22] Let J =]0, 1[, p, q ∈ R and consider the differential operator
Au(x) = x(1 − x)
2
2
u′(x) + q(1 − x)(1 + px)
2u′(x) (0 < x < 1).
If q < 1 and pq + q > −1, then (H0) and (H1) hold true. So, (A, D0(A)) and(A, D∗((A)) generate Feller semigroups on C0(]0, 1[) and C∗(]0, 1[), respectively.
For weights as w(x) = (1 − x)m , with m ≥ 1, the operator (A, Dw0 (A)) also
generates a positive strongly continuous semigroup on Cw0 ([0, 1[) satisfying the Feller
property with respect to C0(]0, 1[) and C∗(]0, 1[).Moreover, by Theorem 3.6 it follows that C2([0, 1])∩ D∗(A) is a core for (A, D∗(A))
and (A, Dw0 ((A)). In particular, taking Proposition 3.9 into account with ϕ(x) = x
near 0 or ϕ(x) = 1 near 1, we obtain
limx→0+ xu′(x) = 0 and lim
x→1−(1 − x)u′(x) = 0.
Example 4.9 Let J =]0, 1[. More in general, we may consider the operator
Au(x) = α(x)u′′(x) + β(x)u′(x) (0 < x < 1)
where α, β ∈ C(]0, 1[) are such that
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S. Milella
(i) 0 < α(x) ≤ x(1 − x)
2for every 0 < x < 1,
(ii) β(x) < 0 on ]0, δ] orβ(x)
α(x)= O(1) as x → 0+,
(iii) β(x) > 0 on [υ, 1[ orβ(x)
α(x)= O(1) as x → 1−, for some δ, υ ∈]0, 1[.
Assumption (i) is natural when the operator A is associated, by a Voronovkaja-typeformula, to a sequence of Bernstein–Schnabl operators (see, e.g., [8] and the referencestherein).
In this case, one may not directly apply Theorem 3.6 or Theorem 3.7, since wedon’t know if the functions pi and qi there exist. Nevertheless, by reasoning as in [7,Sect. 4.1], we get that that C2([0, 1]) ∩ D∗(A) is a core for (A, D∗(A)).
Indeed, set λ(x) = 2α(x)
x(1 − x)(0 < x < 1), consider the operator
Bu(x) = x(1 − x)
2u′′(x) + β(x)
λ(x)u′(x),
defined on the domain D∗(B):={
u ∈ C∗(]0, 1[) ∩ C2(]0, 1[) | limx→0,1
Bu(x) = 0
}.
By (ii) and (iii) we have that conditions (H0) and (H1) are satisfied and, so, boththe operators (A, D∗(A)) and (B, D∗(B)) are generators of Feller semigroups onC∗(]0, 1[).
Moreover A = λB, (λB, D∗(B)) pre-generates a Feller semigroups on C∗(]0, 1[)and its closure is (A, D∗(A)) (for a proof, see Proposition 4.6 in [7]). Accordingly,D∗(B) is dense in D∗(A) for the graph-norm on A.
By Example 4.7, it follows that C2([0, 1]) ∩ D∗(B) is a core for (B, D∗(B)) andhence for (A, D∗(A)).
Remark 4.10 The method used in the previous example suggests that, in some cases,we may avoid the polynomially lower limitation on α that we require in Theorems3.3–3.7.
Example 4.11 (The Black-Scholes operator) [5,6,19] Let J = [0,+∞[ and
Au(x) = σ 2x2
2u′′(x) + μxu′(x) (x > 0)
where σ,μ > 0.
This operator, in the theory of option pricing, is associated to an equation whichmodels the no-arbitrage price of an option. In that context, the parameter σ denotesthe volatility and μ denotes the riskless interest rate.
If we assume that 2μ < σ 2, then (H0) holds true.Condition (H2∞) is obviously satisfied. Hence, the operators (A, D0(A)) and
(A, D∗((A)) generate Feller semigroups on C0([0,+∞[) and C∗([0,+∞[), respec-tively.
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Cores for generators
For polynomial weights as w(x) = (1 + xm)−1, with m ≥ 1, the oper-ator (A, Dw
0 ((A)) also generates a positive strongly continuous semigroup onCw
0 ([0,+∞[) satisfying the Feller property with respect to C0([0,+∞[) andC∗([0,+∞[).
Moreover, from Theorem 3.4 with q(x) = x , it follows that the space of func-tions u ∈ K 2([0,+∞[) which are constant near 0 is a core for (A, D0(A)) and(A, Dw
0 ((A)).Indeed, we have only to prove that
limx→0
xu′(x) = limx→+∞ xu′(x) = 0.
By simple calculation, we obtain W (x) = x− 2μ
σ2 . So, it suffices to apply Proposition3.9 and Proposition 3.10 with ϕ(x) = x .
Remark 4.12 Our results also include known results on the operators
Au(x) = xu′′(x) + β(x)u′(x) on J = [0,+∞[
which has been studied in [10], and
Au(x) = α(x)u′′(x) + β(x)u′(x) on J = R
which has been investigated in [9,21].
Acknowledgments The author wish to thank the anonymous referee for his/her accuracy in refereeingthe paper and for his/her useful comments.
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