ON A CLASS OF SECOND ORDER DIFFERENTIAL OPERATORS ON THE REAL LINE

DIFFERENTIAL OPERATORS ON THE REAL LINE

SABINA MILELLA

The present paper is concerned with degenerate elliptic operators of the form Au(x) = ^σ2₂^(x)u⁰⁰(x) + ^(σσ0₂^)(x)u⁰(x) defined on its maximal domain in some weighted space of continuous functions on the real line. We show that these op- erators are generators of positiveC0-semigroups which may be approximated in terms of integral-type operators. By considering the stochastic differential equa- tion associated with A, we also determine an explicit integral representation of such semigroups.

AMS 2000 Subject Classification: 47D06, 60H10, 41A36.

Key words: positive semigroup, integral operator, weighted continuous function space, approximation by positive operators, stochastic equation.

1. INTRODUCTION

In [6] we studied a wide class of (possibly degenerate) elliptic second order differential operators acting on weighted spaces of continuous functions on the real line. Among other things, we established some conditions un- der which these operators generate strongly continuous positive semigroups.

Moreover, we also showed that these semigroups are transition semigroups associated with some Markov processes on the real line and that they can be approximated in terms of iterates of constructively defined integral-type positive linear operators (see [5] and [6] for more details).

Along the same line, in the present paper we consider differential opera- tors of the form

Au(x) = σ²(x)

2 u⁰⁰(x) +σ(x)σ⁰(x)

2 u⁰(x), x∈R, defined on the space of all functions u ∈ C²(R) satisfying lim

x→±∞w(x)u(x) =

x→±∞lim w(x)(σ²u⁰⁰+σσ⁰u⁰)(x) = 0, whereσ is a continuously differentiable real function such that

(1.1) σ(x)>0 for everyx∈R,

REV. ROUMAINE MATH. PURES APPL.,53(2008),2–3, 189–207

(1.2) σ^(r)(x) =O(|x|^1−r) asx→ ±∞and r= 0,1 and w is a strictly positive, bounded continuous function on R.

We show that such an operator generates a positive C0-semigroup (T(t))t≥0 on C₀^w(R) =n

f ∈C(R) | lim

x→±∞w(x)f(x) = 0o

which is the transi- tion semigroup of a Markov process with state space [−∞,+∞]. Provided w is a polynomial weight, we give some shape-preserving and saturation infor- mation on the semigroup (T(t))t≥0, approximating it by means of iterates of integral-type operators.

By considering the solution of the stochastic differential equation associ- ated withA, we also obtain an explicit integral representation of the semigroup in more general weighted spaces of continuous functions and this allow us to investigate the asymptotic behavior of (T(t))t≥0.

2. THE SEMIGROUP (T_m(t))t≥0 AND ITS APPROXIMATION In the sequel we shall denote by C(R) the space of all real valued con- tinuous functions on R and with C∗(R) (resp. C0(R)) the Banach lattice of functionsf ∈C(R) such that lim

x→±∞f(x)∈R(resp. lim

x→±∞f(x) = 0), endowed with the natural order and the uniform norm k·k_∞.

The symbolC²(R) (resp. U C_b²(R)) will stand for the space of all twice continuously differentiable functions (resp. twice differentiable functions with uniformly continuous and bounded second derivative).

Moreover, we shall denote by K²(R) the subspace of all functions f ∈ C²(R) having compact support.

Ifwis a weight function, i.e.,w∈C(R) is bounded and strictly positive, we shall denote by C₀^w(R) the Banach lattice of all functions f ∈C(R) such thatwf ∈C0(R), endowed with the natural order and the weighted normk·k_w defined by kfk_w:=kwfk_∞(f ∈C₀^w(R)).

Form∈N, let wm(x) := (1 +x^2m)⁻¹, x∈R, and consider the operator A on the space

D_m(A) :=n

u∈C₀^wm(R)∩C²(R)| lim

x→±∞

(σ²u⁰⁰+σσ⁰u⁰)(x) 1 +x^2m = 0o

.

Consider also the operator Ae:=A defined on D(A) :=e n

u∈C∗(R)∩C²(R)| lim

x→±∞(σ²u⁰⁰+σσ⁰u⁰)(x) = 0o .

Theorem 2.1. For every m ≥ 1, the operator (A, D_m(A)) generates a positive C0-semigroup (Tm(t))t≥0 on C₀^wm(R) such that kT_m(t)k ≤ e^ωmt for

every t≥0, where ω_m := sup

x∈R

m(2m−1)σ(x)²x^2m−2+m(σσ⁰)(x)x^2m−1

1 +x^2m .

Moreover, the restrictions of (Tm(t))t≥0 to C0(R) and C∗(R) are Feller semi- groups whose generators are (A, D(e A)e ∩C0(R)) and(A, D(e A)), respectively.e

Proof. We shall prove the assertion by applying Theorem 3.1 in [2].

Following the notation made there, for α:= ^σ₂² and β := ^σσ₂⁰ we set ω := sup

x∈R

|α(x)(2w_m⁰ (x)²−w_m(x)w_m⁰⁰(x))−β(x)w_m(x)w_m⁰ (x)|

wm(x)² .

Moreover, with a fixed x₀∈R, for everyx∈R we set W(x) := exp

− Z x

x0

β(x) α(x)ds

= exp

− Z x

x0

σ⁰(s) σ(s)ds

= σ(x0) σ(x) . We shall prove that ω∈R and

(1)

Z +∞

x0

W(x) Z +∞

x

dtdx α(t)W(t) =

Z _x0

−∞

W(x) Z _x

−∞

dtdx

α(t)W(t) = +∞.

To this end observe that by (1.2) there exists C > 0 such that |σ(x)| ≤ C(1 +|x|) and|σ⁰(x)| ≤C for everyx∈R. Then, by simple calculations, we get ω=ω_m<+∞ and

Z +∞

x0

W(x) Z +∞

x

dsdx α(s)W(s) =

Z +∞

x0

σ(x₀) σ(x)

Z +∞

x

2

σ(s)σ(x₀)dsdx

≥ Z +∞

x0

1 C(1 +|x|)

Z +∞

x

2

C(1 +|s|)dsdx= +∞.

The second part of (1) can be similarly proved, hence the proof is com- plete.

We proceed to approximate the semigroup (Tm(t))t≥0 by means of ite- rates of integral-type operators previously introduced and studied in [5].

LetE(R) be the space of functionsf ∈C(R) such that (2.1)

Z +∞

−∞

|f(ay+b)|e^−y

2

2 dy <+∞ for everya≥0 andb∈R and consider for everyn≥1 the positive linear operatorG_ndefined by setting (2.2) G_n(f) (x) := 1

√2π Z +∞

−∞

f σ(x)

√n y+x+σ(x)σ⁰(x) 2n

e^−y

2 2 dy.

for every f ∈E(R) andx∈R.

Theorem 2.2. Let m≥2 andδ >0. Then in each of the cases

(1) σ∈C²(R\ ]−δ, δ[) and σ⁰⁰(x) =O _|x|¹

as x→ ±∞;

(2) i) lim

x→±∞

σ(x) x = 0,

ii) σ(x)−xσ⁰(x)6= 0for|x| ≥δ, and _σ(x)−xσ^σ(x)0(x) =O(1)asx→ ±∞, for every f ∈C₀^wm(R) and t≥0,

(2.3) T_m(t)f = lim

n→∞G^k(n)_n f in C₀^wm(R),

where(k(n))_n≥1is an arbitrary sequence of positive integers such thatk(n)/n→ t and (Gn)^k(n) denotes the iterate of order k(n) of Gn.

In particular, the limit holds uniformly on compact subsets of R.

Proof. Arguing as in the proof of Theorem 2.10 in [6], we deduce that the space K²(R) is a core for (A, Dm(A)).

On the other hand, for everym≥2, eachG_nmaps continuouslyC₀^wm(R) into itself and there exists Km >0, independent ofn, such that

kG_nk_Cwm

0 (R)≤1 +K_m n

(see Example 2.3.3 and Theorem 2.5 in [5]). Moreover, for everyf ∈U C_b²(R),

n→∞limn(Gn(f)−f) = σ²

2 f⁰⁰+σσ⁰

2 f⁰ inC₀^wm(R)

([5, Theorem 3.5]). Therefore, as a consequence of a theorem of Trotter ([14, Theorem 5.3]), we deduce the assertion.

Below, arguing as in Corollaries 3.3, 3.4 and 4.4 of [6], we list some qualitative information on the semigroup (T_m(t))t≥0. For M ≥ 0 and k >0 we denote by Lip (k, M) the space of all functions f ∈C(R) such that

|f(x)−f(y)| ≤M|x−y|^k for every x, y∈R.

Proposition 2.3. Under the assumptions of the previous theorem, the statements below hold.

(1) Let a∈R∪ {−∞,+∞}and suppose that σ is increasing on [a,+∞[

and decreasing on]− ∞, a]. ThenTm(t)f(x)≥f(x) for every convex function f ∈C₀^wm(R) that is increasing on[a,+∞[and decreasing on ]− ∞, a].

(2)Ifσσ⁰ is affine, thenTm(t)maps affine functions into affine functions.

(3)Ifσ is constant, then for every f ∈C₀^wm(R)the following statements are equivalent:

a)f is convex (resp. affine);

b) T_m(t)f(x)≥f(x) (resp. T_m(t)f(x) =f(x))for every x∈R; c)Tm(t)f is convex (resp. affine).

Moreover, the net (T_m(t)f)t≥0 is increasing for every convex function f ∈C₀^wm(R).

(4)If σ is constant, then Tm(t) maps increasing(resp. decreasing) func- tions into increasing (resp. decreasing) functions.

(5) If σ is constant then for every f ∈Lip (k, M)∩C₀^wm(R) and t≥0, we have Tm(t)f ∈Lip(k, M).

(6)For every f ∈C₀^wm(R) the following statements are equivalent:

a)kT_m(t)(f)−fk_m =o(t) as t→0⁺; b) there exist a, c₁, c₂∈Rsuch that

f(x) =c1

Z x a

1

σ(t)dt+c2, x∈R.

Remark 2.4. It follows from Theorem 2.8 in [1] that (T_m(t))t≥0 is the transition semigroup of a Markov process which depends only on the re- striction of (Tm(t))t≥0 to C∗(R). More precisely, there exists a Markov pro- cess (Ω, F,(P^x)_{x∈[−∞,+∞]},(Z_t)_0≤t≤+∞)with state space [−∞,+∞]and whose paths are continuous almost surely such that, for every x∈R and t≥0,

(i) P^x{Z_t= +∞}=P^x{Z_t=−∞}= 0;

(ii) the distribution P_Z^x

t of the random variable Z_t with respect to P^x pos- sesses finite moments of any order;

(iii) T_m(t)f =R

Ωf^∗(Z_t)dP^x for every m≥1 and f ∈C₀^wm(R), where f^∗ denotes the extension off to [−∞,+∞] vanishing at infinity.

In some particular cases, by using formula (2.3), we can estimate the mean value and the variance of the random variable Z_t.

Leta, b, c∈Rsuch that a >0, b²−4ac <0 andσ(x) :=√

ax²+bx+c, x∈R. By simple calculations, for everyx∈R and n≥1 we have

Gn(e1)(x) =x+σ(x)σ⁰(x)

2n =

1 + a

2n

x+ b 4n and

G_n(e₂)(x) = σ²(n)

n +

x+σ(x)σ⁰(x) 2n

2

=

1 +2a n + a²

4n²

x²+ b 2n

3 + a

2n

x+ b² 16n² + c

n.

Now, observe that for k≥2 we have G^k_n(e₁)(x) =

1 + a 2n

k

x+ b 4n

k−1

X

j=0

1 + a

2n j

= 1 + a

2n k

x− b 2a

1−

1 + a 2n

k .

Therefore, if t≥0 and (k(n))n≥1 is a sequence of positive integers such that k(n)/n→t, since Ex(Zt) =T2(t) (e1) (x) = lim

n→∞(Gn)^k(n)(e1)(x), we obtain Ex(Zt) = e^at2 x− b

2a

1−e^at2

. Moreover, it we set

c₁ := 1 + 2a n + a²

4n², c₂:= b 2n

3 + a

2n

, c₃ := b² 16n² + c

n, then we have

G^k_n(e₂)(x) =c^k₁x²+c₂

k−1

X

j=0

c^k−1−j₁ 1 + a

2n j

x

+c₂ b 4n

k−2

X

j=0 j

X

i=0

cⁱ₁ 1 + a

2n k−2−j

+c₃

k−1

X

j=0

c^j₁

=c^k₁x²+c₂ c^k₁ −

1 + a 2n

k

c₁− 1 + a

2n x

+c₂ b 4n

k−2

X

j=0

1−c^j+1₁ 1−c₁

1 + a

2n k−2−j

+c₃1−c^k₁ 1−c₁

=c^k₁x²+c2

4n² 6an+a²

c^k₁ −

1 + a 2n

k x

−c₂ b 4n

4n² 8an+a²

1 + a

2n k−2

×







 1−

1 + a 2n

−(k−1)

1− 1 + a

2n

−1 −c1

1−c^k−1₁ 1 + a

2n

−(k−1)

−3a 2n− a²

4n²

1 + a 2n

−1









−c₃ 4n²

8an+a²(1−c^k₁), whence E_x(Z_t²) = T₂(t) (e₂) (x) = lim

n→∞(G_n)^k(n)(e₂)(x) = e^2atx² + _a^b(e^2at− e^at2)x−_8a^3b2₂(e^at2 −e^at)−_8a^b2₂(e^at2 −e^2at)−_2a^c (1−e^2at), which imply

V_x(Z_t) =E_x(Z_t²)−(E_x(Z_t))² = (e^2at−e^at)x²+ b

a(e^2at−e^at)x + b²

8a²(e^2at+ e^at−2)− c

2a(1−e^2at).

3. AN INTEGRAL REPRESENTATION OF THE SEMIGROUP (T(t))t≥0

In this section we assume that the operatorA is defined on the domain D_w(A) :=n

u∈C₀^w(R)∩C²(R)| lim

x→±∞w(x)(σ²u⁰⁰+σσ⁰u⁰)(x) = 0o ,

where w∈C0(R) is a twice differentiable weight such that (3.1) ω:= sup

x∈R

σ²(x)(2(w⁰)²−ww⁰⁰)(x)−(σσ⁰)(x)(ww⁰)(x)

2w(x)² <+∞.

Then, by Theorem 3.1 in [2] again, the operator (A, D_w(A)) generates a posi- tive C0-semigroup (T(t))_t≥0 onC₀^w(R) such thatkT(t)k ≤e^ωt for everyt≥0.

By using the scheme introduced in [7] (see also [8] and [13]), we shall obtain an explicit integral representation of such a semigroup. To this aim, fundamental role is played by the stochastic differential equation

(3.2) dX_t=σ(X_t) dB_t+1

2σ(X_t)σ⁰(X_t) dt

associated with the operator A, where X₀ = x ∈ R and (B_t)t≥0 is the one- dimensional Brownian motion starting at 0.

By applying the method of Doss and Sussman (see [11], pp. 295–296), we deduce that the family (X_t^x)t≥0 of real random variables defined as (3.3) X_t^x:=ϕ⁻¹(Bt+ϕ(x)), t≥0,

is a solution of (3.2). Here, ϕ ∈ C²(R) is such that ϕ⁰ = _σ¹ and ϕ⁻¹ stands for the inverse function of ϕ.

Now, letf ∈K²(R) and consider the problem (3.4)







∂u

∂t(x, t) = σ(x)² 2

∂²u

∂x²(x, t) +σ(x)σ⁰(x) 2

∂u

∂x(x, t) x∈R, t >0

u(x,0) =f(x) x∈R.

For every t≥0 andx∈Rset

u(x, t) :=E(f◦X_t^x), where E(f◦X_t^x) denotes the expected value off ◦X_t^x.

It follows from Theorem 8.1.1 in [12] thatuis a solution of problem (3.4), so we are led to consider the positive linear operators V(t) defined as

(3.5) V(t)f(x) := 1

√ 2πt

Z +∞

−∞

f ◦ϕ⁻¹

(u+ϕ(x)) e^−u

2

2tdu, t >0,

for everyf ∈K²(R) and x∈R. We shall prove that, under suitable assump- tions, such a definition can be extended toC₀^w(R). Moreover, V(t) =T(t) for every t >0.

Proposition 3.1. Assume that for every t > 0 and for every compact subset J of Rthere exists h₁ ∈L¹(R) such that

(3.6) e^−u

2 2t

(w◦ϕ⁻¹) (u+ϕ(x)) ≤h₁(u) for every u∈R andx∈J.

Then each V(t) is well defined onC₀^w(R) andV(t) (C₀^w(R))⊂C(R).

Moreover, if for every t >0 there existsh2∈L¹(R) such that

(3.7) w(x)e^−u

2 2t

(w◦ϕ⁻¹) (u+ϕ(x)) ≤h2(u) for everyu, x∈R, then each V(t) is continuous from C₀^w(R) into itself and

(3.8) kV(t)k_Cw

0(R)≤ 1

√

2πt sup

x∈R

Z +∞

−∞

w(x)e^−u

2 2t

(w◦ϕ⁻¹) (u+ϕ(x))du.

Proof. Let f ∈C₀^w(R) and t >0. On account of (3.6), for every x∈R there exists h₁ ∈L¹(R) such that

f ◦ϕ⁻¹

(u+ϕ(x)) e^−u

2 2t

≤ kfk_we^−u

2 2t

(w◦ϕ⁻¹) (u+ϕ(x)) ≤ kfk_wh1(u), hence the integral (3.5) is absolutely convergent, i.e., V (t)f is well defined.

The continuity of V(t)f follows easily from Lebesgue’s dominated con- vergence theorem. Indeed, if (x_n)_n≥1 is a sequence of real number converging tox∈R, then

n→∞lim f ◦ϕ⁻¹

(u+ϕ(x_n)) e^−u

2

2t = f◦ϕ⁻¹

(u+ϕ(x)) e^−u

2 2t

and, by (3.6) again, in correspondence with a compact subsetJofRcontaining {x_n:n≥1}, there existsh₁∈L¹(R) such that

f ◦ϕ⁻¹

(u+ϕ(xn)) e^−u

2 2t

≤ kfk_we^−u

2 2t

(w◦ϕ⁻¹) (u+ϕ(xn)) ≤ kfk_wh1(u) for every u∈Rand n≥1. Hence lim

n→∞V(t)f(x_n) =V(t)f(x).

We pass to prove thatV(t)f ∈C₀^w(R). To this end, note that by (1.1) the functionϕis strictly increasing. Moreover, on account of (1.2), there exist C, δ >0 such thatσ(x)≤C|x|for|x| ≥δ, so that

ϕ(x) =ϕ(δ) + Z x

δ

1

σ(s)ds≥ϕ(δ) + Z x

δ

1

Csds for x≥δ,

hence

x→+∞lim ϕ(x) = +∞.

Analogously, lim

x→−∞ϕ(x) =−∞.

On the other hand, by (3.7) there existsh2 ∈L¹(R) such that, for every u, x∈R,

w(x) f ◦ϕ⁻¹

(u+ϕ(x)) e^−u

2 2t

≤((wf)◦ϕ⁻¹)(u+ϕ(x))h2(u)≤ kfk_wh2(u), so that lim

x→±∞w(x) f ◦ϕ⁻¹

(u+ϕ(x)) = 0 for everyu∈Rand, by Lebesgue’s dominated convergence theorem, we get

x→±∞lim w(x)V(t)f(x) = 1

√2πt lim

x→±∞

Z +∞

−∞

w(x) f◦ϕ⁻¹

(u+ϕ(x)) e^−u

2

2tdu= 0.

Finally, observe that because of (3.7) we have sup

x∈R

√1 2πt

Z +∞

−∞

w(x)e^−u

2 2t

(w◦ϕ⁻¹) (u+ϕ(x))du≤ kh₂k₁

√

2πt <+∞

and, for every x∈R,

|w(x)V(t)f(x)| ≤ 1

√ 2πt

Z +∞

−∞

w(x)

f ◦ϕ⁻¹

(u+ϕ(x)) e^−u

2 2tdu

≤ kfk_w

√ 2πt

Z +∞

−∞

w(x)

(w◦ϕ⁻¹) (u+ϕ(x))e^−u

2 2tdu,

whence V(t) is bounded and (3.8) holds. The assertion is now completely proved.

Proposition 3.2. Under the assumptions of the previous proposition suppose that for every t > 0 there exist g1, g2 ∈ L¹(R) such that for every x, u∈R we have

(i) ∂

∂x

ϕ⁻¹(u+ϕ(x)) 2

+

∂²

∂x²

ϕ⁻¹(u+ϕ(x))

! e^−u

2

2t ≤g₁(u), (ii)w(x)

ϕ⁻¹0

(u+ϕ(x))2

+ ϕ⁻¹00

(u+ϕ(x))

e^−u

2

2t ≤g₂(u). Furthermore assume that for everyu∈Rwe have

(iii) lim

x→±∞w(x)

ϕ⁻¹0

(u+ϕ(x)) 2

+ ϕ⁻¹00

(u+ϕ(x))

= 0.

Then, for every t >0,

V(t) K²(R)

⊂Dw(A).

Proof. Let f ∈ K²(R) and t > 0. As a first step we shall prove that V(t)f is twice differentiable. Then, for everyx, u∈R set

Φ (x, u) := 1

√2πt f◦ϕ⁻¹

(u+ϕ(x)) e^−u

2 2t.

Observe that

∂Φ

∂x (x, u)

= 1

√2πt

f⁰◦ϕ⁻¹

(u+ϕ(x)) ∂

∂x

ϕ⁻¹(u+ϕ(x))

e^−u

2 2t

≤ 1

√ 2πt

f⁰

_∞ 1 + ∂

∂x

ϕ⁻¹(u+ϕ(x)) 2!

e^−u

2 2t

≤ 1

√2πt f⁰

_∞

e^−u

2

2t +g₁(u)

and

∂²Φ

∂x² (x, u)

= 1

√ 2πt

f⁰⁰◦ϕ⁻¹

(u+ϕ(x)) ∂

∂x

ϕ⁻¹(u+ϕ(x)) 2

+ f⁰◦ϕ⁻¹

(u+ϕ(x)) ∂²

∂x²

ϕ⁻¹(u+ϕ(x))

e^−u

2 2t

≤ 1

√2πt f⁰⁰

_∞ ∂

∂x

ϕ⁻¹(u+ϕ(x)) 2

e^−u

2 2t

+ 1

√ 2πt

f⁰

_∞

∂²

∂x²

ϕ⁻¹(u+ϕ(x))

e^−u

2 2t

≤ 1

√2πtmax f⁰

_∞, f⁰⁰

_∞

g₁(u). Therefore, by deriving under the integral, sign we obtain

d

dx(V(t)f) (x) = Z +∞

−∞

∂Φ

∂x(x, u) du and

d²

dx² (V(t)f) (x) = Z +∞

−∞

∂²Φ

∂x² (x, u) du.

It follows from Lebesgue’s dominated convergence theorem that V (t)f ∈ C²(R).

In order to prove the assertion it remains to show that the boundary conditions are fulfilled. Since ϕ⁰(x) = _σ(x)¹ and ϕ⁰⁰(x) =−_σ^σ2^0(x)(x), we get

σ²(x)∂²Φ

∂x² (x, u) +σ(x)σ⁰(x)∂Φ

∂x(x, u)

= σ²(x)

√

2πt f⁰⁰◦ϕ⁻¹

(u+ϕ(x)) ∂

∂x

ϕ⁻¹(u+ϕ(x)) 2

e^−u

2 2t

+σ²(x)

√2πt f⁰◦ϕ⁻¹

(u+ϕ(x)) ∂²

∂x²

ϕ⁻¹(u+ϕ(x)) e^−u

2 2t

+σ(x)σ⁰(x)

√2πt f⁰◦ϕ⁻¹

(u+ϕ(x)) ∂

∂x

ϕ⁻¹(u+ϕ(x)) e^−u

2 2t

= σ²(x)

√

2πt f⁰⁰◦ϕ⁻¹

(u+ϕ(x))

ϕ⁻¹0

(u+ϕ(x)) 2

ϕ⁰(x)2

e^−u

2 2t

+σ²(x)

√2πt f⁰◦ϕ⁻¹

(u+ϕ(x))h ϕ⁻¹00

(u+ϕ(x)) ϕ⁰(x)2

+ ϕ⁻¹0

(u+ϕ(x))ϕ⁰⁰(x) i

e^−u

2 2t

+σ(x)σ⁰(x)

√2πt f⁰◦ϕ⁻¹

(u+ϕ(x)) ϕ⁻¹0

(u+ϕ(x))ϕ⁰(x)e^−u

2 2t

= 1

√

2πt f⁰⁰◦ϕ⁻¹

(u+ϕ(x))

ϕ⁻¹0

(u+ϕ(x)) 2

e^−u

2 2t

+ 1

√2πt f⁰◦ϕ⁻¹

(u+ϕ(x)) ϕ⁻¹00

(u+ϕ(x)) e^−u

2 2t

− σ⁰(x)

√2πt f⁰◦ϕ⁻¹

(u+ϕ(x)) ϕ⁻¹0

(u+ϕ(x)) e^−u

2 2t

+ σ⁰(x)

√2πt f⁰◦ϕ⁻¹

(u+ϕ(x)) ϕ⁻¹0

(u+ϕ(x)) e^−u

2 2t

= 1

√ 2πt

f⁰⁰◦ϕ⁻¹

(u+ϕ(x))

ϕ⁻¹0

(u+ϕ(x)) 2

+ f⁰◦ϕ⁻¹

(u+ϕ(x)) ϕ⁻¹00

(u+ϕ(x)) i

e^−u

2 2t.

Hence, for every x∈R, w(x)

σ²(x) d²

dx²(V(t)f) (x) +σ(x)σ⁰(x) d

dx(V (t)f) (x)

= 1

√2πt

Z +∞

−∞

σ²(x)∂²Φ

∂x² (x, u) +σ(x)σ⁰(x)∂Φ

∂x(x, u) 2

e^−u

2 2tdu

≤ 1

√ 2πt

Z +∞

−∞

w(x)

f⁰⁰◦ϕ⁻¹

(u+ϕ(x))

ϕ⁻¹0

(u+ϕ(x))2

e^−u

2 2tdu

+ 1

√ 2πt

Z +∞

−∞

w(x)

f⁰◦ϕ⁻¹

(u+ϕ(x)) ϕ⁻¹00

(u+ϕ(x)) e^−u

2 2tdu

≤ 1

√ 2πt

Z +∞

−∞

w(x) h

f⁰⁰ ∞

ϕ⁻¹0

(u+ϕ(x)) 2

+ f⁰

_∞ ϕ⁻¹00

(u+ϕ(x)) i

e^−u

2

2tdu≤ 1

√2πtmax f⁰

_∞, f⁰⁰

_∞ ·

· Z +∞

−∞

w(x)h ϕ⁻¹0

(u+ϕ(x))2

+ ϕ⁻¹00

(u+ϕ(x)) i

e^−u

2 2tdu.

Thus, taking (ii) and (iii) into account and applying once more the Lebesgue’s dominated convergence theorem, we obtain

x→±∞lim w(x)

σ²(x) d²

dx² (V(t)f) (x) +σ(x)σ⁰(x) d

dx(V(t)f) (x)

= 0.

We can now present the main result of this section.

Theorem 3.3. Under the assumptions of the previous proposition sup- pose that e₂∈C₀^w(R) and

(i) lim

t→0⁺

√w(x) 2πt

R+∞

−∞

ϕ⁻¹(u+ϕ(x))−x e^−u

2

2tdu= 0, (ii) lim

t→0⁺

√w(x) 2πt

R+∞

−∞

h

ϕ⁻¹(u+ϕ(x))2

−x²i e^−u

2

2tdu= 0

uniformly with respect to x ∈ R. Moreover, assume that there exists M > 0 such that, for every t∈]0,1[,

(iii) ^√1

2πt sup

x∈R

R+∞

−∞

w(x)e^−u

2 2t

(w◦ϕ⁻¹)(u+ϕ(x))du≤M.

Then for every t >0, f ∈C₀^w(R) andx∈Rwe have (3.9) T(t)f(x) = 1

√2πt Z +∞

−∞

f◦ϕ⁻¹

(u+ϕ(x)) e^−u

2 2tdu and

kT(t)k_Cw

0(R)≤ 1

√2πt sup

x∈R

Z +∞

−∞

w(x)e^−u

2 2t

(w◦ϕ⁻¹) (u+ϕ(x))du.

Proof. First, note that for every t >0, x∈Rand k= 0,1,2 we have V(t)(e_k)(x)−e_k(x) = 1

√ 2πt

Z +∞

−∞

h

ϕ⁻¹(u+ϕ(x))k

−x^ki e^−u

2 2tdu,

where, as usual, we set ek(x) :=x^k, x∈ R. Accordingly, since V(t)(e0) = e0

and taking (i) and (ii) into account, we get lim

t→0⁺kV(t)(e_k)−e_kk_w = 0 fork= 0,1,2.

Moreover, because of (iii), the net (V(t))0<t<1 is equicontinuous and so, since {e₀, e₁, e₂} is a Korovkin subset in C₀^w(R) (see [3, Example 2.3.3], [4, Exam- ple 4.9]), for every f ∈C₀^w(R) we have

(3.10) lim

t→0⁺V(t)f =f inC₀^w(R). Now, let f ∈K²(R)⊂D_w(A) and set

u(x, t) :=V(t)f(x) x∈R, t >0.

By Proposition 3.2, the function u(·, t) ∈D_w(A) for everyt > 0. Moreover, by (3.10) and Theorem 8.1.1 in [12], u solves the problem







∂u

∂t (x, t) = σ²(x) 2

∂²u

∂x²(x, t) +σ(x)σ⁰(x) 2

∂u

∂x(x, t), t >0, x∈R, lim

t→0⁺u(·, t) =f inC₀^w(R),

hence V(t)f(x) =u(x, t) =T(t)f(x) for everyt >0 andx∈R.

Since both operators V(t) and T(t) are linear and bounded on C₀^w(R) and K²(R) is dense in (C₀^w(R),k·k_w), this equality also holds forf ∈C₀^w(R) and the assertion is proved.

Remark 3.4. If σ ∈ R and wm(x) := (1 +x^2m)⁻¹ or wm(x) := (e^mx+ e^−mx)⁻¹, m≥1, x∈R, we can easily show that all the hypotheses of Theo- rem 3.3 are satisfied, so for every t >0, f ∈C₀^wm(R) andx∈Rwe have

T(t)f(x) = 1

√ 2πt

Z +∞

−∞

f(σu+x) e^−u

2 2tdu, which is the well known representation of the heat semigroup.

The representation formula (3.9) allows us to study the asymptotic be- havior of the semigroup (T(t))t≥0.

Proposition 3.5. For everyf ∈C∗(R) and x∈R we have

t→+∞lim T(t)f(x) = 1 2

y→+∞lim f(y) + lim

y→−∞f(y)

.

Proof. Givenf ∈C∗(R) and x∈R, note that T(t)f(x) = 1

√π Z +∞

−∞

f◦ϕ⁻¹√

2tv+ϕ(x) e^−v2dv

for every t > 0. Accordingly, since lim

y→±∞ϕ⁻¹(y) = ±∞, it follows from Lebesgue’s dominated convergence theorem that

t→+∞lim

√1 π

Z +∞

0

f◦ϕ⁻¹√

2tv+ϕ(x)

e^−v2dv= 1 2 lim

y→+∞f(y) and

t→+∞lim

√1 π

Z 0

−∞

f◦ϕ⁻¹√

2tv+ϕ(x)

e^−v2dv= 1 2 lim

y→−∞f(y).

Thus the assertion holds.

3.1. Example Let σ(x) :=√

1 +x²,x ∈R and, form ≥1, let wm(x) := (1 +x^2m)⁻¹, x∈R. As we proved in Theorem 2.1, for everym≥1 the operator (A, D_m(A)) generates a positive C0-semigroup (Tm(t))_t≥0 on C₀^wm(R). In this case, the stochastic differential equation associated with A is

(3.11) dXt=

q

1 +X_t²dBt+1 2Xtdt whose solution satisfying X0 =x∈R is

X_t^x := sinh(Bt+ log(x+p

1 +x²)), t≥0.

Now, for t >0,f ∈K²(R) and x∈Rlet us define (3.12) V(t)f(x) := 1

√ 2πt

Z +∞

−∞

(f ◦sinh) (u+ log(x+p

1 +x²))e^−u

2 2tdu.

Propositions 3.1 and 3.2 imply the results below.

Proposition3.6. Letm≥1.Then for every t >0 the operator V(t) is well defined on C₀^wm(R), maps continuously C₀^wm(R) into itself and

(3.13)

kV(t)k_C^wm

0 (R)≤ 1

√

2πt sup

x∈R

Z +∞

−∞

1+

h

sinh(u+log(x+√

1+x²)) i2m

1+x^2m e^−u

2 2tdu.

Proof. Lett >0. Given a compact subset J of R, set M_J := max

maxx∈J(x+p

1 +x²), max

x∈J

1 x+√

1 +x²

and

h₁(u) := e^−u

2 2t +

M_J 2

2m

e^u+ e^−u2m

e^−u

2

2t, u∈R.

Then h1∈L¹(R) and, for everyu∈Rand x∈J, e^−u

2 2t

(w_m◦ϕ⁻¹) (u+ϕ(x)) =h

1 + ϕ⁻¹(u+ϕ(x))2mi e^−u

2 2t

= e^−u

2

2t + 1

2^2m h

(x+p

1 +x²)e^u−(x+p

1 +x²)⁻¹e^−ui2m

e^−u

2

2t ≤h₁(u). So, condition (3.6) is fulfilled.

On the other hand, set (3.14) Mm:= max

( sup

x∈R

x+√ 1 +x²

2m√

1 +x^2m , sup

x∈R

1 (x+√

1 +x²)^2m√

1 +x^2m )

and

h₂(u) := e^−u

2 2t +

M_m 2

2m

e^u+ e^−u2m

e^−u

2 2t. We have h₂∈L¹(R) and, for everyu, x∈R,

w_m(x)e^−u

2 2t

(wm◦ϕ⁻¹) (u+ϕ(x)) = 1 +h

sinh(u+ log(x+√

1 +x²))i2m

1 +x^2m e^−u

2 2t

=

1 +₂2m¹

h (x+√

1 +x²)e^u−(x+√

1 +x²)⁻¹e^−u i2m

1 +x^2m e^−u

2 2t

≤e^−u

2

2t + 1

2^2m

"

(x+√

1 +x²)e^u−(x+√

1 +x²)⁻¹e^−u

2m√

1 +x^2m

#2m

e^−u

2

2t ≤h₂(u). Then, by Proposition 3.1 the assertion holds.

Proposition 3.7. Let m≥1. Then, for every t >0, V(t) K²(R)

⊂Dm(A).

Proof. Lett >0. It suffices to verify all hypotheses of Proposition 3.2.

To this purpose, set g(u) :=

M_m

2 e^u+ e^−u 2

e^−u

2

2t +M_m e^u+ e^−u e^−u

2

2t, u∈R, where Mm is defined by (3.14). Clearlyg∈L¹(R).

Now, observe that for every x, u∈Rwe have

∂

∂x

ϕ⁻¹(u+ϕ(x))

=

∂

∂x

sinh(u+ log(x+p

1 +x²))

= 1 2

x+√ 1 +x²

√1 +x² e^u+ e^−u (x+√

1 +x²)√ 1 +x²

≤ M_m

2 e^u+ e^−u .

Then

∂

∂x

ϕ⁻¹(u+ϕ(x)) 2

+

∂²

∂x²

ϕ⁻¹(u+ϕ(x))

! e^−u

2 2t

≤ M_m

2 e^u+ e^−u 2

e^−u

2 2t +

∂²

∂x² h

sinh(u+ log(x+p

1 +x²))i

e^−u

2 2t

= Mm

2 e^u+ e^−u 2

e^−u

2 2t +

sinh(u+ log(x+p

1 +x²)) 1 (1 +x²)

−cosh(u+ log(x+p

1 +x²)) x q

(1 +x²)³

e^−u

2

2t ≤g(u). Therefore, condition (i) of Proposition 3.2 is satisfied.

As regards (ii), for everyx, u∈R we have w(x)

ϕ⁻¹0

(u+ϕ(x)) 2

+ ϕ⁻¹00

(u+ϕ(x))

e^−u

2 2t

=

cosh²(u+ log(x+√

1 +x²)) +

sinh(u+ log(x+√

1 +x²))

1 +x^2m e^−u

2 2t

≤ cosh²(u+ log(x+√

1 +x²)) + cosh(u+ log(x+√

1 +x²))

1 +x^2m e^−u

2

2t ≤g(u). Finally, for every u∈R,

x→±∞lim w(x)

ϕ⁻¹0

(u+ϕ(x)) 2

+ ϕ⁻¹00

(u+ϕ(x))

= lim

x→±∞

cosh²(u+ log(x+√

1 +x²)) +

sinh(u+ log(x+√

1 +x²))

1 +x^2m = 0

hence (iii) is also satisfied. So, the assertion is proved.

At this point we may give an explicit integral representation of the semi- group (T_m(t))_t≥0.

Theorem3.8. Letm≥2. Denote by(Tm(t))_t≥0the semigroup generated by (A, D_m(A)). Then

Tm(t)f(x) = 1

√ 2πt

Z +∞

−∞

(f ◦sinh) (u+ log(x+p

1 +x²))e^−u

2 2tdu.

for every t >0, f ∈C₀^wm(R) and x∈R. Moreover, kT(t)k_Cwm

0 (R) ≤ 1

√2πt sup

x∈R

Z +∞

−∞

1 +h

sinh(u+ log(x+√

1 +x²))i2m

1 +x^2m e^−u

2 2tdu.

Proof. We shall apply Theorem 3.3. To this end let us remark that by (3.13) and (3.14) we have

kV(t)k_Cwm

0 (R)≤1 + 1

√ 2πt

M_m 2

2mZ +∞

−∞

e^u+ e^−u2m

e^−u

2 2tdu for every t >0.

In particular, for 0< t <1,

√1 2πt

Z +∞

−∞

e^u+ e^−u2m

e^−u

2

2tdu= 1

√ 2πt

Z +∞

−∞

2m

X

j=0

2m j

e^jue^−(2m−j)ue^−u

2 2tdu

= 1

√ 2πt

2m

X

j=0

2m j

Z +∞

−∞

e^{(2j−2m)u−u}

2 2tdu

=

2m

X

j=0

2m j

e^{(2j−2m)2t/2} <

2m

X

j=0

2m j

e^2(j−m)2,

so condition (iii) of Theorem 3.3 is fulfilled.

We also get that for t >0,x∈Rand k= 0,1,2 we have V(t)e_k(x) = 1

√ 2πt

Z +∞

−∞

h

sinh(u+ log(x+p

1 +x²)) ik

e^−u

2 2tdu

= 1

2^k√ 2πt

Z +∞

−∞



(x+p

1 +x²)e^u− e^−u

x+√

1 +x²





k

e^−u

2 2tdu

= 1

2^k√ 2πt

Z +∞

−∞

k

X

j=0

k j

(−1)^k−j x+p

1 +x² j

e^ju

× e^−(k−j)u

x+√

1 +x²k−je^−u

2 2tdu

= 1

2^k√ 2πt

k

X

j=0

k j

(−1)^k−j x+p

1 +x²

2j−kZ +∞

−∞

e^{(2j−k)u−u}

2 2tdu

= 1 2^k

k

X

j=0

k j

(−1)^k−j x+p

1 +x²2j−k

e^(2j−k)2t/2

and

e_k(x) = 1 2^k

x+p

1 +x²− 1 x+√

1 +x² k

= 1 2^k

k

X

j=0

k j

(−1)^k−j(x+p

1 +x²)^j 1 (x+√

1 +x²)^k−j

= 1 2^k

k

X

j=0

k j

(−1)^k−j x+p

1 +x²2j−k

.

Then

V(t)e_k(x)−e_k(x) =

k

X

j=0

k j

(−1)^k−j 2^k

x+p

1 +x²2j−k

(e^(2j−k)2t/2−1), which imply

sup

x∈R

|V(t)ek(x)−e_k(x)|

1 +x^2m ≤

e^(2j−k)2t/2−1 2^k

k

X

j=0

k j

sup

x∈R

x+√

1 +x² 2j−k

1 +x^2m . Therefore, conditions (i) and (ii) of Theorem 3.3 are also satisfied and this completes the proof.

Acknowledgements. The author wishes to thank Prof. F. Altomare and Prof.

I. Ra¸sa for having turned her attention to the integral representation by stochastic methods dealt with in this paper.

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Received 15 October 2007 University of Bari

Department of Mathematics Via E. Orabona, 4

70125 Bari, Italy smilella@dm.uniba.it