J. Math. Anal. Appl. 400 (2013) 273–284

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A modified method of stochastic regularization for an ill-posedCauchy problemYifeng LuSchool of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, People’s Republic of China

a r t i c l e i n f o

Article history:Received 27 March 2012Available online 8 November 2012Submitted by Xu Zhang

Keywords:Ill-posed Cauchy problemStochastic regularizationFractional powerFunctional calculus

a b s t r a c t

This paper is concerned with the ill-posed Cauchy problem associated with a denselydefined linear operator in a Banach space. We investigate a method of stochasticregularization for this ill-posed Cauchy problem by using fractional powers and functionalcalculus for the sectorial operators.

© 2012 Elsevier Inc. All rights reserved.

1. Introduction

Ill-posed Cauchy problems arise in a variety of important applications in science and industry, and have received a greatdeal of attention (see [1–8]). In this paper, we consider the following ill-posed Cauchy problem:

u′(t) = Au(t), 0 < t ≤ T ,u(0) = x, (1.1)

where −A is the infinitesimal generator of a holomorphic semigroup S(t) of α with α ∈ (0, π/2] on a Banach space X (seeDefinition 2.1). A function u ∈ C([0, T ], X) is said to be a solution of (1.1) if u ∈ C1((0, T ], X), u(t) ∈ D(A) (the domain ofA) for 0 < t ≤ T , and (1.1) is satisfied.

As is known, problem (1.1) is usually ill-posed in the Hadamard sense. That is, the solution does not always exist; evenif a unique solution exists on [0, T ], it does not depend continuously on the initial value. In fact, problem (1.1) has a uniqueentire solution for each initial value x ∈ D, where D is a dense subspace in X (see [9]). In this paper, we only need to considerthe problem on the interval [0, T ]. In the Appendix, we show that problem (1.1) has a unique solution u(t) on [0, T ] if andonly if the initial value x ∈ R(S(T )), and the solution u(t) = S(t)−1x. Since S(t)−1

t≥0 is not a family of bounded linearoperators on X , the solution of problem does not depend continuously on the initial value. In other words, a small amountof error in the ideal initial value can lead to enormous errors in the solution.

A widely used approach to the study of ill-posed problems is to find corresponding well-posed problems such that thesolution can approximate the solution of the ill-posed problems.We can particularlymention the quasi-reversibilitymethodintroduced by Lattès and Lions in [5] (also cf. [10–15,7]), the quasi-boundary value method [16–18], and the stochasticregularization method [19]. The numerical analysis of the stochastic regularization method has been given in [20–23].

The aimof this paper is to regularize ill-posed Cauchy problem (1.1) by the stochastic regularizationmethod.We continuethe line of investigation started in [19].

E-mail address: luyifenghust@yahoo.com.cn.

0022-247X/$ – see front matter© 2012 Elsevier Inc. All rights reserved.doi:10.1016/j.jmaa.2012.11.007

274 Y. Lu / J. Math. Anal. Appl. 400 (2013) 273–284

In [19], Dalecky and Goncharuk introduced the stochastic regularization method. Problem (1.1) was approximated byduε(t) = Auε(t)dt + εAuε(t)dWt , 0 < t ≤ T ,uε(0) = x, (1.2)

where Wt is a one-dimensional standard Brownian motion, and −A is the infinitesimal generator of a holomorphicsemigroup of angle α(π/4 < α ≤ π/2).

In this paper, we modify the stochastic regularization method, and approximate problem (1.1) byduε(t) = Auε(t)dt + εAbuε(t)dWt , 0 < t ≤ T ,uε(0) = x, (1.3)

where ε > 0,−A is the infinitesimal generator of a holomorphic semigroup of angle α (0 < α ≤ π/2), and Ab (1/2 <b < π/2(π − 2α)) is defined as the fractional power. In fact, our method includes the method introduced by Dalecky andGoncharuk in [19] as a particular case.

One of the main improvements is the fact that the restriction of the angle of the homomorphic semigroup generated by−A is relaxed from (π/4, π/2] to (0, π/2]. Our result is more convenient in application to differential operators.

Another improvement appears in the estimate of the solution operator of the approximate equation. In [19] the estimateis not convenient to use. We obtain a better one, and it illustrates clearly the asymptotic behavior of the solution for largevalues of time and also the relationship between the solution and the regularization parameter ε. It is worth remarkingthat one of the difficulties appears as the exponential integral with a fractional polynomial function as the exponent. Toovercome such difficulty, we represent the solution operator as a multiplication of two bounded operators which are easyto estimate.

This paper is organized as follows. In Section 2, we provide some preliminaries, which include notation, notions likeholomorphic semigroups, functional calculus for sectorial operators and its properties, and fractional powers. In Section 3,we define a stochastic evolution family and study its properties. In Section 4, we show that the stochastic evolution familydefined in Section 3 is the solution operator of the approximate problem. The main result of this paper is given in Section 5;we show that the stochastic evolution family is a family of regularizing operators for problem (1.1). In the Appendix, wegive some properties of the inverse of the holomorphic semigroup and the solution of (1.1).

2. Preliminaries

Throughout the paper, (X, ∥ · ∥) is a Banach space, and B(X) is the space of all bounded linear operators on X . ByD(B), R(B), σ (B), ρ(B), and R (λ, B)(λ ∈ ρ(B)) we denote the domain, the range, the spectrum, the resolvent set and theresolvent of the operator B, respectively. Let (Ω,F , P) be a complete filtered probability space, andWt be a scalar Brownianmotion whose natural filtration is given by F := Ftt≥0.

For 0 ≤ β ≤ π , let

Sβ :=

z ∈ C : z = 0 and | arg z| < β, if β ∈ (0, π],(0,∞), if β = 0,

and

Sβ(r, R) := z ∈ Sβ : r < |z| < R.

We first give the definition of a holomorphic semigroup. The basic theory of C0-semigroups can be found in [24]; alsosee [25,26].

Definition 2.1. Let 0 < α ≤ π/2. If a C0-semigroup T (z)z≥0 has an analytic extension into Sα satisfying limSα′∋z→0 T (z)x =

x for each x ∈ X and α′∈ (0, α), then T (z)z∈Sα is called a holomorphic semigroup of angle α. Its generator is the generator

of T (z)z≥0. Moreover, the holomorphic semigroup of angle α is said to be bounded if for each α′∈ (0, α), there exists

Mα′ > 0 such that ∥T (z)∥ ≤ Mα′ for z ∈ Sα′ .

Lemma 2.2. Let 0 < α ≤ π/2. Then the following statements are equivalent:(a) B is the infinitesimal generator of a bounded holomorphic semigroup of angle α.(b) Sα+π/2 ⊂ ρ(B), and for each α′

∈ (0, α), there exists Mα′ > 0 such that ∥R(z, B)∥ ≤ Mα′/|z| for z ∈ Sα′+π/2.

Lemma 2.3 ([14]). Let −A be the generator of a bounded holomorphic semigroup of angle (0 < α ≤π2 ), 0 ∈ ρ(A), and let

Aε = A − εAb, where ε > 0 and b ∈ (1, ππ−2α ). Then for any β ∈ (0, π2 − (π2 − α)b), Aε is the generator of a holomorphic

semigroup Vε(t) of angle β . Moreover,

∥Vε(t)∥ ≤ M expCε1/(1−b)t for t ≥ 0,

where M and C are positive constants independent of ε.

Wenow turn to a short introduction to functional calculus for sectorial operators. For details, we refer the reader to [27–30].

Y. Lu / J. Math. Anal. Appl. 400 (2013) 273–284 275

An operator A on X is called sectorial of angle β ∈ [0, π) (for short: A ∈ Sect(β)) if

(1) σ(A) ⊂ Sβ and(2) M(A, β ′) := sup∥zR(z, A)∥ : z ∈ C \ Sβ ′ < ∞ for all β ′

∈ (β, π).

We call

βA := min0 ≤ β ≤ π : A ∈ Sect(β)

the spectral angle of A.We employ the following subspaces of the space H(Sϕ) of all holomorphic functions on Sϕ .

H∞(Sϕ) := f ∈ H(Sϕ) : ∥f ∥∞ < ∞,

where ∥f ∥∞ = sup|f (z)| : z ∈ Sϕ.

H∞

0 (Sϕ) := f ∈ H∞(Sϕ) : ∃s > 0, C ≥ 0 such that |f (z)| ≤ C |z|s(1 + |z|2s)−1, ∀z ∈ Sϕ,

and

H∞

(0)(Sϕ) := f ∈ H∞(Sϕ) : f is decaying at ∞ and bounded at 0.

Let A ∈ Sect(β), f ∈ H∞

(0)(Sϕ), 0 < β < β ′ < ϕ < π ; we can define

f (A) :=1

2π i

Γ

f (z)R(z, A)dz, (2.1)

where Γ is an appropriate curve surrounding σ(A). If f is holomorphic at 0, the curve Γ is chosen as the positively orientedboundary of Sβ ′ ∪Bδ(0), where δ > 0 is small enough that f is holomorphic in a neighborhood of Bδ(0). If f is not holomorphicat 0 and 0 ∈ ρ(A), the curve Γ is chosen as the positively oriented boundary of Sβ ′ \ Bδ′(0), where δ′ is a positive constantsuch that Bδ′(0) ⊂ ρ(A).

Lemma 2.4. Let A ∈ Sect(β) on X, ϕ ∈ (β, π) and f ∈ H∞

(0)(Sϕ). The mapping

(f → f (A)) : H∞

(0)(Sϕ) → B(X)

defined by (2.1) is an algebra homomorphism.

Remark 2.5. Obviously, A ∈ Sect(β) (0 < β < π/2) if and only if −A is the infinitesimal generator of a boundedholomorphic semigroup of angle π/2 − β . Moreover, e−tA defined via (2.1) is the holomorphic semigroup generated by−A.

Finally, we give a short introduction to fractional powers (see [25,26,29,24]).

Definition 2.6. Let A ∈ Sec(β) for some β ∈ [0, π), and let 0 ∈ ρ(A). For 0 < α < π/β , the fractional power of A is definedas follows:

A−α:=

12π i

Γβ′

λ−αR(λ, A)dλ,

where the path Γβ ′ runs in the resolvent set of A from ∞eiβ′

to ∞e−iβ ′

, β ′∈ (β, π), avoiding the negative real axis and the

origin, and λ−α is taken to be positive for real positive values of λ. Define Aα := (A−α)−1 and A0:= I .

Lemma 2.7. Let A ∈ Sec(β) for some β ∈ (0, π); then for every α ∈ (0, π/βA), the operator Aα is sectorial with βAα = αβA.

3. A stochastic evolution family

In this section, we define a stochastic evolution family, and study its properties.

Definition 3.1. Let −A be the infinitesimal generator of a bounded holomorphic semigroup of angle α with α ∈ (0, π/2],and 0 ∈ ρ(A); then a stochastic evolution family Uε(t, s) is given by the formulas

Uε(s, t) :=

I, 0 ≤ s = t ≤ T ,1

2π i

Γ (θ,δ)

f (λ, t, s)R(λ, A)dλ, 0 ≤ s < t ≤ T , (3.1)

where f (λ, t, s) := exphλ −12ε

2hλ2b + ε1Whλb, h = t − s,1Wh = Wt − Ws, ε = 0, b ∈ (1/2, π/2(π − 2α)), θ ∈

(π/2 − α, π/4b), λ2b and λb are taken as the principal branch, and the path Γ (θ, δ) is the positively oriented boundary ofSθ \ Bδ(0), δ being a positive constant such that Bδ(0) ⊂ ρ(A).

276 Y. Lu / J. Math. Anal. Appl. 400 (2013) 273–284

For any 0 ≤ s ≤ t < ∞, the function f (λ, t, s) ∈ H∞

(0)(S π4b ) P-a.s., so Uε(t, s) is almost surely well defined and bounded.We need to estimate the norm of Uε(t, s). Themain idea of estimating its norm is to represent Uε(t, s) as amultiplicationof two bounded operators which are easy to estimate. For any 0 ≤ s ≤ t < ∞, f1(λ, t, s) := exp−h( 12ε

2φλ2b − λ) ∈

H∞

(0)(S π4b ), and f2(λ, t, s) := exp−[12ε

2(1 − φ)hλ2b − ε1Whλb] ∈ H∞

(0)(S π4b ) P-a.s., where φ ∈ (0, 1). We can thereforedefine

U1(h) :=1

2π i

Γ

f1(λ, t, s)R(λ, A)dλ, (3.2)

U2(t, s) :=1

2π i

Γ

f2(λ, t, s)R(λ, A)dλ, (3.3)

where Γ = Γ (θ, δ) is given as in Definition 3.1. From Lemma 2.4, we have Uε(t, s) = U1(h)U2(t, s).From Lemma 2.3 it follows that U1(h) is the holomorphic semigroup generated by −

12ε

2φA2b+ A, and

∥U1(h)∥ ≤ M1 expC1ε2/(1−2b)h, for h ≥ 0, (3.4)

whereM1 and C1 are positive constants independent of ε and h.We obtain the estimate of the norm of U2(t, s) in the following lemma.

Lemma 3.2. Let ε > 0, 0 ≤ s < t and U2(t, s) be defined by (3.3); then

∥U2(t, s)∥ ≤ M2eC2(1Wh)

2

h , P-a.s., (3.5)

where M2 and C2 are positive constants independent of ε and h.

Proof. If1Wh = 0, by Remark 2.5 and Lemma2.7,we obtainU2(t, s) = T2b( 12ε2(1−φ)h), where T2b(h)denotes the bounded

holomorphic semigroup generated by−A2b. Therefore ∥U2(t, s)∥ ≤ C3, where C3 is a positive constant independent of ε andh.

If1Wh < 0, by Lemma 2.4, Remark 2.5 and Lemma 2.7, we obtain U2(t, s) = T2b( 12ε2(1−φ)h)Tb(−ε1Wh), where Tb(h)

denotes the bounded holomorphic semigroup generated by −Ab. Therefore ∥U2(t, s)∥ ≤ C4, where C4 is a positive constantindependent of ε and h.

It remains only to show the case where 1Wh > 0. Let K =

12ε

2h(1 − φ) and L = ε1Wh; then for almost every

Brownian sample path, the function exp−K 2z2 + Lz ∈ H∞

(0)(S π4 ). Since Ab∈ Sect(b(π2 − α)), one can define an operator

12π i

Γbθ,δ

e−K2z2+LzR(z, Ab)dz,

where Γbθ,δ is the positively oriented boundary of Sbθ ∪Bδ(0), δ > 0. Note that e−K2z2+Lz is an entire function in the z-plane;thus the curve does not need to avoid the negative real axis and the origin.

We first show that

U2(t, s) =1

2π i

Γbθ,δ

e−K2z2+LzR(z, Ab)dz. (3.6)

It is clear that

R(z, Ab) =1

2π i

Γ

(z − λb)−1R(λ, A)dλ, (3.7)

where the path Γ is given as in Definition 3.1.By Cauchy’s integral theorem, the path Γbθ,δ can be deformed to Γbθ ′,δ (

π2 − α < θ < θ ′ < π

4b ) so that for everyλ ∈ Γ , λb ∈ Sbθ ′ . By Cauchy’s integral theorem, Fubini’s theorem and (3.7), we have

12π i

Γbθ ′,δ

e−K2z2+LzR(z, Ab)dz =1

2π i

Γbθ ′,δ

e−K2z2+Lz

1

2π i

Γ

(z − λb)−1R(λ, A)dλ

dz

=1

2π i

Γ

1

2π i

Γbθ ′,δ

e−K2z2+Lz(z − λb)−1dz

R(λ, A)dλ

=1

2π i

Γ

e−12 ε

2h(1−φ)λ2b+ε1WhλbR(λ, A)dλ

= U2(t, s).

Y. Lu / J. Math. Anal. Appl. 400 (2013) 273–284 277

Next, we estimate the norm of U2(t, s) via (3.6). We observe that

U2(t, s) =1

2π i

KΓbθ,δ

e−µ2+

LK µ

1KRµK, Ab

dµ.

which follows on making the change of variables µ = Kz. Since the function e−µ2+

LK µ 1

K R(µ

K , Ab) is holomorphic in

−Sbθ (min1, Kδ,max1, Kδ), by Cauchy’s integral theorem, the path KΓbθ,δ can be deformed to Γbθ,1, i.e., Γbθ,1 consistsof the three parts Γ ′

1 = reibθ : ∞ ≥ r ≥ 1,Γ ′

2 = eiψ : bθ ≤ ψ ≤ 2π − bθ,Γ ′

3 = re−ibθ: 1 ≤ r ≤ ∞. Then

∥U2(t, s)∥ =

12π i

Γbθ,1

e−µ2+

LK µ

1KRµK, Ab

dµ

≤

12π

Γ ′1

+

Γ ′2

+

Γ ′3

e−µ2

+LK µ

1KRµK, Ab

dµ

≤

12π(I1 + I2 + I3).

Since Ab∈ Sect(b(π2 − α)), then there exists a constantM such thatR µ

K, Ab

≤M|µ|

K

=KM|µ|

, µ ∈ Γbθ,1. (3.8)

We deduce from (3.8) that

I1 ≤ M

∞

1e−r2 cos 2bθ+ L

K r cos bθ drr

≤ M

∞

1e−

r√cos 2bθ− L cos bθ

2K√cos 2bθ

2+

L2(cos bθ)2

4K2 cos 2bθ dr.

Making the change of variables u = r√cos 2bθ −

L cos bθ2K

√cos 2bθ

, we obtain

I1 ≤M

√cos 2bθ

eL2(cos bθ)2

4K2 cos 2bθ

∞

√cos 2bθ− L cos bθ

2K√cos 2bθ

e−u2du

≤M

√cos 2bθ

eL2(cos bθ)2

4K2 cos 2bθ

∞

−∞

e−u2du.

Since

∞

−∞e−u2du =

√π , then

I1 ≤

√πM

√cos 2bθ

eL2(cos bθ)2

4K2 cos 2bθ

=

√πM

√cos 2bθ

e(cos bθ)2

2(1−φ) cos 2bθ(1Wh)

2

h .

The same estimate holds for I3.It follows from (3.8) that

I2 ≤ M 2π−bθ

bθe− cos 2ψ+

LK cosψdψ

≤ Me1+L2

8K2

2π−bθ

bθe−

√2 cosψ−

L2√2K

2

dψ

≤ (2π − 2bθ)Me1+1

4(1−φ)(1Wh)

2

h .

Combining the above estimates, we obtain

∥U2(t, s)∥ ≤ M2eC2(1Wh)

2

h , P-a.s., (3.9)

where M2 and C2 are positive constants independent of ε and h.

278 Y. Lu / J. Math. Anal. Appl. 400 (2013) 273–284

From Lemma 3.2, we have the following corollary.

Corollary 3.3. Let ε > 0, 0 ≤ s < t and U2(t, s) be defined by (3.3); then for x ∈ X,

limε→0

U2(t, s)x = x, P-a.s.

Proof. By Lemma 3.2, U2(t, s) is uniformly bounded in ε > 0. Therefore it suffices to verify that for x ∈ D(A),

limε→0

U2(t, s)x = x, P-a.s.

Let x ∈ D(A); since

U2(t, s)x =1

2π i

Γ

f2(λ, t, s)R(λ, A)dλx

=1

2π i

Γ

f2(λ, t, s)1 + λ

R(λ, A)dλ(I + A)x,

where Γ is given as in Definition 3.1, Lebesgue’s dominated convergence theorem implies

limε→0

U2(t, s)x = limε→0

12π i

Γ

f2(λ, t, s)1 + λ

R(λ, A)dλ(I + A)x

=1

2π i

Γ

11 + λ

R(λ, A)dλ(I + A)x

= (I + A)−1(I + A)x= x

for almost every Brownian sample path. The proof is complete.

By (3.2) and Lemma 3.2, we obtain the estimate of the norm of Uε(t, s).

Theorem 3.4. Let ε > 0, 0 ≤ s < t and Uε(t, s) be as defined in Definition 3.1; then

∥Uε(t, s)∥ ≤ M3 exp

C1ε

21−2b h + C2

(1Wh)2

h

, P-a.s., (3.10)

where M3, C1 and C2 are some constants independent of ε and h.

Remark 3.5. In [19], they obtained an estimate as follows:

∥Uε(t, s)∥ ≤a1ε√hexp

a2

√hε

+ h−ν

2+ b1 exp

b2

(ε2 + b3)h + εh

12 −ν

,

where a1, a2, b1, b2, b3 are positive constants independent of ε and h, ν is a positive constant such that1Wh√h

< cν,hh−ν

for almost every Brownian sample path. By comparison with the estimate in [19], (3.10) is more concise, and illustrates therelationship between the solution and the regularization parameter ε more clearly.

Theorem 3.6. Let ε > 0, 0 ≤ s ≤ τ ≤ t and Uε(t, s) be defined in Definition 3.1; then

Uε(t, s) = Uε(t, τ )Uε(τ , s), P-a.s.

Proof. It follows from Lemma 2.4.

Theorem 3.7. Let ε > 0, 0 ≤ s < t and Uε(t, s) be as defined in Definition 3.1; then for x ∈ D(Ab),

limt→s

Uε(t, s)x = x, P-a.s.

Proof. Since U1(h) is the strongly continuous holomorphic semigroup generated by −12ε

2φA2b+A, then U1(h) is uniformly

bounded in h ∈ (0, T ]. Therefore it remains to show the continuity of U2(t, s) on D(Ab). Following from the proof of

Y. Lu / J. Math. Anal. Appl. 400 (2013) 273–284 279

Lemma 3.2, we have

U2(t, s) =1

2π i

Γbθ,1

e−µ2+

LK µ

1KRµK, Ab

dµ

where Γbθ,1, K and L are given in the proof of Lemma 3.2. Let x ∈ D(Ab); by Cauchy’ integral theorem, one can obtain

U2(t, s)x − x =1

2π i

Γbθ,1

e−µ2+

LK µ

1KR

µ

K, Ab

−

1µI

xdµ

=1

2π i

Γbθ,1

e−µ2+

LK µ

1µRµK, Ab

Abxdµ.

Then

∥U2(t, s)x − x∥ =

12π i

Γbθ,1

e−µ2+

LK µ

1µRµK, Ab

Abxdµ

≤

12π i

Γbθ,1

e−µ2+

LK µ

1µRµK, Ab

dµ

∥Abx∥.

We obtain, similarly to the estimate of U2(t, s) in Lemma 3.2,

∥U2(t, s)x − x∥ ≤ M2KeC2(1Wh)

2

h ∥Abx∥

= M2

12ε2h(1 − φ)eC2

(1Wh)2

h ∥Abx∥, P-a.s.,

where M2, C2 are given in Lemma 3.2. The law of iterated logarithms for Brownian motion implies that

limh→0

√heC2

(1Wh)2

h = 0, P-a.s.

Thereby for x ∈ D(Ab),

limt→s

U2(t, s)x = x, P-a.s.

Theorem 3.8. Let ε > 0, 0 ≤ s < t and Uε(t, s) be as defined in Definition 3.1, and denote by S(t), Tb(t) and T2b(t) the stronglycontinuous holomorphic semigroups generated by −A,−Ab and −A2b, respectively; then for almost every Brownian sample path,

Uε(t, s) =

S(h)−1Tb(ε1Wh)

−1T2b

12ε2h, if 1Wh ≥ 0,

S(h)−1Tb(−ε1Wh)T2b

12ε2h, if 1Wh < 0.

Moreover Uε(t, s) is invertible, and

Uε(t, s)−1=

T2b

12ε2h−1

Tb(ε1Wh)S(h), if 1Wh ≥ 0,

T2b

12ε2h−1

Tb(−ε1Wh)−1S(h), if 1Wh < 0,

with domain

D(Uε(t, s)−1) = R(Uε(t, s)).

Proof. If1Wh ≥ 0, by Lemma 2.4, Remark 2.5 and Lemma 2.7, we obtain

Tb(ε1Wh)S(h)Uε(t, s) = T2b

12ε2h

∈ B(X).

280 Y. Lu / J. Math. Anal. Appl. 400 (2013) 273–284

Thus,

Uε(t, s) = S(h)−1Tb(ε1Wh)−1T2b

12ε2h,

Uε(t, s)−1= T2b

12ε2h−1

Tb(ε1Wh)S(h),

and R(T2b( 12ε2h)) ∈ D(Tb(ε1Wh)

−1), R(Tb(ε1Wh)−1T2b( 12ε

2h)) ∈ D(S(h)−1).Similarly, one can obtain the assertion for the case where1Wh < 0.

Theorem 3.9. Under the assumptions of Definition 3.1, then for almost every Brownian sample path,

limε→0

Uε(t, s)S(h)x = x, x ∈ X . (3.11)

Proof. It follows from Lemma 3.2 and Corollary 3.3, if we notice that Uε(t, s)S(h) = U2(t, s)when φ = 0.

4. The solution of the approximating equation

In this section, we will consider the relationship between the approximating equation (1.3) and the stochastic evolutionfamily (3.1).

Theorem 4.1. Assume ε > 0; the approximating equation (1.3) has a unique solution which is given by Uε(t, 0)x for x ∈ X,where Uε(t, 0) is the stochastic evolution family defined in Definition 3.1.

Proof. Existence. It is obvious that the stochastic function f (λ, t, 0) is a geometric Brownianmotionwhich is the solution toa simple scalar stochastic differential equation with initial value f (λ, 0, 0) = 1. Therefore we convert the abstract equation(1.3) to a scalar stochastic differential equation via functional calculus for operator A. By Itô’s formula for the differential off (λ, t, s), we have

df (λ, t, 0) = f (λ, t, 0)(λdt + ελbdWt)

or by the definition of a stochastic differential,

f (λ, t, 0)− f (λ, 0, 0) =

t

0λf (λ, τ , 0)dτ +

t

0ελbf (λ, τ , 0)dWτ .

Then for any s ∈ (0, t),

f (λ, t, 0)− f (λ, s, 0) =

t

sλf (λ, τ , 0)dτ +

t

sελbf (λ, τ , 0)dWτ .

Computing the increment Uε(t, 0)x − Uε(s, 0)x,

Uε(t, 0)x − Uε(s, 0)x =1

2π i

Γ

[f (λ, t, 0)− f (λ, s, 0)]R(λ, A)xdλ

=1

2π i

Γ

λ

t

sf (λ, τ , 0)dτ

R(λ, A)xdλ

+1

2π i

Γ

ελb

t

sf (λ, τ , 0)dWτ

R(λ, A)xdλ.

Changing the order of integration gives

Uε(t, 0)x − Uε(s, 0)x =

t

s

12π i

Γ

λf (λ, τ , 0)R(λ, A)xdλdτ + ε

t

s

12π i

Γ

λbf (λ, τ , 0)R(λ, A)xdλdWτ .

Since1

2π i

Γ

λf (λ, τ , 0)R(λ, A)xdλ = AUε(τ , 0)x,

12π i

Γ

λbf (λ, τ , 0)R(λ, A)xdλ = AbUε(τ , 0)x,

Y. Lu / J. Math. Anal. Appl. 400 (2013) 273–284 281

and the operators AUε(τ , 0) and AbUε(τ , 0) are almost surely bounded, we obtain

Uε(t, 0)x − Uε(s, 0)x =

t

sAUε(τ , 0)xdτ + ε

t

sAbUε(τ , 0)xdWτ . (4.1)

Hence, Uε(t, 0)x is a solution of the approximating equation (1.3).Uniqueness. Suppose there exist another evolution family Vε(t, 0) such that Vε(t, 0)x is also a solution of the

approximating equation (1.3). Then for any x ∈ D(A)

D(Ab), Vε(t, 0)x ∈ D(A)

D(Ab)(t ≥ 0) and

Vε(t, 0)x − x =

t

0AVε(τ , 0)xdτ +

t

0εAbVε(τ , 0)xdWτ . (4.2)

Since ( 12ε2A2b

− A)Uε(t, s), εAbUε(t, s) ∈ B(X), P-a.s., by Itô’s formula,

Uε(t, s)− Uε(t, 0) =

s

0(ε2A2b

− A)Uε(t, τ )dτ −

s

0εAbUε(t, τ )dWτ . (4.3)

For x ∈ D(A)

D(Ab) and t = s, now we consider the function s → Uε(t, s)Vε(s, 0)x. By Itô’s formula, (4.2) and (4.3) imply

Uε(t, s)Vε(s, 0)x − Uε(t, 0)Vε(0, 0)x =

s

0

(ε2A2b

− A)Uε(t, τ )Vε(τ , 0)x + Uε(t, τ )AVε(τ , 0)x

− ε2AbUε(t, τ )AbVε(τ , 0)xdτ

+

s

0

−εAbUε(t, τ )Vε(τ , 0)x + Uε(t, τ )εAbVε(τ , 0)x

dWτ .

Since Uε(t, τ ) commutes with A and Ab on D(A)

D(Ab), we obtain

Uε(t, s)Vε(s, 0)x = Uε(t, 0)Vε(0, 0)x = Uε(t, 0)x.

Thus Uε(t, s)Vε(s, 0)x does not depend on s. Theorem 3.7 implies that Uε(t, s)Vε(s, 0)x is continuous at s = t . Hence for anys ∈ [0, t] and x ∈ D(A)

D(Ab),

Uε(t, s)Vε(s, 0)x = Uε(t, 0)x.

Therefore Vε(t, 0)x = Uε(t, 0)x holds for x ∈ D(A)

D(Ab). The condition of boundedness of Uε(t, 0) and D(A)

D(Ab) isdense in X , guaranteeing that Vε(t, 0)x = Uε(t, 0)x holds for all x ∈ X .

Hence, Uε(t, 0)x is the unique solution of the approximating equation (1.3).

Theorem 4.1 shows that (1.3) has a unique solution, and Theorem 3.4 implies that Uε(t, 0) ∈ B(X), P-a.s. Therefore (1.3)is well-posed.

5. Regularization for the problem

In this section, we show that Uε(t, 0) defined in Definition 3.1 is a family of regularizing operators for problem (1.1). Tobe precise, we start with the definition of families of regularizing operators.

Definition 5.1. A family Rε,t; ε > 0, t ∈ [0, T ] ∈ B(X) is called a family of regularizing operators for problem (1.1) if foreach solution u(t)(0 ≤ t ≤ T ) of (1.1) with initial element x, for any δ > 0, there exists ε(δ) > 0 such that

(a) ε(δ) → 0 as δ → 0,(b) ∥Rε(δ),txδ − u(t)∥ → 0 as δ → 0 for each t ∈ [0, T ], where ∥xδ − x∥ ≤ δ.

The main result of this paper is as follows:

Theorem 5.2. Uε(t, 0) defined in Definition 3.1 is a family of regularizing operators for problem (1.1).

Proof. Let u(t) (0 ≤ t ≤ T ) be a solution of (1.1) with initial value x, and ∥xδ − x∥ ≤ δ. The approximating equation (3.1)with initial value xδ has a unique solution uε,δ := Uε(t, 0)xδ .

When t = 0, it is clear that

limδ→0

∥Uε(0, 0)xδ − u(0)∥ = limδ→0

∥xδ − x∥ → 0.

When t ∈ (0, T ], we have

∥Uε(t, 0)xδ − u(t)∥ ≤ ∥Uε(t, 0)xδ − Uε(t, 0)x∥ + ∥Uε(t, 0)x − u(t)∥:= E1 + E2.

282 Y. Lu / J. Math. Anal. Appl. 400 (2013) 273–284

We first estimate E2. Note that x = S(t)u(t) (see Theorem A.3), where S(t) is the holomorphic semigroup generated by −A.It follows from Theorem 3.9 that for almost every Brownian sample path,

E2 = ∥Uε(t, 0)x − u(t)∥ = ∥Uε(t, 0)S(t)u(t)− u(t)∥ → 0 as δ → 0, (5.1)

whenever ε → 0 as δ → 0. As for E1, it follows from Theorem 3.4 that for almost every Brownian sample path,

E1 ≤ δ∥Uε(t, 0)∥ ≤ δM3eC1ε2

1−2b t+C2(1Wt )2

t .

Choose

ε = (−TC1(ln δγ )−1)1−2b

2 , 0 < δ < 1, 0 < γ < 1;

then ε → 0 as δ → 0 and for almost every Brownian sample path,

E1 ≤ M3eC2(1Wt )2

t δeC1ε2

2b−1 T= M3eC2

(1Wt )2t δ1−γ → 0 as δ → 0. (5.2)

Combining (5.1) with (5.2), we obtain for almost every Brownian sample path,

∥Uε(t, 0)xδ − u(t)∥ → 0 as δ → 0, t ∈ (0, T ].

Thereby Uε(0, t) is a family of regularizing operators for problem (1.1).

Remark 5.3. In the general case where −A is the generator of a holomorphic semigroup with 0 ∈ ρ(A), one can replace Aby A− ξ , where ξ ∈ R is a constant such that ξ −A is the generator of a bounded holomorphic semigroup and 0 ∈ ρ(A− ξ);then the results in this paper hold true for such a case.

Example 5.1. We consider the following Cauchy problem (cf. [14]):∂

∂tu(t, x) = P(D)u(t, x), for (t, x) ∈ (0, T ] × Rn,

u(0, x) = u0(x), for x ∈ Rn,

(5.3)

where P : Rn→ C is a polynomial of order 2m. The principal part of P(ξ) is denoted by P2m(ξ). Assume that P(D) is strongly

elliptic. Then (see, e.g., [31]) −P(D) with domain W 2m,p(Rn) is the generator of a holomorphic semigroup of angle α onLp(R)n(1 < p < ∞), where

α = arctan

sup|ξ |=1

|ReP2m(ξ)|

sup|ξ |=1

|ImP2m(ξ)|

.

Obviously, problem (5.3) is ill-posed.When α ≤ π/4, The method in [19] is inapplicable. However, the modified method in this paper is always applicable for

all α ∈ (0, π/2].

Acknowledgments

The author thanks the referees for some helpful remarks, comments and suggestions.

Appendix

We give some properties of the inverse of a holomorphic semigroup and the solution of problem (1.1)

Lemma A.1 ([9, p. 69]). Suppose S(z)z∈Sα is a strongly continuous holomorphic semigroup; then S(z) is injective and has denserange for all z ∈ Sα .

It follows from Lemma A.1 that the operator S(z)−1 (z ∈ Sα) is closed and densely defined with domain D(S(z)−1) =

R(S(z)).Let Pz,α be a parallelism, where two of its vertexes are the origin and z, and its sides are parallel to either eiα or e−iα , and

let ∂Pz,α be the boundary of Pz,α , and D(A∞) :=

∞

n=0 D(An).

Lemma A.2. Let −A be the infinitesimal generator of a holomorphic semigroup S(z) of angle α; then:

(i) For z ∈ Sα , and λ ∈ Pz,α \ ∂Pz,α ,

R(S(z)) ⊂ R(S(λ)) ⊂ D(A∞).

Y. Lu / J. Math. Anal. Appl. 400 (2013) 273–284 283

(ii) For z ∈ Sα, x ∈ R(S(z)) the mapping λ → S(λ)−1x is holomorphic in Pz,α/∂Pz,α , and

ddλ

S(λ)−1x = AS(λ)−1x.

Proof. (i) Since z − λ ∈ Pz,α \ ∂Pz,α ⊂ Sα , then R(S(z)) ⊂ R(S(λ)) follows from S(z) = S(λ)S(z − λ). For x ∈ X, z ∈ Sα ,

AS(z)x = A1

2π i

Γ

e−zλR(λ, A)xdλ

=1

2π i

Γ

λe−zλR(λ, A)xdλ−

1

2π i

Γ

e−zλdλ

x

=1

2π i

Γ

λe−zλR(λ, A)xdλ,

where Γ = ∂Sθ (π/2 − α < θ < π/2 − | arg z|) the boundary of ∂Sθ oriented in the positive sense, andΓe−zλdλ = 0.

Then we obtain ∥AS(z)x∥ < ∞, that is R(S(z)) ⊂ D(A). Similarly R(S(z)) ⊂ D(A2). Repeating the process, thenR(S(z)) ⊂ D(A∞)with

AnS(z)x =1

2π i

Γ

λne−zλR(λ, A)xdλ,

for any x ∈ X, n ∈ N and z ∈ Sα .(ii) By (i), then for x ∈ R(S(z)), there exists y ∈ X such that x = S(z)y. Therefore,

S(λ)−1x = S(λ)−1S(z)y = S(λ)−1S(λ)S(z − λ)y = S(z − λ)y.

Since S(z − λ) is holomorphic in Pz,α \ ∂Pz,α , then for z ∈ Sα the mapping λ → S(λ)−1x is holomorphic in Pz,α \ ∂Pz,α and

ddλ

S(λ)−1x =ddλ

S(z − λ)y = AS(z − λ)y = AS(λ)−1x.

The next theorem follows from Lemma A.2.

Theorem A.3. Let −A be the infinitesimal generator of a holomorphic semigroup S(z) of angle α. The Cauchy problem (1.1) hasa unique solution if and only the initial value x ∈ R(S(T )). The solution u(t) = S(t)−1x, and u(t) ∈ R(S(T − t)) for t ∈ [0, T ].

Proof. If the initial value x ∈ R(S(T )), then from Lemma A.2 it follows that u(t) = S(t)−1x is the unique solution of (1.1)and u(t) ∈ R(S(T − t)) for t ∈ [0, T ].

If (1.1) has a solution u(t)(t ∈ [0, T ]), then u(T − t)(t ∈ [0, T ]) is obviously a solution ofv′(t) = −Av(t), 0 < t ≤ T ,v(0) = u(T ). (A.1)

By the uniqueness of solutions of (A.1), one can obtain that S(t)u(T ) = u(T − t)(t ∈ [0, T ]), which implies thatu(t) ∈ R(S(T − t)) for t ∈ [0, T ] and S(T )u(T ) = u(0) = x by using the strong continuity of the C0-semigroup. Thusx ∈ R(S(T )) and S(t)u(t) = S(t)S(T − t)u(T ) = S(T )u(T ) = x for t ∈ [0, T ]. By Lemma A.1, S(t) is invertible for t ≥ 0; thenu(t) = S(t)−1x for t ∈ [0, T ]. The uniqueness of the solution of (1.1) follows from the uniqueness of the C0-semigroup.

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