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458 A. Lunardi

Proof. Let K = {ϕ ∈ C

): ∀x ∈ R

,ϕ(x) ≥ 0} be the cone of the nonnegative

functions in C

). By Remark 3.2(iv), if ϕ ∈ C

) \{0} is such that ϕ(x) ≥ 0

for each x, then inf V (s)ϕ =infG(s + T,s)ϕ(x) > 0. In other words, V (s)maps

K \{0} into the interior part of K. Theorem 4.1 implies that the spectral radius

1ofV (s) is a simple eigenvalue, and it is the unique eigenvalue of V (s)onthe

unit circle. The associated spectral projection P (s)=I −Q(s), with Q(s) deﬁned

above, may be expressed as

P (s)ϕ = m

ϕ 1 ,ϕ∈ C

for some m

in the dual space of C

). To prove that m

ϕ =



ϕ(y)μ

(dy)for

some measure μ

we use the Stone–Daniell Theorem (e.g., [4, Thm. 4.5.2]): it is

enough to check that m

ϕ ≥ 0ifϕ ≥ 0, and that for each sequence (ϕ

) ⊂ C

)

such that (ϕ

(x)) is decreasing and converges to 0 for each x ∈ R

,wehave

lim

n→∞

=0.Inthiscase,μ

is a probability measure for every s, because

P (s)1 = 1.

By the general spectral theory, P (s) = lim

λ→1

−

,whereV

:= (λ−1)(λI −

V (s))

−1

. In its turn, (λI − V (s))

−1

∞

k=0

V (s)

/λ

k+1

maps nonnegative func-

tions into nonnegative functions because V (s) does. Therefore, m

ϕ 1 = P (s)ϕ ≥ 0

for each ϕ ≥ 0.

Let now ϕ

↓ 0. We claim that V (s)ϕ

converges to 0 uniformly. Indeed,

since the measures {p

s+T,s,x

: x ∈ R

} are tight, for each ε>0thereisR>0

such that



\B(0,R)

s+T,s,x

(dy) ≤ ε,foreachx ∈ R

. On the other hand, ϕ

converges to 0 uniformly on B(0,R) by the Dini Monotone Convergence Theorem,

so that for n large, say n ≥ n

,wehaveϕ

(y) ≤ ε,for|y|≤R. Therefore, for

n ≥ n

we have

0 ≤ V (s)ϕ

(x)=



B(0,R)

(y)p

s+T,s,x

(dy)+



\B(0,R)

(y)p

s+T,s,x

(dy)

≤ ε + ϕ



∞

for each x ∈ R

.SinceV (s)ϕ

converges to 0 uniformly, then P (s)V (s)ϕ

V (s)P (s)ϕ

converges to 0 uniformly. But V (s) is the identity on the range of P (s).

Then, P (s)ϕ

converges uniformly to 0, which implies that lim

n→∞

=0.

Let us prove that {μ

: s ∈ R} is a T -periodic evolution system of measures.

Since s → P (s)isT -periodic, then μ

= μ

s+T

for each s ∈ R.Moreover,since

V (t)G(t, s)=G(t, s)V (s), then P (t)G(t, s)ϕ = G(t, s)P (s)ϕ,foreachϕ ∈ C

This means



G(t, s)ϕdμ

1 = G(t, s)





ϕdμ



,ϕ∈ C

and since G(t, s)1 = 1,then



G(t, s)ϕdμ



ϕdμ

,ϕ∈ C

so that {μ

: s ∈ R} is an evolution system of measures. 

Compactness and Asymptotic Behavior 459

Corollary 4.3. Assume that G(t, s) is compact in C

) for t>s.Then:

(i) There exists a unique T -periodic evolution system of measures {μ

: s ∈ R};

(ii) Setting ω

=logr/T ,wherer is deﬁned in (4.1),foreachω>ω

there exists

M = M (ω) > 0 such that

G(t, s)ϕ −



ϕdμ



∞

≤ Me

ω(t−s)

ϕ

∞

,t>s,ϕ∈ C

), (4.3)

while for ω<ω

there is no M such that (4.3) holds.

Proof. Let {μ

: s ∈ R} be the T -periodic evolution system of measures given

by Proposition 4.2. Since P (s)ϕ =



ϕdμ

1, estimate (4.2) implies (4.3). Since

there exist eigenvalues of V (s) with modulus r, (4.3) cannot hold for ω<ω

Indeed, if V (s)ϕ = rϕ then P (s)ϕ =0andG(s + nT, s)ϕ = r

ϕ = e

ϕ for

each n ∈ N,sothatG(s + nT, s)ϕ −m

ϕ

∞

= G(s + nT, s)ϕ

∞

= e

ϕ

∞

If {ν

: s ∈ R} is another T -periodic evolution system of measures, ﬁx t ∈ R

and ϕ ∈ C

). Since G(t, s)ϕ −



ϕdμ

goes to zero uniformly as s →−∞,

then

0 = lim

s→−∞





G(t, s)ϕ −



ϕdμ



dν

= lim

s→−∞





ϕdν

−



ϕdμ



Since s →



ϕdν

−



ϕdμ

is T -periodic and goes to 0 as s →−∞,itvanishes

in R. By the arbitrariness of ϕ, ν

= μ

for every s ∈ R. 

Once we have an evolution system of measures {μ

: s ∈ R}, G(t, s)isex-

tendable to a contraction (still denoted by G(t, s)) from L

,μ

)toL

,μ

)

for t>s. Compactness and asymptotic behavior results in C

) imply compact-

ness and asymptotic behavior results in such L

spaces, as the next proposition

shows.

Proposition 4.4. Let G(t, s) be compact in C

) for t>s. Then for every p ∈

(1, ∞), G(t, s):L

,μ

) → L

,μ

) is compact for t>s. Moreover, for every

ω ∈ (ω

, 0) and p ∈ (1, ∞) there exist M = M(ω, p) > 0 such that

G(t, s)ϕ −



ϕdμ



,μ

)

≤ Me

ω(t−s)

ϕ

,μ

)

,t>s,ϕ∈ L

,μ

(4.4)

and for every ω<ω

there is no M such that (4.4) holds. Here ω

is given by

Corollary 4.3(ii).

Proof. Let us prove that G(t, s):L

,μ

) → L

,μ

)iscompactfort>s.

We have G(t, s)=G(t, (t + s)/2)G((t + s)/2,s)whereG((t + s)/2,s) is bounded

from L

∞

)toC

) by Theorem 2.2(iv), and G(t, (t + s)/2) is compact in

). Therefore, G(t, s)iscompactinL

∞

Now, if μ

and μ

are probability measures and a linear operator is bounded

from L

,μ

)toL

,μ

)andcompactfromL

∞

,μ

)toL

∞

,μ

), then

it is compact from L

,μ

)toL

,μ

), for every p ∈ (1, ∞) (the proof is the

same as in [13, Prop. 4.6], where only one probability measure was considered).

460 A. Lunardi

Let us prove (4.4). Since L

∞

,μ

)=L

∞

,μ

)=L

∞

,dx)by[8,

Prop. 5.2], interpolating (4.3) and G(t, s)ϕ − m

ϕ

,μ

)

≤ 2ϕ

,μ

)

obtain that G(t, s)−m



L(L

,μ

),L

,μ

))

decays exponentially as (t−s) →∞.

However, the decay rate that we obtain by interpolation depends on p.Toprove

(4.4) it is enough to show that the spectrum of the operators V (s)inL

,μ

)

does not depend on s, and coincides with the spectrum in C

). Since V (s)=

G(s+T,s), then V (s)iscompactinL

,μ

). Therefore, its L

spectrum (except

zero) consists of eigenvalues, that are independent of s. They are independent of

p too, as well as the associated spectral projections, by [2, Cor. 1.6.2].

The statement follows now as in the case of evolution operators in a ﬁxed

Banach space as in the mentioned references [7, 11, 6]. Note however that our

Banach spaces L

,μ

)varywiths, so that the classical theory cannot be used

verbatim. For the reader’s convenience we give the proof below.

Let t −s = σ + kT,withk ∈ N and σ ∈ [0,T). We have

G(t, s)ϕ − m

ϕ

,μ

)

= G(t, t − σ)V (s)

(I − P (s))ϕ

,μ

)

≤V (s)

(I −P (s))ϕ

,μ

)

= [V (s)(I − P (s))]

ϕ

,μ

)

For ω>ω

let ε>0 be such that log(r + ε) ≤ ω,andletk(s) ∈ N be such that

[V (s)(I −P (s))]

ϕ

,μ

)

≤ (r +ε)

for each k>k(s). Therefore, if the integer

part [(t − s)/T ] is larger than k(s)wehave

G(t, s)ϕ − m

ϕ

,μ

)

≤ (r + ε)

ϕ

,μ

)

≤ e

(t−s)ω

ϕ

,μ

)

for each ϕ ∈ L

,μ

). Using the obvious inequality

G(t, s)ϕ − m

ϕ

,μ

)

≤ 2ϕ

,μ

)

for [(t − s)/T ] ≤ k(s),

we arrive at

G(t, s)ϕ − m

ϕ

,μ

)

≤ M

(t−s)ω

ϕ

,μ

)

,ϕ∈ L

,μ

)

for some M

> 0. It remains to show that M

can be taken independent of s.Since

V is T -periodic, we may take k(s)=k(s + T ) and hence M

= M

s+T

for every

s ∈ R. Therefore it is enough to show that M

can be taken independent of s for

s ∈ [0,T). For 0 ≤ s<T and t ≥ T we have m

G(T,s)ϕ =



G(T,s)ϕdμ

ϕ, hence

G(t, s)ϕ − m

ϕ

,μ

)

= (G(t, T ) − m

)G(T,s)ϕ

,μ

)

≤ M

ω(t−T )

G(T,s)ϕ

,μ

)

≤ M

|ω|T

ω(t−s)

ϕ

,μ

)

So, we can take M

= M

|ω|T

for 0 ≤ s<T. (4.4) follows. 

Compactness and Asymptotic Behavior 461

References

[1] G. Da Prato, A. Lunardi, Ornstein-Uhlenbeck operators with time periodic coeﬃ-

cients,J.Evol.Equ.7 (2007), no. 4, 587–614.

[2] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge,

1989.

[3] Y. Du, Order structure and topological methods in nonlinear partial diﬀerential

equations. Vol. 1. Maximum principles and applications. Series in Partial Diﬀer-

ential Equations and Applications, 2. World Scientiﬁc Publishing Co. Pte. Ltd.,

Hackensack, NJ, 2006.

[4] R.M. Dudley, Real Analysis and Probability. Cambridge studies in advanced math-

ematics 74. Cambridge University Press, Cambridge, 2002.

[5] A. Friedman, Partial diﬀerential equations of parabolic type. Prentice Hall, 1964.

[6] M. Fuhrman, Bounded solutions for abstract time-periodic parabolic equations with

nonconstant domains,Diﬀ.Int.Eqns.4 (1991), no. 3, 493–518.

[7] D. Henry, Geometric theory of semilinear parabolic equations, Lect. Notes in Math.

840, Springer-Verlag, New York, 1981.

[8] M. Kunze, L. Lorenzi, A. Lunardi, Nonautonomous Kolmogorov parabolic equations

with unbounded coeﬃcients. Trans. Amer. Math. Soc. 362 (2010), no. 1, 169–198.

[9] O.A. Ladyz’enskaya, V.A. Solonnikov, N.N. Ural’tseva, Linear and Quasilinear Equa-

tions of Parabolic Type, Amer. Math. Soc., Rhode Island, Providence, 1968.

[10] L. Lorenzi, A. Lunardi, A. Zamboni, Asymptotic behavior in time periodic parabolic

problems with unbounded coeﬃcients,J.Diﬀ.Eqns.249 (2010), 3377–3418.

[11] A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems,

Birkh¨auser, Basel, 1995.

[12] G. Metafune, D. Pallara, M. Wacker, Feller semigroups in R

, Semigroup Forum 65

(2002), no. 2, 159–205.

[13] G. Metafune, D. Pallara, M. Wacker, Compactness properties of Feller semigroups,

Studia Math. 153 (2002), no. 2, 179–206.

Alessandra Lunardi

Dipartimento di Matematica

Universit`a di Parma

Viale G.P. Usberti, 53/A

I-43124 Parma, Italy

e-mail: alessandra.lunardi@unipr.it

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 463–477

 2011 Springer Basel AG

Gradient Estimates and Domain Identiﬁcation

for Analytic Ornstein-Uhlenbeck Operators

Jan Maas and Jan van Neerven

Dedicated to Herbert Amann on the oc casion of his 70th birthday

Abstract. Let P be the Ornstein-Uhlenbeck semigroup associated with the

stochastic Cauchy problem

dU(t)=AU(t) dt + dW

(t),

where A is the generator of a C

-semigroup S on a Banach space E, H is

a Hilbert subspace of E,andW

is an H-cylindrical Brownian motion. As-

suming that S restricts to a C

-semigroup on H,weobtainL

-bounds for

P (t). We show that if P is analytic, then the invariance assumption is

fulﬁlled. As an application we determine the L

-domain of the generator of

P explicitly in the case where S restricts to a C

-semigroup on H which is

similar to an analytic contraction semigroup. The results are applied to the

1D stochastic heat equation driven by additive space-time white noise.

Mathematics Subject Classiﬁcation (2000). 47D07 (35R15, 42B25, 60H15).

Keywords. Ornstein-Uhlenbeck operators, gradient estimates, domain identi-

ﬁcation.

1. Introduction

Consider the stochastic Cauchy problem

dU(t)=AU(t) dt + dW

(t),t 0,

U(0) = x.

(SCP)

Here A generates a C

-semigroup S =(S(t))

t0

on a real Banach space E, H

is a real Hilbert subspace continuously embedded in E, W

is an H-cylindrical

Brownian motion on a probability space (Ω, F P ), and x ∈ E.Aweak solution is

The authors are supported by VIDI subsidy 639.032.201 (JM) and VICI subsidy 639.033.604

(JvN) of the Netherlands Organisation for Scientiﬁc Research (NWO).

464 J. Maas and J. van Neerven

a measurable adapted E-valued process U

=(U

(t))

t0

such that t → U

(t)is

integrable almost surely and for all t  0andx

∗

∈ D(A

∗

) one has

U

(t),x

∗

 = x, x

∗

 +



U

(s),A

∗

ds + W

(t)i

∗

almost surely.

Here i : H→ E is the inclusion mapping. A necessary and suﬃcient condition for

the existence of a weak solution is that the operators I

: L

(0,t; H) → E,

g :=



S(s)ig(s) ds,

are γ-radonifying for all t  0. If this is the case, then s → S(t−s)i is stochastically

integrable on (0,t) with respect to W

and the process U

is given by

(t)=S(t)x +



S(t − s)idW

(s),t 0.

For more information and an explanation of the terminology we refer to [30].

Assuming the existence of the solution U

, on the Banach space C

(E)of

all bounded continuous functions f : E → R one deﬁnes the Ornstein-Uhlenbeck

semigroup P =(P (t))

t0

P (t)f (x):=Ef(U

(t)),t 0,x∈ E. (1.1)

The operators P (t) are linear contractions on C

(E) and satisfy P (0) = I and

P (s)P (t)=P (s + t) for all s, t  0. For all f ∈ C

(E) the mapping (t, x) →

P (t)f (x) is continuous, uniformly on compact subsets of [0, ∞) ×E.

If the operator I

∞

: L

(0, ∞; H) → E deﬁned by

∞

g :=



∞

S(t)ig(t) dt

is γ-radonifying, then the problem (SCP) admits a unique invariant measure μ

∞

This measure is a centred Gaussian Radon measure on E, and its covariance

operator equals I

∞

∗

∞

. Throughout this paper we shall assume that this measure

exists; if (SCP) has a solution, then this assumption is for instance fulﬁlled if S

is uniformly exponentially stable. The reproducing kernel Hilbert space associated

with μ

∞

is denoted by H

∞

. The inclusion mapping H

∞

→ E is denoted by i

∞

Recall that Q

∞

:= i

∞

∗

∞

= I

∞

∗

∞

.ItiswellknownthatS restricts to a C

contraction semigroup on H

∞

[5] (the proof for Hilbert spaces E extends without

change to Banach spaces E), which we shall denote by S

∞

By a standard application of Jensen’s inequality, the semigroup P has a

unique extension to a C

-contraction semigroup to the spaces L

(E,μ

∞

), 1 

p<∞. By slight abuse of notation we shall denote this semigroup by P again. Its

inﬁnitesimal generator will be denoted by L. In order to give an explicit expression

for L it is useful to introduce, for integers k,l  0, the space F C

k,l

(E) consisting

of all functions f ∈ C

(E)oftheform

f(x)=ϕ(x, x

∗

,...,x, x

∗

)

Ornstein-Uhlenbeck Operators 465

with f ∈ C

)andx

∗

,...,x

∗

∈ D(A

∗l

). With this notation one has that

F C

2,1

(E)isacoreforL, and on this core one has

Lf(x)=

tr D

f(x)+x, A

∗

Df(x).

Here,

f(x)=

n=1

∂ϕ

∂x

(x, x

∗

,...,x, x

∗

) ⊗ i

∗

Df(x)=

n=1

∂ϕ

∂x

(x, x

∗

,...,x, x

∗

) ⊗ x

∗

denote the Fr´echet derivatives into the directions of H and E, respectively.

2. Gradient estimates: the H-invariant case

Our ﬁrst result gives a pointwise gradient bound for P under the assumption that

S restricts to a C

-semigroup on H which will be denoted by S

. As has been

shown in [17, Corollary 5.6], under this assumption the operator D

is closable as

a densely deﬁned operator from L

(E,μ

∞

)toL

(E,μ; H) for all 1  p<∞.The

domain of its closure is denoted by D

Proposition 2.1 (Pointwise gradient bounds). If S restricts to a C

-semigroup on

H, then for all 1 <p<∞ there exists a constant C  0 such that for all t>0

and f ∈ F C

1,0

(E) we have

√

t|D

P (t)f (x)|  Cκ(t)(P (t)|f|

(x))

1/p

where κ(t):=sup

s∈[0,t]

S

(s)

L (H)

Proof. The proof follows the lines of [25, Theorem 8.10] and is inspired by the

proof of [10, Theorem 6.2.2], where the null controllable case was considered.

The distribution μ

of the random variable U

(t) is a centred Gaussian Radon

measure on E.LetH

denote its RKHS and let i

: H

→ E be the inclusion

mapping. As is well known and easy to prove, cf. [9, Appendix B] one has





S(t −s)ig(s) ds : g ∈ L

(0,t; H)



with

h

=inf



g

(0,t;H)

: h =



S(t −s)ig(s) ds



The mapping

: i

∗

→·,x

∗

,x

∗

∈ E

∗

deﬁnes an isometry from H

onto a closed subspace of L

(E,μ

). For h ∈ H

shall write φ

(x):=(φ

h)(x).

466 J. Maas and J. van Neerven

Fix h ∈ H.SinceS restricts to a C

-semigroup S

on H we may consider

the function g ∈ L

(0,t; H)givenbyg(s)=

S(s)h.FromtheidentityS(t)h =



S(t − s)g(s) ds we deduce that S(t)h ∈ H

and

S(t)h

 g

(0,t;H)



S(s)h

ds 

κ(t)

h

. (2.1)

Fix a function f ∈ F C

1,0

(E), that is, f(x)=ϕ(x, x

∗

,...,x, x

∗

)with

ϕ ∈ C

)andx

∗

,...,x

∗

∈ E

∗

. It is easily checked that for all t>0we

have P (t)f ∈ F C

1,0

(E); in particular this implies that P (t)f ∈ D

). By the

Cameron-Martin formula [3],



P (t)f (x + εh) −P (t)f(x)







f(S(t)(x + εh)+y) − f(S(t)x + y)



dμ

(y)



εS(t)h

− 1)f(S(t)x + y) dμ

(y),

where for h ∈ H

we write

(x):=exp(φ

(x) −

h

It is easy to see that for each h ∈ H

the family



εh

− 1)



0<ε<1

is uniformly

bounded in L

(E,μ

), and therefore uniformly integrable in L

(E,μ

). Passage to

the limit ε ↓ 0 in the previous identity now gives

P (t)f (x),h]=



f(S(t)x + y)φ

S(t)h

(y) dμ

(y).

By H¨older’s inequality with

= 1 and the Kahane-Khintchine inequality,

which can be applied since φ

S(t)h

is a Gaussian random variable,

|[D

P (t)f (x),h]|







|f(S(t)x + y)|

dμ

(y)







|φ

S(t)h

(y)|

dμ

(y)



 K





|f(S(t)x + y)|

dμ

(y)







|φ

S(t)h

(y)|

dμ

(y)



= K

(P (t)|f|

(x))

S(t)h

Using (2.1) we ﬁnd that

√

t[D

P (t)f (x),h]

 K

κ(t)(P (t)|f|

(x))

h

and by taking the supremum over all h ∈ H of norm 1 we obtain the desired

estimate. 

Corollary 2.2. If S restricts to a C

-semigroup on H, then for all 1 <p<∞ the

operators D

P (t), t>0, extend uniquely to bounded operators from L

(E,μ

∞

)

to L

(E,μ

∞

; H), and there exists a constant C  0 such that for any t>0,

√

tD

P (t)

L (L

(E,μ

∞

),L

(E,μ

∞

;H))

 Cκ(t).

Ornstein-Uhlenbeck Operators 467

Proof. Integrating the inequality of the proposition and using the fact that μ

∞

an invariant measure for P we obtain



√

P (t)f 

(E,μ

∞

)

 C

κ(t)



P (t)|f |

(x) dμ

∞

(x)

= C

κ(t)



|f|

(x) dμ

∞

(x)=C

κ(t)

f

(E,μ

∞

)

. 

3. Gradient estimates: the analytic case

Analyticity of the semigroup P on L

(E,μ

∞

) has been investigated by several

authors [15, 16, 18, 24]. The following result of [18] is our starting point. Recall

that in the deﬁnition of an analytic C

-contraction semigroup, contractivity is

required on an open sector containing the positive real axis.

Proposition 3.1. For any 1 <p<∞ the following assertions are equivalent:

(1) P is an analytic C

-semigroup on L

(E,μ

∞

);

(2) P is an analytic C

-contraction semigroup on L

(E,μ

∞

);

(3) S restricts to an analytic C

-contraction semigroup on H

∞

;

(4) Q

∞

∗

acts as a bounded operator in H.

A more precise formulation of (4) is that there should exist a bounded op-

erator B : H → H such that iBi

∗

= Q

∞

∗

for all x

∗

∈ E

∗

. The identity

∞

∗

+ AQ

∞

= −ii

∗

implies that B + B

∗

= −I.

In what follows we shall simply say that ‘P is analytic’ to express that the

equivalent conditions of the proposition are satisﬁed for some (and hence for all)

1 <p<∞.

The next result has been shown in [24] for p = 2 and was extended to 1 <

p<∞ in [25].

Proposition 3.2. If P is analytic, then F C

2,1

(E) is a core for the generator L of

P in L

(E,μ

∞

),andonthiscoreL is given by

L = D

∗

Our ﬁrst aim is to show that analyticity of P implies that H is S-invariant.

For self-adjoint P this was proved in [7, 18].

Theorem 3.3. If P is analytic, then S restricts to a bounded analytic C

-semigroup

on H.

Proof. Consider the linear mapping

V : i

∗

∞

∗

→ i

∗

∈ E

∗

. (3.1)

It is shown in [17] that i

∗

∞

∗

= 0 implies i

∗

= 0, so that this mapping is well

deﬁned, and that the closability of D

implies the closability of V as a densely

deﬁned operator from H

∞

to H. With slight abuse of notation we denote its closure

by V again and let D(V ) the domain of the closure.