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Communications in Partial Differential Equations

ISSN: 0360-5302 (Print) 1532-4133 (Online) Journal homepage: https://www.tandfonline.com/loi/lpde20

The Global Random Attractor for a Class ofStochastic Porous Media Equations

Wolf-Jürgen Beyn , Benjamin Gess , Paul Lescot & Michael Röckner

To cite this article: Wolf-Jürgen Beyn , Benjamin Gess , Paul Lescot & Michael Röckner (2010)The Global Random Attractor for a Class of Stochastic Porous Media Equations, Communicationsin Partial Differential Equations, 36:3, 446-469, DOI: 10.1080/03605302.2010.523919

To link to this article: https://doi.org/10.1080/03605302.2010.523919

Published online: 22 Dec 2010.

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Communications in Partial Differential Equations, 36: 446–469, 2011Copyright © Taylor & Francis Group, LLCISSN 0360-5302 print/1532-4133 onlineDOI: 10.1080/03605302.2010.523919

The Global Random Attractor for a Classof Stochastic PorousMedia Equations

WOLF-JÜRGEN BEYN1, BENJAMIN GESS1,PAUL LESCOT2, AND MICHAEL RÖCKNER1�3

1Faculty of Mathematics, University of Bielefeld, Bielefeld, Germany2Laboratoire de Mathématiques Raphaël Salem, CNRS, UMR 6085,Université de Rouen, Rouen, France3Department of Mathematics and Statistics, Purdue University,West Lafayette, Indiana, USA

We prove new L2-estimates and regularity results for generalized porous mediaequations “shifted by” a function-valued Wiener path. To include Wiener paths withmerely first spatial (weak) derivates we introduce the notion of “�-monotonicity” forthe non-linear function in the equation. As a consequence we prove that stochasticporous media equations have global random attractors. In addition, we show that (inparticular for the classical stochastic porous media equation) this attractor consistsof a random point.

Keywords Random attractor; Random dynamical systems; Stochastic partialdifferential equations; Stochastic porous medium equation.

Mathematics Subject Classification Primary 76S05, 60H15; Secondary 37L55,35B41.

Introduction

In recent years there has been quite an interest in random attractors for stochasticpartial differential equations. We refer e.g., to [8–10, 14, 15, 19, 26, 35], but thislist is far from complete. The study of a new class of stochastic partial differentialequations, namely stochastic porous media equations was initiated in [16] andfurther developed in [17], as well as in a number of subsequent papers (see Section 1below for a more complete list). So far, however, random attractors for stochasticporous media equations have not been investigated.

The purpose of this paper is to analyze or even determine the random attractor(in the sense of [12, 14, 15]) of a stochastic porous medium equation over a bounded

Received February 17, 2010; Accepted July 8, 2010Address correspondence to Michael Röckner, Faculty of Mathematics, University of

Bielefeld, Universitätsstraße 25, O-33615 Bielefeld, Germany; E-mail: roeckner@math.uni-bielefeld.de

446

The Global Random Attractor 447

open set � ⊂ �d of type

dXt = ����Xt��dt +QdWt� t ≥ s� (0.0)

where t� s ∈ �, � is the Laplacian, � � � → � is continuous, ��0� = 0, and �

satisfies certain coercivity conditions and �Wt�t≥0 is a function valued Wiener processon a probability space ��� � P�.

To state our results precisely, we need to recall some of the underlying notionsand describe the set-up. This we shall do in Section 1 below. Here we only brieflydescribe some of the main analytic results we have obtained and which are crucialfor the probabilistic part, more precisely, for the proof of the existence of a global(compact) random attractor for (0.0).

As explained in detail in the next section a fundamental property to beestablished is the cocycle property for the random dynamical system (cf. [2, 22])given by the solutions to (0.0) for all ∈ (outside a set of �-measure zero),all times s� t ∈ � and all initial conditions x ∈ H (=the Hilbert space carrying thesolution-paths to (0.0)).

Therefore, we have to restrict to additive noise and transform equation (0.0) bythe usual change of variables

Zt �= Xt −QWt��

to the equation

dZt = ���Zt +QWt���dt� t ≥ s� (0.1)

for ∈ fixed, i.e., to a deterministic partial differential equation with time-dependent nonlinear coefficient and fixed parameter ∈ . The analysis of thisequation is hence purely analytic. Our main results are the regularity Lemma 3.3 andthe estimate on the L2-norm of the solution to (0.1) in Theorem 3.1. These results arecrucial for the existence proof of a random attractor for (0.0) and in particular thelatter gives an explicit control of the -dependence. To get this estimate on the L2-norm of the solution to (0.1) we introduce the new notion of “�-weak monotonicity”(cf. Hypothesis 1.1 below) for the function �, which seems to be exactly appropriatefor our purposes. We distinguish two cases, namely QWt ∈ H

2�p+10 ��� and the much

harder case when QWt ∈ H1�p+10 ���. For details we refer to Sections 2 and 3 below.

We would, however, like to emphasize that these analytic results are of interest inthemselves and bear potential for further applications besides merely the analysis ofrandom attractors.

On the basis of the estimates obtained in Sections 2 and 3 we can then usea standard result from [15] to prove the existence of a global (compact) randomattractor for (0.0) in Section 4.

In Section 5 under a different (more restrictive, see Remark 5.1) set ofassumptions on � we prove that the random attractor exists and is just a randompoint by a different, but very direct technique. We conclude this paper by someshort remarks on computational methods in Section 6.

448 Beyn et al.

1. Basic Notions and Framework

Equation (0.0) has recently been extensively studied within the so-called variationalapproach to SPDE (cf. e.g., [4–7, 17, 25, 32, Example 4.1.11, 33, 34, 38], we also referto [3, 37] and the references therein as background literature for the deterministiccase). The underlying Gelfand triple is

V ⊂ H ⊂ V ∗� (1.1)

where V �= Lp+1���, p ≥ 1, H �= H10 ���

∗, with H10 ��� being the Sobolev space

of order one on � with Dirichlet boundary conditions. We emphasize that thedualization in (1.1) is with respect to H , i.e., precisely

V ⊂ H ≡ H∗ �=H10 ���� ⊂ V ∗�

where the identification of H and H∗ is given by the Riesz isomorphism, �u�2H10�=∫

���u�2�dd�, u ∈ H1

0 ���, and �·�H is its dual norm. Here �·��d denotes Euclidiannorm on �d and below ·� ·�d shall denote the corresponding inner product. By�·�p we will denote the Lp-norm. Further, for r ∈ �� p ≥ 1 let Hr�p

0 ��� denote theusual Sobolev space of order r in Lp��� with Dirichlet boundary conditions.

We take Q and the Wiener process Wt of the following special type. W =� �1�� � � � � �m�� is a Brownian motion on �m defined on the canonical Wiener space��� � ��t�� P�, i.e., �= C��+��m�, Wt�� �= �t�, and ��t� is the correspondingnatural filtration. As usual we can extend Wt (and �t) for all t ∈ � (cf. e.g., [32,p. 99]). Q � �m → H is defined by

Qx =m∑j=1

xj�j� x = �x1� � � � � xm� ∈ �m�

for fixed �1� � � � � �m ∈ �10��� �⊂L2��� ⊂ H�. Here �1

0��� denotes the set of allcontinuously differentiable functions with compact support in �.

The existence and uniqueness of solutions for (0.0) under monotonicity andcoercivity conditions on � is well-known even under much more general conditionsthan will be used here (see [5, 33]). We will always assume the continuous function� � � → � to satisfy the following conditions:

(A1) Weak monotonicity: For all t� s ∈ �

���t�−��s���t − s� ≥ 0�

(A2) Coercivity: There are p ∈ �1���� a ∈ �0���� c ∈ �0��� such that for all s ∈ �

��s�s ≥ a�s�p+1 − c�

(A3) Polynomial boundedness: There are c1� c2 ∈ �0��� such that for all s ∈ �

���s�� ≤ c1�s�p + c2�

where p is as in (A2).

The Global Random Attractor 449

Here and below the notion of solution to (0.0) is the usual one (cf. [32,Definition 4.1]). We recall that in particular

�∫ T

0�Xt�p+1

p+1 dt < � for all T > 0� (1.2)

with p as in (A2), (A3).In order to obtain the existence of a random attractor we need slightly more

restrictive dissipativity and coercivity conditions on �. We will prove existenceunder two sets of assumptions. In the first case we need to assume strongerregularity of the noise, i.e., QWt ∈ C2

0���, while in the second we allow QWt ∈ C10���,

but require stronger assumptions on the non-linearity �.

Hypothesis 1.1. Assume �j ∈ C20���, 1 ≤ j ≤ m, thus QWt ∈ C2

0���. Let further � �� → �, ��0� = 0 be a function such that

(A1)′ �-Weak monotonicity: For all t� s ∈ �

���t�−��s���t − s� ≥ ���t�− ��s��2�

(A2)′ �-Coercivity: For p� a� c as in (A2) and for all s ∈ �

��s�2 ≥ a�s�p+1 − c�

Remark 1.2. Note that we do not assume � (hence �) to be strictly monotone.Furthermore, we note that the first inequality in (A2)′ follows from (A1)′ since��0� = 0 = ��0�.

Remark 1.3. In case of a continuously differentiable nonlinearity �, (moreprecisely, it suffices to assume that � ∈ H1�1

loc ���) it is easy to find a candidate for �.Namely, we simply define

��s� �=∫ s

0

√�′�r�dr� s ∈ �� (1.3)

Then by Hölder’s inequality (A1)′ holds. Therefore, to ensure that also (A2)′ holdswe only need to assume that for some a ∈ �0���� c ∈ �0���� p ∈ �1���(∫ s

0

√�′�r�dr

)2

≥ a�s�p+1 − c ∀s ∈ �� (1.4)

Conversely, this produces a lot of examples for � satisfying (A1)′, (A2)′, (A3).Simply, take � � � → � continuously differentiable and non-decreasing with ��0� =0 and such that for some a ∈ �0���� c� c1� c2 ∈ �0���� p ∈ �1����

�2�s� ≥ a�s�p+1 − c� �′�s� ≤ c1�s�p−12 + c2 ∀s ∈ ��

Then define

��s� �=∫ s

0��′�r��2dr� s ∈ ��

450 Beyn et al.

In particular, ��s� �= s�s�p−1 arises this way (cf. also Section 5 below). In this casewe have ��s� =

(2√p

p+1

)s�s� p−1

2 .

Hypothesis 1.4. Let �j ∈ C10���� 1 ≤ j ≤ m. Assume further that � ∈ C1���,

satisfying (1.4) such that

�′�r� > 0 for almost all r ∈ �� (1.5)

and that for some c̃1 ∈ �0���

�′�s� ≤ c̃1��s�p−1 + 1� ∀s ∈ �� (1.6)

where p is as in (1.4).

Example 1.5. Let � > 0 and

��r� �=

�r + ��3� for r ≤ −�0� for �r� < �

�r − ��3� for r ≥ ��

Then � ∈ H1�1loc ��� and by Remark 1.3 Hypothesis 1.1 holds with p = 3. Hypothesis

1.4, however, does not hold.

In Remark 1.3 we have already obtained

Lemma 1.6. Assume �j ∈ C20���� 1 ≤ j ≤ m. Then Hypothesis 1.4 implies Hypothesis

1.1 with

��s� �=∫ s

0

√�′�r�dr� s ∈ ��

Remark 1.7.

(i) There is a set 0 ⊂ of full measure such that for each p ≥ 1, ∈ 0,�QWt���pp , ���QWt����pp and (if QWt ∈ C2

0 ) ���QWt����pp are of at mostpolynomial growth as �t� → �.

(ii) We shall largely follow the strategy of [15], in which similar assumptions on Q,hence on the noise QW are made. The condition that each �i should be in C1

0���(C2

0��� resp.) can be easily relaxed to QWt ∈ H1�p+10 ��� (QWt ∈ H

2�p+10 ��� resp.)

and is imposed here for the sake of simplicity only.

In the following let �1 denote the constant appearing in Poincaré’s inequality,i.e., for all f ∈ H1�2

0 ���

�1

∫�f�x�2dx ≤

∫���f�x��2dx�

For t ≥ s and x ∈ H , X�t� s� x� will denote the value at time t of the solution Xt of(0.0) such that Xs = x.

The Global Random Attractor 451

We now recall the notions of a random dynamical system and a randomattractor. For more details confer [2, 14, 15]. Let ���� ���� ��t�t∈�� be a metricdynamical system over a complete probability space ��� ���, i.e., �t� � �→ �t��is ����⊗ �/� -measurable, �0 = id, �t+s = �t � �s and �t is �-preserving, for alls� t ∈ �.

Definition 1.8. Let �H� d� be a complete separable metric space. A randomdynamical system (RDS) over �t is a measurable map

� � �+ ×H × → H

�t� x�� �→ ��t� �x

such that ��0� � = id and � satisfies the cocycle property, i.e.,

��t + s� � = ��t� �s� � ��s���for all t� s ∈ �+ and all ∈ . � is said to be a continuous RDS if �-a.s. x �→��t� �x is continuous for all t ∈ �+.

With the notion of an RDS at our disposal we can now recall the stochasticgeneralization of notions of absorption, attraction and -limit sets.

Definition 1.9. Let �H� d� be as in Definition 1.7

(i) A set-valued map K � → 2H is called measurable if for all x ∈ H themap �→ d�x�K��� is measurable, where for nonempty sets A�B ∈ 2H weset d�A� B� = supx∈A infy∈B d�x� y� and d�x� B� = d��x�� B�. A measurable set-valued map is also called a random set.

(ii) Let A, B be random sets. A is said to absorb B if �-a.s. there exists anabsorption time tB�� ≥ 0 such that for all t ≥ tB��

��t� �−t�B��−t� ⊆ A���

A is said to attract B if

d���t� �−t�B��−t�� A��� −−→t→� 0� �-a.s.

(iii) For a random set A we define the -limit set to be

A�� �=⋂T≥0

⋃t≥T

��t� �−t�A��−t��

Definition 1.10. A random attractor for an RDS � is a compact random set Asatisfying �-a.s.

(i) A is invariant, i.e., ��t� �A�� = A��t� for all t > 0.(ii) A attracts all deterministic bounded sets B ⊆ H .

The following proposition yields a sufficient criterion for the existence of arandom attractor of an RDS �. We also refer to [27] for a different approach basedon ergodicity, which however seems less suitable for our case.

452 Beyn et al.

Proposition 1.11 (cf. [15, Theorem 3.11]). Let � be an RDS and assume the existenceof a compact random set K absorbing every deterministic bounded set B ⊆ H . Thenthere exists a random attractor A, given by

A�� = ⋃B⊆H� B bounded

B���

From now on we take H �= H10 ���

∗ with metric determined by its norm � · �H .Since we aim to apply Proposition 1.11 to prove the existence of a random attractorfor (0.0), we first need to define the RDS associated to (0.0). We take ��� ���to be the canonical two-sided Wiener space, i.e., = C0����m� and �t to be theWiener shift given by �t �= �t + ·�− �t�. As in [15, pp. 375–377] we considerY�t� s� x� �= X�t� s� x�−QWt. Then we have for all s ∈ �� x ∈ H��-a.s.:

Y�t� s� x� = x −QWs +∫ t

s���Y�r� s� x�+QWr�dr� ∀t ≥ s�

We can rewrite this as an -wise equation:

Zt�� = x −QWs��+∫ t

sA�r� Zr���dr� ∀t ≥ s� (1.7)

where A�r� v� �= ���v+QWr���. Since for each fixed ∈ , A � V → V ∗ ishemicontinuous, monotone, coercive and bounded we can apply [32, Theorem 4.2.4]to obtain the unique existence of a solution

Z�t� s� x�� ∈ Lp+1loc ��s���� V� ∩ C��s����H� (1.8)

to (1.7) for all x ∈ H , ∈ , s ∈ � and its continuous dependence on the initialcondition x. We now define in analogy to [14]

S�t� s� �x �= Z�t� s� x��+QWt��� s� t ∈ �� s ≤ t

(1.9)��t� �x �= S�t� 0� �x = Z�t� 0� x� �+QWt��� t ≥ 0�

By uniqueness for (0.0) S�t� s� �x is a version of X�t� s� x���, for each x ∈ H , s ∈�. For fixed s� � x we at times abbreviate S�t� s� �x by St and Z�t� s� x�� by Zt.By the pathwise uniqueness of the solution to equation (1.7) we have for all ∈ ,r� s� t ∈ �, s ≤ r ≤ t�

S�t� s� � = S�t� r� �S�r� s� � (1.8′)

S�t� s� � = S�t − s� 0� �s�� (1.8′′)

The joint measurability of � � �+ ×H × → H follows from the construction ofthe solutions Zt (for details we refer to [21]). Hence � defines an RDS. We canthus apply Proposition 1.11 to prove the existence of a random attractor for �. Forthis we need to prove the existence of a compact set K��, which absorbs everybounded deterministic set in H , �-almost surely. This set will be chosen as K�� �=BL2�0� ����

H, where BL2�0� �� denotes the ball with center 0 and radius � in L2.

The Global Random Attractor 453

Note that since ��t� �−t� = S�t� 0� �−t� = S�0�−t� �, this amounts to provingpathwise bounds on S�0�−t� �x in the L2-norm, where we use the compactness ofthe embedding L2��� ↪→ H . In order to get such estimates we consider norms �·�Ha

on H such that �·�Ha↑ �·�L2 , as a ↓ 0. These are defined as the dual norms (via the

Riesz isomorphism) of the norms

H10 ��� � u �→

(a∫���u�2d�+

∫u2d�

)1/2

�

Then for s ≤ t we have (see e.g., [34, Theorem 2.6 and Lemma 2.7(i), (ii)]) for a �= 1n

�Zt�2H 1n

= �Zs�2H 1n

+ 2∫ t

s

⟨��Sr�� n

(1− 1

n�

)−1

Zr − nZr

⟩dr� (1.9)

where for f� g � � → � measurable we set

f� g �=∫�fg d�

if �fg� ∈ L1���. We shall use (1.9) in a crucial way several times below.

2. Estimates for �St�H and Bounded Absorption

In this section we construct bounded random sets that absorb trajectories withrespect to the weak norm � · �H . For these estimates only the standard assumptions(A1)–(A3) are needed.

Theorem 2.1. Let ∈ �0���, with ≤ a2 , if p = 1. Then there exists a function

p� �1 � �× → �+ such that p� �1 �t� ·� is � -measurable for each t ∈ �, p� �1 �·� ) is

continuous and of at most polynomial growth for �t� → �� ∈ and for all x ∈ H , ∈ 0 (cf. Remark 1.7(i)), s ∈ �:

�Z�t2� s� x���2H ≤ �Z�t1� s� x���2H − ∫ t2

t1

�Z�r� s� x���22 dr

+∫ t2

t1

p� �1 �r� �dr� for all s ≤ t1 ≤ t2� (2.1)

Proof. We fix x�� s and set Zr �= Z�r� s� x��, Sr �= S�r� s� �x for r ≥ s. Allconstants appearing in the proof below are, however, independent of x� and s!

Since for s ≤ t1 ≤ t2∥∥Zt2

∥∥2H= ∥∥Zt1

∥∥2H− 2

∫ t2

t1

Zr���Sr�dr�

we have for dr-a.e. r ∈ �s��� by (A2)

d

dr�Zr�2H = −2Zr���Sr�

= −2Sr −QWr���Sr�

454 Beyn et al.

= −2Sr���Sr� + 2QWr���Sr�≤ −2a

∫��Sr �p+1d�+ 2

∫���QWr��Sr�� + c� d��

By Young’s inequality, for arbitrary � > 0 and some C� �=C��p��� C1� C2 ∈ � wehave by (A3) ∫

��QWr��Sr��d� ≤

∫�

(C��QWr �p+1 + ����Sr��

p+1p

)d�

≤ �C1�Sr�p+1p+1 + C��QWr�p+1

p+1 + �C2����

where ��� �= ∫�d�. Thus by choosing � = a

C1we obtain for dr-a.e. r ∈ �t1� t2�

d

dr�Zr�2H ≤ −a�Sr�p+1

p+1 + C��QWr�p+1p+1 + 2����c + C3��

where C3 �= aC2C1

.Now, if p > 1, then for each > 0 we can find a C such that for all y ∈ � one

has a�y�p+1 ≥ 2 �y�2 − C . If p = 1, then we have the same, provided ∈ �0� a2 �. We

obtain

a�Sr�p+1p+1 ≥ 2 �Sr�22 − ���C = 2 �Zr +QWr�22 − ���C

≥ �Zr�22 − 2 �QWr�22 − ���C �

Hence for

p� �1 �r� � �=

{2 �QWr�22 + ���C + C��QWr�p+1

p+1 + 2����c + C3�� if ∈ 0

0� otherwise

we obtain for dr-a.e. r ∈ �t1� t2�

d

dr�Zr�2H ≤ − �Zr�22 + p

� �1 �r� ��

and the assertion follows. �

Corollary 2.2. Let ∈ �0���, with ≤ a2 if p = 1 and let t ∈ �. Then there exists an

� -measurable function q� �t�1 � → �� such that for all x ∈ H , ∈ 0 and s ≤ t

�Z�t� s� x���2H ≤ q� �t�1 ��+ e

−

e2�t−s� �Z�s� s� x���2H � (2.2)

Proof. Since the embedding L2 ↪→ H is continuous, there is a constant c > 0 suchthat �v�H ≤ c �v�2, for all v ∈ L2. Hence by Theorem 2.1

d

dr��Zr�2H� ≤ −

c2�Zr�2H + p

� �1 �r� � dr-a.e. on �s� t��

Hence by Gronwall’s Lemma the assertion follows with q �t1 �� �=∫ t

−� e−

c2�t−r�

p1�r� �dr . �

The Global Random Attractor 455

Corollary 2.3 (Bounded Absorption). Let t ∈ �. Then there is an � -measurablefunction q

�t�1 � → � such that for each � > 0 there is an s��� ≤ t such that for all

∈ 0, x ∈ H with �x�H ≤ �

Z�t� s� x�� ∈ �BH�0� q�t�1 ���� for all s ≤ s���

i.e., there exists a bounded random set absorbing �Zt� at time t.

Proof. Let �= a2 . By Corollary 2.2, we have for ̃ �=

c2

�Zt�2H ≤ e− ̃�t−s� �Zs�2H + q� �t�1

≤ 2e− ̃�t−s���x�2H + �QWs�2H�+ q� �t�1

≤ 2�2e− �t−s� + 2e− ̃�t−s� �QWs�2H + q� �t�1 �

for all t ≥ s� Hence the result follows with

q�t�1 �= 1+ q

� �t�1 + 2 sup

s≤t�e− ̃�t−s� �QWs�2H�

and s��� ≤ t chosen so that 2�2e− ̃�t−s� ≤ 1 for all s ≤ s���. �

We will need the following auxiliary estimate.

Corollary 2.4. There is an � -measurable function q � → �+ such that for each � >0 there exists s��� ≤ −1 such that for all ∈ 0� x ∈ H with �x�H ≤ �∫ 0

−1�S�r� s� �x�22 dr ≤ q�� for all s ≤ s����

Proof. Using (2.1) in Theorem 2.1 with t1 = −1� t2 = 0 and then using Corollary2.3 for t = −1 yields for = a

2 and s ≤ s���, where s��� ≤ −1 is as in Corollary 2.3,

∫ 0

−1�S�r� s� �x�22 dr ≤ 2 �Z�−1� s� x���2H + 2

∫ 0

−1p� �1 �r� �dr

+ 2 ∫ 0

−1�QWr���22 dr

≤ q���

where q�� �= 2 q�−1�1 ��+ 2

∫ 0−1 p

� �1 �r� �dr + 2

∫ 0−1 �QWr���22 dr� �

3. Estimate for �St�2 and Compact Absorption

In this section we utilize the stronger assumptions from Hypothesis 1.1 and 1.4 inorder to obtain absorption with respect to the strong norm � · �2. By the compactembedding L2��� ↪→ H this will prove the required compactness properties.

Theorem 3.1. Suppose that either Hypothesis 1.1 or Hypothesis 1.4 holds. Let � > 0,with � ∈ �0� ��1

2 � if p = 1. Then there is a function p���2 � �× → � such that p���2 �t� ·�

456 Beyn et al.

is � -measurable for each t ∈ �, p���2 �·� ) is continuous and of at most polynomialgrowth for �t� → �� ∈ and for all x ∈ L2���� ∈ 0 (cf. Remark 1.7(i)), s ∈ �:

�Z�t2� s� x���22 ≤ �Z�t1� s� x���22 − �∫ t2

t1

�Z�r� s� x���22 dr

+∫ t2

t1

p���2 �r� �dr for all s ≤ t1 ≤ t2� (3.1)

In particular, t → Zt is strongly right continuous in L2���.

Proof. Again we fix x�� s and use the abbreviation Zr �= Z�r� s� x��, Sr �=S�r� s� �x for r ∈ �s���. But all constants appearing in the proof below areindependent of x� and s.

Case 1. Assume Hypothesis 1.1.Let t1 ≥ s such that Zt1

∈ L2��� and t2 ≥ t1. (1.9) implies

∥∥Zt2

∥∥2H 1

n

= ∥∥Zt1

∥∥2H 1

n

+ 2∫ t2

t1

⟨��Sr�� n

(1− 1

n�

)−1

Sr − nSr

⟩dr

− 2∫ t2

t1

⟨��Sr�� �

(1− 1

n�

)−1

QWr

⟩dr� (3.2)

We now recall a formula given in [34, Lemma 5.1(ii)]. Let q� q′ > 1 such that 1q+

1q′ = 1, f ∈ Lq���, g ∈ Lq′��� and pn��� d�̃� the kernel corresponding to �1− 1

n��−1

(cf. [34, Lemma 5.1(i)]). Using the symmetry of �1− 1n��−1 in L2��� we obtain

⟨f� g −

(1− 1

n�

)−1

g

⟩= 1

2

∫�

∫��f��̃�− f�����g��̃�− g����pn��� d�̃�d�

+∫�

(1−

(1− 1

n�

)−1

1)fg d��

Using this and proceeding analogously to the calculation following formula (5.6) in[34] yields for dr-a.e. r ∈ �s���⟨

��Sr�� n

(1− 1

n�

)−1

Sr − nSr

⟩= −n

⟨��Sr�� Sr −

(1− 1

n�

)−1

Sr

⟩= −n

2

∫�

∫����Sr��̃��−��Sr������Sr��̃�− Sr����pn��� d�̃�d�

− n∫�

(1−

(1− 1

n�

)−1

1)��Sr�Srd�

≤ −n

2

∫�

∫����Sr��̃��− ��Sr�����

2pn��� d�̃�d�− n∫�

(1−

(1− 1

n�

)−1)��Sr�

2d�

The Global Random Attractor 457

= −n⟨��Sr��

(1−

(1− 1

n�

)−1)��Sr�

⟩= −��n����Sr�� ��Sr���

where ���n�����n��� is the closed coercive form on L2��� with ���n�� = H10 ���

and generator n�1− �1− 1n��−1� = ��1− 1

n��−1. We obtain from (3.2):

∥∥Zt2

∥∥2H 1

n

+ 2∫ t2

t1

��n����Sr�� ��Sr��dr

≤ ∥∥Zt1

∥∥2H 1

n

− 2∫ t2

t1

⟨��Sr�� �

(1− 1

n�

)−1

QWr

⟩dr� (3.3)

Next we prove an upper bound for the second term on the right-hand side of(3.3). Note that we shall make use of the assumption QWt ∈ C2

0 here. Using Young’sinequality and (A3), for all � > 0 and some C�� C1� C2 > 0 we obtain for dr-a.e. r ∈�s���

∣∣∣∣⟨��Sr�� �(1− 1n�

)−1

QWr

⟩∣∣∣∣=∣∣∣∣⟨��Sr��(1− 1

n�

)−1

�QWr

⟩∣∣∣∣≤ �

∫����Sr��

p+1p d�+ C�

∫�

∣∣∣∣((1− �

n

)−1

�QWr

)∣∣∣∣p+1

d�

≤ �C1�Sr�p+1p+1 + C���QWr�p+1

p+1 + C2�

Hence

∥∥Zt2

∥∥2H 1

n

+ 2∫ t2

t1

��n����Sr�� ��Sr��dr

≤ ∥∥Zt1

∥∥22+ 2

∫ t2

t1

[�C1�Sr�p+1

p+1 + C���QWr�p+1p+1 + C2

]dr < �� (3.4)

We note that by (1.8) the right-hand side of (3.4) is indeed finite. Since��n����Sr�� ��Sr�� is increasing in n, we conclude that supn∈� ��n����Sr�� ��Sr�� < �for dr-a.e. r ∈ �t1���. By (A2)′ and (A3) we know that for some c1� c2 ≥ 0

��s�2 ≤ ��s�s ≤ c1�s�p+1 + c2�s��

Since Sr ∈ Lp+1��� this implies ��Sr� ∈ L2��� for dr-a.e. r ∈ �t1���. We now recallthe following result from the theory of Dirichlet forms: Let ������� be theclosed coercive form on L2��� given by ��f� g� = ∫

��f� �g�dd� for f� g ∈ ��� =

H10 ���. From [28, Chapter I, Theorem 2.13] we know for f ∈ L2���, that f ∈

��� = H10 ��� iff supn∈� ��n��f� f� < �. Moreover, limn→� ��n��f� g� = ��f� g� =∫

��f� �g�dd� for all f� g ∈ ���. Hence we obtain for dr-a.e. r ∈ �t1��� that

458 Beyn et al.

��Sr� ∈ ��� = H10 ��� and that

limn→���n����Sr�� ��Sr�� = ����Sr�� ��Sr�� =

∫�����Sr��2�d d��

Using Fatou’s lemma and taking n → � in (3.4) yields∥∥Zt2

∥∥22+ 2

∫ t2

t1

∫�����Sr��2�dd� dr

≤ ∥∥Zt1

∥∥22+ 2�C1

∫ t2

t1

�Sr�p+1p+1dr +

∫ t2

t1

�C���QWr�p+1p+1 + C2�dr� (3.5)

Since Zs = x −QWs ∈ L2���, for all t1 ≥ s we obtain Zt1∈ L2��� and thus (3.5)

holds for all t2 ≥ t1 ≥ s. particular Ys ∈ L2��� for all s ≥ S and (3.5) holds with Sreplaced by s, for all s ≥ S.

Choosing � = a�12C1

, applying Poincaré’s inequality and using the fact that if p >

1 for each � > 0 we can find C̃� ≥ 0 such that for all y ∈ � one has a�1�y�p+1 ≥2��y�2 − C̃�; the same is true for p = 1, if � ∈ �0� ��1

2 �. We obtain from (A2)′ that

∥∥Zt2

∥∥22≤ ∥∥Zt1

∥∥22− 2�1

∫ t2

t1

���Sr��22dr + a�1

∫ t2

t1

�Sr�p+1p+1dr

+∫ t2

t1

�C���QWr�p+1p+1 + C2�dr

≤ ∥∥Zt1

∥∥22− a�1

∫ t2

t1

�Sr�p+1p+1dr +

∫ t2

t1

�C���QWr�p+1p+1 + C2 + c�dr

≤ ∥∥Zt1

∥∥22− 2�

∫ t2

t1

�Sr�22dr +∫ t2

t1

�C���QWr�p+1p+1 + C2 + c + C̃��dr�

Now

�Zr�22 = �Sr −QWr�22 ≤ 2(�Sr�22 + �QWr�22

)�

whence ∥∥Zt2

∥∥22≤ ∥∥Zt1

∥∥22− �

∫ t2t1�Zr�22dr +

∫ t2t1p�2�r� �dr� (3.6)

for � > 0 arbitrary and

p���2 �r� � �=

{C���QWr�p+1

p+1 + C2 + c + C̃� + 2� �QWr�22 � if ∈ 0

0� else.

To obtain right continuity of Zt in L2��� first note that by (3.5) applied fort1 = s and continuity of Zt in H we obtain weak continuity in L2���. Now for tn ↓ tby (3.5) applied to t1 = t we obtain

lim supn→�

∥∥Ztn

∥∥22≤ �Zt�22 �

which implies the right continuity of Zt in L2���.

The Global Random Attractor 459

Case 2. Assume Hypothesis 1.4.Let � be as defined in Lemma 1.6 and again let t1 ≥ s such that Zt1

∈ L2��� andt2 ≥ t1. In order to prove (3.1) in the case QWt ∈ C1

0��� we need to be more carefulwhen bounding the second term on the right hand side of (3.3). For this we needthe regularity result proved in Lemma 3.3 below, which implies that for every � > 0there exist constants C�� C̃� �=C��p�� C̃��p�� such that for dr-a.e. r ∈ �s���

−⟨��Sr�� �

(1− 1

n�

)−1

QWr

⟩=⟨���Sr�� �

(1− 1

n�

)−1

QWr

⟩

≤ �����Sr��p+1p

p+1p

+ C�

∥∥∥∥�(1− 1n�

)−1

QWr

∥∥∥∥p+1

p+1

≤ �����Sr��p+1p

p+1p

+ C̃���QWr�p+1p+1� (3.6′)

Now using Lemma 3.3 and (3.6′) with � = 1 in (3.3) yields for some constantsc� C ∈ �

∥∥Zt2

∥∥2H 1

n

+ 2∫ t2

t1

��n����Sr�� ��Sr��dr

≤ ∥∥Zt1

∥∥2H 1

n

+ 2∫ t2

t1

[����Sr��

p+1p

p+1p

+ C̃1��QWr�p+1p+1

]dr

≤ c∥∥Zt1

∥∥22+ C

∫ t2

t1

���QWr�p+1p+1 + 1�dr < ��

Now we can proceed as after (3.4) to deduce ��Sr� ∈ ��� = H10 ��� and

limn→���n����Sr�� ��Sr�� =

∫�����Sr��2�dd��

for dr-a.e. r ∈ �s���. Since �′�r� > 0, ��s� = ∫ s

0

√�′�r�dr is C1��� with continuous

inverse �−1. Thus

��x� =∫ x

0�′�r�dr =

∫ x

0

√�′�r�

√�′�r�dr

=∫ x

0�′�r�

√�′�r�dr =

∫ ��x�

0

√�′��−1�r��dr = F���x���

where F�s� �= ∫ s

0

√�′��−1�r��dr . Since F ∈ C1���, ��Sr� ∈ H1

0 ��� for dr-a.e. r ∈�s��� and F ′���Sr�����Sr� =

√�′�Sr����Sr� ∈ L1��� (by (1.4)), we have ��Sr� =

F���Sr�� ∈ H1�10 ��� for dr-a.e. r ∈ �s��� with

���Sr� =√�′�Sr����Sr� ∈ L1���� (3.7)

By (A2)′ and (1.6) there are some constants C1� C2 such that

�′�r�2p+1p−1 ≤ C1��r�

2 + C2�

460 Beyn et al.

Using (3.7) and then Young’s and Poincaré’s inequalities, for some constant C(which may change from line to line) we have for dr-a.e. r ∈ �s���

����Sr��p+1p

p+1p

=∫�����Sr��

p+1p d� =

∫��√�′�Sr����Sr��

p+1p d�

=∫���′�Sr����Sr��

p+1p d� ≤ ����Sr��22 + C

∫���′�Sr��2

p+1p−1d�

≤ ����Sr��22 + C���Sr��22 + C ≤ C����Sr��22 + C� (3.8)

We can now go on with bounding the second term on the right hand side of (3.3)as follows:(3.6′) and (3.8) imply that for dr-a.e. r ∈ �s���⟨

��Sr�� �

(1− 1

n�

)−1

QWr

⟩≤ �����Sr��

p+1p

p+1p

+ C̃���QWr�p+1p+1

≤ �C1����Sr��22 + �C2 + C̃���QWr�p+1p+1� (3.9)

Using this with � = 1C1

in (3.3) and letting n → � yields for some constant C

∥∥Zt2

∥∥22+∫ t2

t1

����Sr��22dr ≤∥∥Zt1

∥∥22+ 2C

∫ t2

t1

�1+ ��QWr�p+1p+1�dr� (3.10)

Now we can proceed as done in the proof of Case 1 after (3.5). �

Remark 3.2. As indicated before the arguments in the proof can easily begeneralized to noise QWt ∈ H

2�p+10 ��� �QWt ∈ H

1�p+10 ��� resp.).

Lemma 3.3. Assume Hypothesis 1.4 and let x ∈ L2���, s ∈ � and ∈ . Then

��S�·� s� �x� ∈ Lp+1p

loc ��s����H1� p+1

p

0 � and there exist constants c > 0� C ∈ �,independent of x� s and , such that

�Z�t2� s� x���22 + c∫ t2

t1

����S�r� s� �x��p+1p

p+1p

dr

≤ �Z�t1� s� x���22 + C∫ t2

t1

���QWr���p+1p+1 + 1�dr� ∀t2 ≥ t1 ≥ s�

Proof. We use the Galerkin approximation and the notation used in the proofof unique existence of a solution to (1.7) in [32, Theorem 4.2.4]). Let �ei�i ∈ ��be the orthonormal basis of H consisting of eigenfunctions of � on L2��� withDirichlet boundary conditions. Then ei ∈ C���� ∩H1

0 ��� ⊆ V . Furthermore, letHn = span�e1� � � � � en� and define Pn � V

∗ → Hn ⊆ C���� ∩H10 ��� by

Pny �=n∑i=1

V ∗y� eiV ei�

Note that via the embedding L2��� ⊆ H ⊆ V ∗, Pn�L2��� � L2��� → Hn is just the

orthogonal projection in L2��� onto Hn. Let t1 ≥ s such that Zt1∈ L2���, let Zn

t

The Global Random Attractor 461

denote the solution of

Znt = PnZt1

+∫ t

t1

PnA�r� Znr �dr� ∀t ≥ t1

and let Snt �= Znt +QWt. By the chain rule, for all t2 ≥ t1

�Znt2�22 = �PnZt1

�22 + 2∫ t2

t1

A�r� Znr �� Z

nr dr

= �PnZt1�22 + 2

∫ t2

t1

���Snr �� Snr dr − 2∫ t2

t1

���Snr ��QWrdr� (3.11)

By the same argument as for (3.7) and with � as defined in Lemma 1.6 we get

���Snr �� Snr = −���Snr �� �Snr = −

⟨√�′�Snr ����S

nr �� �S

nr

⟩= −����Snr ��22

and using Young’s inequality

−���Snr ��QWr = ���Snr �� �QWr≤ �����Snr ��

p+1p

p+1p

+ C���QWr�p+1p+1�

for all � > 0 and some C� ∈ �. By (3.11) this yields

�Znt2�22 ≤ �PnZt1

�22 − 2∫ t2

t1

����Snr ��22dr

+ 2�∫ t2

t1

����Snr ��p+1p

p+1p

dr + 2C�

∫ t2

t1

��QWr�p+1p+1dr� (3.12)

By the same argument as for (3.8) we realize

����Snr ��p+1p

p+1p

≤ C1����Snr ��22 + C2�

for some constants C1� C2. Using this in (3.12), with � = 12C1

yields for some c> 0,C ∈ �

�Znt2�22 + c

∫ t2

t1

����Snr ��p+1p

p+1p

dr ≤ �Zt1�22 + C

∫ t2

t1

���QWr�p+1p+1 + 1�dr� (3.13)

Both C1� C2 and c� C are independent of x� s and .

Hence we obtain the existence of a �̄ ∈ Lp+1p ��t1� t2�� H

1� p+1p

0 � such that (selectinga subsequence if necessary)

��Snr � ⇀ �̄�

462 Beyn et al.

in Lp+1p ��t1� t2�� H

1� p+1p

0 � and thus in Lp+1p ��t1� t2�� L

p+1p ����. By the proof of unique

existence of a solution we also know that (again selecting a subsequence if necessary)

���Snr � ⇀ ���Sr��

in Lp+1p ��t1� t2�� V

∗� and by definition of �� � V → V ∗ this is equivalent to ��Snr � ⇀

��Sr�, in Lp+1p ��t1� t2�� L

p+1p ����. Hence �̄ = ��Sr�. An analogous argument applied

to Znt2yields Zn

t2⇀ Zt2

in L2���. Letting n → � in (3.13) we arrive at

�Zt2�22 + c

∫ t2

t1

����Sr��p+1p

p+1p

dr ≤ �Zt1�22 + C

∫ t2

t1

���QWr�p+1p+1 + 1�dr� (3.14)

Since Zs = x −QWs ∈ L2���, for all t1 ≥ s we obtain Zt1∈ L2��� and thus (3.14)

holds for all t2 ≥ t1 ≥ s. �

Corollary 3.4 (Compact Absorption). There is an � -measurable function � � →�+ such that for each � > 0 there exists s��� ≤ −1 such that for all x ∈ H with �x�H ≤� and all ∈ 0

�S�0� s� �x�2 ≤ ���� for all s ≤ s����

Remark 3.5. This is analogous to [15, Lemma 5.5, p. 380].

Proof. (3.1) in Theorem 3.1 with t2 = 0 ≥ t1 ≥ s implies

�Z0�22 ≤∥∥Zt1

∥∥22− �

∫ 0

t1

(�Zr�22 + p

���2 �r� �

)dr�

Integrating over t1 ∈ �−1� 0� yields

�Z0�22 ≤∫ 0

−1

(�Zr�22 + �p���2 �r� ��

)dr

≤∫ 0

−1�2 �Sr�22 + 2 �QWr�22 + �p���2 �r� ���dr�

Hence using Corollary 2.4 and recalling that Z0 = S�0� s� �x we obtain theassertion. �

4. Existence of the Global Random Attractor

Theorem 4.1. The random dynamical system associated with (0.0) and defined by (1.9)admits a random attractor.

Proof. We show that the assumptions of Proposition 1.11 are satisfied. Since theembedding L2��� ↪→ H is compact, for each ∈ the set

K�� �= BL2�0� ����H

is nonempty and compact in H .

The Global Random Attractor 463

For the reader’s convenience, we prove that it is a random set (cf.Definition 1.9(i)) in the Polish space H . According to [13, Proposition 2.4], it isenough to check that for each open set O ⊂ H , CO �= � ∈ �O ∩ K�� �= ∅� ismeasurable. But

O ∩ K�� = O ∩ BL2�0� ����H = O ∩�BL2�0� ����

= O ∩ L2��� ∩�BL2�0� �����

For C ⊆ L2��� and x ∈ L2��� let dL2�x� C� �= infy∈C �x − y�2. If O ∩ L2��� = ∅,then CO = ∅ is measurable and if O ∩ L2��� �= ∅, then

CO = � ∈ �dL2

(0� O ∩ L2���

) ≤ ����

is measurable as � is.Let B be a bounded subset of H . Then B ⊂ �BH�0� ��, for some � > 0. By

Corollary 3.4 there exists a tB �= −s��� ≥ 1 such that for all x ∈ B, t ≥ tB and ∈0

��t� �−t��x� = S�t� 0� �−t�x = S�0�−t� �x ≤ ����

Hence for all t ≥ tB� ∈ 0, ��t� �−t��B� ⊂ K��, i.e. the random compact set Kabsorbs all deterministic bounded sets.

Now we may apply Proposition 1.11 to get the existence of a global compactattractor A, given by:

A�� = ⋃B⊂H�B bounded

B��H

�

where B�� �=⋂

T≥0

⋃t≥T ��t� �−t�B denotes the -limit set of B. �

Remark 4.2. By [15, Proposition 4.5] the existence of a random attractor asconstructed in the proof of Theorem 4.1 implies the existence of an invariantMarkov measure · ∈ �H� for � (in the sense of [15, Definition 4.1]), supported byA. Hence using [11] there exists an invariant measure for the Markovian semigroupdefined by Ptf�x� = ��f�S�t� 0� x��� and it is given by

�B� =∫ �B�P�d��

where B ⊆ H is a Borel set. If the invariant measure for Pt is unique, then theinvariant Markov measure · for � is unique and given by

= limt→���t� �−t� �

5. Attraction by a Single Point

So far we obtained the existence of the random attractor A for (0.0), but we didnot deduce any information about its finer structure. Under a stronger monotonicitycondition which was first introduced in [17], we will now prove that A consistsof a single random point. While we had to restrict to noise of regularity at least

464 Beyn et al.

H1�p+10 ��� before, we can now allow more general noise. Let Q be a Hilbert–

Schmidt operator from L2��� → H and Wt be a cylindrical Brownian Motion onL2���. Then QWt is an R �= QQ∗-Wiener process on H . Let ek be an orthonormalbasis of eigenvectors of R with corresponding eigenvalues k. Assume further∑�

k=1√ k�ek�V < �. Then QWt defines an almost surely continuous process in V .

Now the associated RDS to (0.0) can be defined as before.Define � � � → � to be a continuous function such that there exist some

constants c ≥ 0, p ∈ �1���, ! > 0 such that

���s�� ≤ c�1+ �s�p�(5.1)

�s − t����s�−��t�� ≥ !�s − t�p+1� s� t ∈ ��

It has been shown in [17] that (5.1) holds if � ∈ C1���, ��0� = 0 and if there existconstants �� ! > 0 such that

�p+ 1�2

4!�s�p−1 ≤ �′�s� ≤ ��1+ �s�p−1�� s ∈ �� (5.2)

This, for example is true for ��s� = s�s�p−1. By Remark 1.3 it is easy to see that (5.2)implies the weaker monotonicity assumption (A1)′. Also note that (5.1) implies thecoercivity property (A2). Thus (A1)–(A3) are satisfied and we can define Zt� St andthe RDS � as before (cf. (1.9)).

Remark 5.1.

(i) Obviously, for � as in Example 1.5 the conditions in (5.1) do not hold.(ii) Let ��r� �= ∫ r

0 e− 1

�s�ds. Then � ∈ C1���, 0 < �′�r� ≤ 1 and Hypothesis 1.4(hence also Hypothesis 1.1) holds with p = 1. But obviously (5.1) above doesnot hold.

Theorem 5.2. Assume (5.1). Then

�S�t� s1� �x − S�t� s2� �y�2H

≤{�S�s2� s1� �x − y�1−pH + !�

p+12

1 �p− 1��t − s2�

}− 2p−1

≤{!�

p+12

1 �p− 1��t − s2�

}− 2p−1

�

for s1 ≤ s2 < t, ∈ and x� y ∈ H . In particular for each t ∈ �,lims→−� S�t� s� �x = !t�� exists independently of x and uniformly in x�.

Proof. Let s1 ≤ s2 < t. Then for all s2 ≤ s ≤ t

S�t� s1� �x − S�t� s2� �y = S�s� s1� �x − S�s� s2� �y

+∫ t

s���S�r� s1� �x�− ���S�r� s2� �y�dr�

The Global Random Attractor 465

By Itô’s-Formula and since �u�p+1p+1 ≥ �

p+12

1 �u�p+1H , for all s2 ≤ s ≤ t:

�S�t� s1� �x − S�t� s2� �y�2H= �S�s� s1� �x − S�s� s2� �y�2H

+ 2∫ t

sV∗ ���S�r� s1� �x�− ���S�r� s2� �y�� S�r� s1� �x − S�r� s2� �yV dr

= �S�s� s1� �x − S�s� s2� �y�2H− 2

∫ t

sV∗ ��S�r� s1� �x�−��S�r� s2� �y�� S�r� s1� �x − S�r� s2� �yV dr

≤ �S�s� s1� �x − S�s� s2� �y�2H − 2!∫ t

s�S�r� s1� �x − S�r� s2� �y�p+1

p+1dr

≤ �S�s� s1� �x − S�s� s2� �y�2H − !̃∫ t

s�S�r� s1� �x − S�r� s2� �y�p+1

H dr� (5.3)

where for notational convenience we have set !̃ �= 2!�p+12

1 . Thus, formally�S�t� s1� �x − S�t� s2� �y�2H is a subsolution of the ordinary differential equation

h′�t� = −!̃h�t� p+12 � ∀t ≥ s2

(5.4)h�s2� = �S�s2� s1� �x − y�2H�

Let

h��t� ={��S�s2� s1� �x − y�H + ��1−p + !̃

2�p− 1��t − s2�

}− 2p−1

� t ≥ s2�

h� is a solution of (5.4) with h��s2� = ��S�s2� s1� �x − y�H + ��2, which suggests�S�t� s1� �x − S�t� s2� �y�2H ≤ h��t�. This will be proved next.

Let ���t� �= h��t�− �S�t� s1� �x − S�t� s2� �y�2H and "� =inf �t ≥ s2� 0 ≥ ���t��. Using 0 < ���s2� and continuity of �� we realize "� > s2.Further note that by definition we have h��t� ≥ �S�t� s1� �x − S�t� s2� �x�2H on�s2� "�� and that

h��t� ≤ ��S�s2� s1� �x − y�H + ��2 =� c��

Assume "� < �. Then ���"�� ≤ 0 and for all s2 ≤ s ≤ t ≤ "�, by the mean valuetheorem and (5.3):

���t� = h��t�− �S�t� s1� �x − S�t� s2� �y�2H≥ ���s�− !̃

∫ t

s�h��r�

p+12 − (�S�r� s1� �x − S�r� s2� ��2H

) p+12 �dr

≥ ���s�− !̃

(p+ 12

)c

p−12

�

∫ t

s���r�dr�

466 Beyn et al.

Using the Gronwall Lemma we obtain

���"�� ≥ ���s2�e−!̃� p+1

2 �cp−12

� �"�−s2� > 0�

This contradiction proves "� = � and since this is true for all � > 0 we conclude:

�S�t� s1� �x − S�t� s2� �y�2H

≤{��S�s2� s1� �x − y�H�1−p +

!̃

2�p− 1��t − s2�

}− 2p−1

≤ �S�s2� s1� �x − y�2H ∧{!̃

2�p− 1��t − s2�

}− 2p−1

≤{!̃

2�p− 1��t − s2�

}− 2p−1

�

for each t > s2. �

Theorem 5.3. Assume (5.1). The random dynamical system given by ��t� �x =S�t� 0� �x has a compact global attractor A�� consisting of one point

A�� = �!0����

Proof. Since !0�� is measurable, A�� is a random compact set. We need to checkinvariance and attraction for A��. Let t > 0. Then for any x ∈ H , by continuity ofx �→ S�t� 0� �x and (1.8′), (1.8′′)

��t� �A�� ={S�t� 0� � lim

s→−� S�0� s� �x}={lims→−� S�t� s� �x

}={lims→−� S�0� s − t� �t�x

}= �!0��t�� = A��t��

Since the convergence in Theorem 5.2 is uniform with respect to x ∈ H , for anybounded set B ⊆ H we have (again using (1.8′′))

d���t� �−t�B�A��� = supx∈B

�S�t� 0� �−t�x − !0���H= sup

x∈B�S�0�−t� �x − !0���H → 0�

for t → �. Hence A�� attracts all deterministic bounded sets. �

It is easy to see that the convergence lims→−� S�t� s� �x = !t�� implies theexistence and uniqueness of an invariant measure for the associated Markoviansemigroup, defined by Ptf�x� �= ��f�S�t� 0� ·�x�� (cf. [17]). This invariant measureis given by = � � !−1

0 . In fact we can deduce much more. Since evidently !0 ismeasurable with respect to �− (as defined in [11]), by [11] �= limt→� ��t� �−t� exists �-a.s. and defines an invariant measure for the random dynamical system� (for more details on invariant random measures cf. [15]). Moreover by

The Global Random Attractor 467

[13, Theorem 2.12] every invariant measure for � is supported by A = �!0�, i.e., ��!0���� = 1 for �-a.a. . Hence we have proved the following

Corollary 5.4. There exists a unique invariant random measure · ∈ �H� for therandom dynamical system � and it is given by

= �!0��� �-a.s.

6. Concluding Remarks on Computational Approaches

The porous medium equation considered here is a model case for a general typeof equations that include more details of the permeable medium and that hasimportant applications to the simulation of oil reservoirs. We refer to [1] for suchan application and for an up-to-date finite element method that can be used forsolving the deterministic version of (1.1). One of the major difficulties here is toaccount for the spatial variations (represented by the functions �j in the operatorQ) by introducing different scales in the finite element subspace. For the quasilinearsteady state equation suitable finite element approximations have been set up, cf.[30, 31] and the references therein.

It seems, however, that computational methods for random attractors in infinitedimensional systems (except for the case of a singleton) are well beyond today’scomputational capabilities.

There are a few approaches to approximate random attractors in stochasticordinary differential equations [23, 24, 36]. These are based on the subdivision andbox covering techniques developed over the last years by Dellnitz and coworkers(see [18] for a survey). However, these methods are essentially still limited to lowerdimensions. In order to proceed to high-dimensional or even infinite-dimensionalcases (see e.g., [39]) one will need reduction principles as they are well establishedin the theory of inertial manifolds for deterministic PDEs. The correspondingproperties of squeezing and flattening (cf. [20, 29]) have been generalized to randomdynamical systems in [26]. It is also shown in [26] that squeezing is a strongercondition than flattening, but that the latter one is sufficient to establish theexistence of a compact random attractor. The determining modes occurring in theseproperties should form the basis of a reduced space to which numerical methodsapply.

Acknowledgments

W.-J. B was supported by the DFG through SFB-701. B. G. was supported byDFG-Internationales Graduiertenkolleg Stochastics and Real World Models, theSFB-701 and the BiBoS-Research Center. M. R. was supported by the DFGthrough SFB-701 and IRTG 1132 as well as the BIBOS-Research Center.

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