Wolf-Jürgen Beyn
email: beyn@math.uni-bielefeld.de

Benjamin Gess
email: bgess@math.uni-bielefeld.de

Paul Lescot
email: paul.lescot@univ-rouen.fr

Michael Röckner
email: roeckner@math.uni-bielefeld.de

The global random attractor for a class of stochastic porous media equations

Keywords: 76S05, 60H15, Secondary: 37L55, 35B41 Stochastic porous medium equation, Random attractor, Random dynamical systems, Stochastic partial differential equations

We prove new L 2 -estimates and regularity results for generalized porous media equations "shifted by" a function-valued Wiener path. To include Wiener paths with merely first spatial (weak) derivates we introduce the notion of "ζ-monotonicity" for the non-linear function in the equation. As a consequence we prove that stochastic porous media equations have global random attractors. In addition, we show that (in particular for the classical stochastic porous media equation) this attractor consists of a random point.

Introduction

In recent years there has been quite an interest in random attractors for stochastic partial differential equations. We refer e.g. to [START_REF] Caraballo | Non-autonomous and random attractors for delay random semilinear equations without uniqueness[END_REF], [START_REF] Caraballo | Stability and random attractors for a reaction diffusion equation with multiplicative noise[END_REF], [START_REF] Chuesov | Averaging of attractors and inertial manifolds for parabolic PDE with random coefficients[END_REF], [START_REF] Crauel | Random attractors[END_REF], [START_REF] Crauel | Attractors for random dynamical systems[END_REF], [START_REF] Flandoli | Random attractors for the 3D-stochastic Navier-Stokes equation with multiplicative white noise[END_REF], [START_REF] Kloeden | Flattening, squeezing and the existence of random attractors[END_REF], [START_REF] Schmalfuss | Measure attractors and random attractors for stochastic partial differential equations[END_REF], but this list is far from being complete. The study of a new class of stochastic partial differential equations, namely stochastic porous media equations was initiated in [START_REF] Da Prato | Weak solutions to stochastic porous media equations[END_REF] and further developed in [START_REF] Da Prato | Strong solutions of stochastic generalized porous media equations: existence, uniqueness and ergodicity[END_REF], as well as in a number of subsequent papers (see Sect. 1 below for a more complete list). So far, however, random attractors for stochastic porous media equations have not been investigated.

The purpose of this paper is to analyze or even determine the random attractor (in the sense of [START_REF] Crauel | Global random attractors are uniquely determined by attracting deterministic compact sets[END_REF], [START_REF] Crauel | Random attractors[END_REF], [START_REF] Crauel | Attractors for random dynamical systems[END_REF]) of a stochastic porous medium equation over a bounded open set Λ ⊂ R d of type (0.0) dX t = ∆(Φ(X t ))dt + QdW t , t ≥ s, where t, s ∈ R, ∆ is the Laplacian, Φ : R → R is continuous, Φ(0) = 0, and Φ satisfies certain coercivity conditions and (W t ) t≥0 is a function valued Wiener process on a probability space (Ω, F, P ).

To state our results precisely, we need to recall some of the underlying notions and describe the set-up. This we shall do in Section 1 below. Here we only briefly describe some of the main analytic results we have obtained and which are crucial for 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 be established is the cocycle property for the random dynamical system (cf. [START_REF] Arnold | Random dynamical systems[END_REF][START_REF] Imkeller | The conjugacy of stochastic and random differential equations and the existence of global attractors[END_REF]) given by the solutions to (0.0) for all ω ∈ Ω (outside a set of P-measure zero), all times s, t ∈ R and all initial conditions x ∈ H (= the Hilbert space carrying the solution-paths to (0.0)). Therefore, we have to restrict to additive noise and transform equation (0.0) by the usual change of variables Z t := X t -QW t (ω)

to the equation (0.1) dZ t = ∆Φ(Z t + QW t (ω))dt, t ≥ s,

for ω ∈ Ω fixed, i.e. to a deterministic partial differential equation with time dependent nonlinear coefficient and fixed parameter ω ∈ Ω. The analysis of this equation is hence purely analytic. Our main results are the regularity Lemma 3.3 and the estimate on the L 2 -norm of the solution to (0.1) in Theorem 3.1. These results are crucial for the existence proof of a random attractor for (0.0) and in particular the latter gives an explicit control of the ω-dependence. To get this estimate on the L 2 -norm of the solution to (0.1) we introduce the new notion of "ζ-weak monotonicity" (cf. Hypothesis 1.1 below) for the function Φ, which seems to be exactly appropriate for our purposes. We distinguish two cases, namely QW t ∈ H 2,p+1 0 (Λ) and the much harder case when QW t ∈ H 1,p+1 0 (Λ). For details we refer to Sections 2 and 3 below. We would, however, like to emphasize that these analytic results are of interest in themselves and bear potential for further applications besides merely the analysis of random attractors. On the basis of the estimates obtained in Sections 2 and 3 we can then use a standard result from [START_REF] Crauel | Attractors for random dynamical systems[END_REF] to prove the existence of a global (compact) random attractor for (0.0) in Section 4.

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

Basic notions and framework

Equation (0.0) has recently been extensively studied within the so-called variational approach to SPDE (cf. e.g. [START_REF] Barbu | Existence and uniqueness of non negative solutions to the stochastic porous media equation[END_REF], [START_REF] Barbu | Existence of strong solutions for stochastic porous media equation under general monotonicity conditions[END_REF], [START_REF] Barbu | Finite time extinction for solutions to fast diffusion stochastic porous media equations[END_REF], [START_REF] Barbu | Stochastic porous media equation and self-organized criticality[END_REF], [START_REF] Da Prato | Strong solutions of stochastic generalized porous media equations: existence, uniqueness and ergodicity[END_REF], [START_REF] Kim | On the stochastic porous media equation[END_REF], [START_REF] Prévôt | A concise course on stochastic partial differential equations[END_REF]Example 4.1.11], [START_REF] Ren | Stochastic generalized porous media and fast diffusion equations[END_REF], [START_REF] Röckner | Non-monotone stochastic generalized porous media equations[END_REF], [START_REF] Wang | Harnack inequality and applications for stochastic generalized porous media equations[END_REF], we also refer to [START_REF] Aronson | The porous medium equation[END_REF], [START_REF] Vázquez | The porous medium equation[END_REF] and the references therein as background literature for the deterministic case). The underlying Gelfand triple is

(1.1) V ⊂ H ⊂ V * ,
where V := L p+1 (Λ), p ≥ 1, H := H 1 0 (Λ) * , with H 1 0 (Λ) being the Sobolev space of order one on Λ with Dirichlet boundary conditions. We emphasize that the dualization in (1.1) is with respect to H, i.e. precisely

V ⊂ H ≡ H * (= H 1 0 (Λ)) ⊂ V * ,
where the identification of H and H * is given by the Riesz isomorphism, u 2 We take Q and the Wiener process W t of the following special type. W = (β (1) , . . . , β (m) ) is a Brownian motion on R m defined on the canonical Wiener space (Ω, F, (F t ), P ), i.e. Ω := C(R + , R m ), W t (ω) := ω(t), and (F t ) is the corresponding natural filtration. As usual we can extend W t (and F t ) for all t ∈ R (cf. e.g. [32, p. 99]). Q : R m → H is defined by

H 1 0 := Λ |∇u| 2 R d dξ, u ∈ H 1 0 (Λ),
Qx = m j=1 x j ϕ j , x = (x 1 , . . . , x m ) ∈ R m , for fixed ϕ 1 , . . . , ϕ m ∈ C 1 0 (Λ) (⊂ L 2 (Λ) ⊂ H).
Here C 1 0 (Λ) denotes the set of all continuously differentiable functions with compact support in Λ.

The existence and uniqueness of solutions for (0.0) under monotonicity and coercivity conditions on Φ is well-known even under much more general conditions than will be used here (see [START_REF] Ren | Stochastic generalized porous media and fast diffusion equations[END_REF], [START_REF] Barbu | Existence of strong solutions for stochastic porous media equation under general monotonicity conditions[END_REF]). We will always assume the continuous function Φ : R → R to satisfy the following conditions:

(A1) Weak monotonicity: For all t, s ∈ R (Φ(t) -Φ(s))(t -s) ≥ 0. (A2) Coercivity: There are p ∈ [1, ∞), a ∈ (0, ∞), c ∈ [0, ∞) such that for all s ∈ R Φ(s)s ≥ a|s| p+1 -c. (A3) Polynomial boundedness: There are c 1 , c 2 ∈ [0, ∞) such that for all s ∈ R |Φ(s)| ≤ c 1 |s| p + c 2 ,
where p is as in (A2).

Here and below the notion of solution to (0.0) is the usual one (cf. [START_REF] Prévôt | A concise course on stochastic partial differential equations[END_REF]Definition 4.1]). We recall that in particular

(1.2) E T 0 X t p+1 p+1 dt < ∞ for all T > 0,
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 Φ. We will prove existence under two sets of assumptions. In the first case we need to assume stronger regularity of the noise, i.e. QW t ∈ C 2 0 (Λ), while in the second we allow QW t ∈ C 1 0 (Λ), but require stronger assumptions on the non-linearity Φ.

Hypothesis 1.1. Assume ϕ j ∈ C 2 0 (Λ), 1 ≤ j ≤ m, thus QW t ∈ C 2 0 (Λ). Let further ζ : R → R, ζ(0) = 0 be a function such that (A1)' ζ-Weak monotonicity: For all t, s ∈ R (Φ(t) -Φ(s))(t -s) ≥ (ζ(t) -ζ(s)) 2 .
(A2)' ζ-Coercivity: For p, a, c as in (A2) and for all s ∈ R ζ(s) 2 ≥ a|s| p+1c. Remark 1.2. Note that we do not assume ζ (hence Φ) to be stricly monotone. Furthermore, we note that the first inequality in (A2)' follows from (A1)' since Φ(0) = 0 = ζ(0). Remark 1.3. In case of a continuously differentiable nonlinearity Φ, (more precisely, it suffices to assume that Φ ∈ H 1,1 loc (R)) it is easy to find a candidate for ζ. Namely, we simply define

(1.3) ζ(s) := s 0 Φ ′ (r)dr, s ∈ R.
Then by Hölder's inequality (A1)' holds. Therefore, to ensure that also (A2)' holds we only need to assume that for some a

∈ (0, ∞), c ∈ [0, ∞), p ∈ [1, ∞) (1.4)
s 0 Φ ′ (r)dr 2 ≥ a|s| p+1 -c ∀s ∈ R.
Conversely, this produces a lot of examples for Φ satisfying (A1)',(A2)',(A3). Simply, take ζ : R → R continuously differentiable and non-decreasing with ζ(0) = 0 and such that for some a ∈ (0,

∞); c, c 1 , c 2 ∈ [0, ∞), p ∈ [1, ∞), ζ 2 (s) ≥ a|s| p+1 -c, ζ ′ (s) ≤ c 1 |s| p-1 2 + c 2 ∀s ∈ R.
Then define

Φ(s) := s 0 (ζ ′ (r)) 2 dr, s ∈ R.
In particular, Φ(s) := s|s| p-1 arises this way (cf. also Section 5 below). In this case we have

ζ(s) = 2 √ p p+1 s|s| p-1 2 . Hypothesis 1.4. Let ϕ j ∈ C 1 0 (Λ), 1 ≤ j ≤ m. Assume further that Φ ∈ C 1 (R), satisfying (1.4) such that (1.5) Φ ′ (r) > 0 for almost all r ∈ R,
and that for some c1 ∈ [0, ∞)

(1.6) Φ ′ (s) ≤ c1 (|s| p-1 + 1) ∀s ∈ R,
where p is as in (1.4).

Example 1.5. Let δ > 0 and

Φ(r) :=      (r + δ) 3 , for r ≤ -δ 0 , for |r| < δ (r -δ) 3 , for r ≥ δ.
Then Φ ∈ H (ii.) We shall largely follow the strategy of [START_REF] Crauel | Attractors for random dynamical systems[END_REF], in which similar assumptions on Q, hence on the noise QW are made. The condition that each ϕ i should be in

C 1 0 (Λ) (C 2 0 (Λ) resp.) can be easily relaxed to QW t ∈ H 1,p+1 0 (Λ) (QW t ∈ H 2,p+1 0 (Λ) resp.
) and is imposed here for the sake of simplicity only.

In the following let λ 1 denote the constant appearing in Poincaré's inequality, i.e. for all f ∈ H 1,2 0 (Λ)

λ 1 Λ f (x) 2 dx ≤ Λ |∇f (x)| 2 dx.
For t ≥ s and x ∈ H, X(t, s, x) will denote the value at time t of the solution X t of (0.0) such that X s = x.

We now recall the notions of a random dynamical system and a random attractor. For more details confer [START_REF] Arnold | Random dynamical systems[END_REF][START_REF] Crauel | Random attractors[END_REF][START_REF] Crauel | Attractors for random dynamical systems[END_REF]. Let ((Ω, F, P), (θ t ) t∈R ) be a metric dynamical system over a complete probability space (Ω, F, P), i.e. (t, ω) → θ t (ω) is B(R) ⊗ F/F-measurable, θ 0 = id, θ t+s = θ t • θ s and θ t is P-preserving, for all s, t ∈ R. Definition 1.8. Let (H, d) be a complete separable metric space. A random dynamical system (RDS) over θ t is a measurable map

ϕ : R + × H × Ω → H (t, x, ω) → ϕ(t, ω)x
such that ϕ(0, ω) = id and ϕ satisfies the cocycle property, i.e.

ϕ(t + s, ω) = ϕ(t, θ s ω) • ϕ(s, ω),
for all t, s ∈ R + and all ω ∈ Ω. ϕ is said to be a continuous RDS if P-a.s. x → ϕ(t, ω)x is continuous for all t ∈ R + .

With the notion of an RDS at our disposal we can now recall the stochastic generalization of notions of absorption, attraction and Ω-limit sets. Definition 1.9. Let (H, d) be as in Definition 1.7 (ii.) Let A, B be random sets. A is said to absorb B if P-a.s. there exists an absorption time t B (ω) ≥ 0 such that for all t ≥ t B (ω)

(i.) A set-valued map K : Ω → 2 H is called measurable if for all x ∈ H the map ω → d(x, K(ω)) is measurable,
ϕ(t, θ -t ω)B(θ -t ω) ⊆ A(ω). A is said to attract B if d(ϕ(t, θ -t ω)B(θ -t ω), A(ω)) ---→ t→∞ 0, P-a.s. . (iii.
) For a random set A we define the Ω-limit set to be

Ω A (ω) := T ≥0 t≥T ϕ(t, θ -t ω)A(θ -t ω).
Definition 1.10. A random attractor for an RDS ϕ is a compact random set A satisfying P-a.s.

i. A is invariant, i.e. ϕ(t, ω)A(ω) = A(θ t ω) for all t > 0.

ii. A attracts all deterministic bounded sets B ⊆ H.

The following proposition yields a sufficient criterion for the existence of a random attractor of an RDS ϕ. We also refer to [START_REF] Kuksin | On random attractors for systems of mixing type[END_REF] for a different approach based on ergodicity, which however seems less suitable for our case.

Proposition 1.11 (cf. [START_REF] Crauel | Attractors for random dynamical systems[END_REF], Theorem 3.11). Let ϕ be an RDS and assume the existence of a compact random set K absorbing every deterministic bounded set B ⊆ H. Then there exists a random attractor A, given by

A(ω) = B⊆H, B bounded Ω B (ω).
From now on we take H := H 1 0 (Λ) * with metric determined by its norm • H . Since we aim to apply Proposition 1.11 to prove the existence of a random attractor for (0.0), we first need to define the RDS associated to (0.0). We take (Ω, F, P) to be the canonical two-sided Wiener space, i.e. Ω = C 0 (R, R m ) and θ t to be the Wiener shift given by θ t ω := ω(t+•)-ω(t).

As in [15, pp. 375-377] we consider Y (t, s, x) := X(t, s, x) -QW t . Then we have for all s ∈ R, x ∈ H, P-a.s.:

Y (t, s, x) = x -QW s + t s ∆Φ(Y (r, s, x) + QW r )dr, ∀t ≥ s.
We can rewrite this as an ω-wise equation:

(1.7) Z t (ω) = x -QW s (ω) + t s A ω (r, Z r (ω))dr, ∀t ≥ s,
where

A ω (r, v) := ∆Φ(v + QW r (ω)). Since for each fixed ω ∈ Ω, A ω : V → V * is hemicon- tinuous,
Z(t, s, x, ω) ∈ L p+1 loc ([s, ∞); V ) ∩ C([s, ∞), H) to (1.7) for all x ∈ H, ω ∈ Ω, s ∈ R
and its continuous dependence on the initial condition x. We now define in analogy to [START_REF] Crauel | Random attractors[END_REF]

S(t, s, ω)x := Z(t, s, x, ω) + QW t (ω), s, t ∈ R; s ≤ t ϕ(t, ω)x := S(t, 0, ω)x = Z(t, 0, x, ω) + QW t (ω), t ≥ 0. (1.9)
By uniqueness for (0.0) S(t, s, ω)x is a version of X(t, s, x)(ω), for each x ∈ H, s ∈ R. For fixed s, ω, x we at times abbreviate S(t, s, ω)x by S t and Z(t, s, x, ω) by Z t . By the pathwise uniqueness of the solution to equation (1.7) we have for all ω ∈ Ω, r, s, t ∈ R, s ≤ r ≤ t, S(t, s, ω) = S(t, r, ω)S(r, s, ω)

(1.8') S(t, s, ω) = S(t -s, 0, θ s ω). (1.8")
The joint measurability of ϕ : R + ×H ×Ω → H follows from the construction of the solutions Z t (for details we refer to [START_REF] Gess | Random Attractor for SPDE driven by general additive noise[END_REF]). Hence ϕ defines an RDS. We can thus apply Proposition 1.11 to prove the existence of a random attractor for ϕ. For this we need to prove the existence of a compact set K(ω), which absorbs every bounded deterministic set in H, P-almost surely. This set will be chosen as

K(ω) := B L 2 (0, κ(ω))
H , where B L 2 (0, κ) denotes the ball with center 0 and radius κ in L 2 . Note that since ϕ(t, θ -t ω) = S(t, 0, θ -t ω) = S(0, -t, ω), this amounts to proving pathwise bounds on S(0, -t, ω)x in the L 2 -norm, where we use the compactness of the embedding L 2 (Λ) ֒→ H. In order to get such estimates we consider norms • Ha on H such that • Ha ↑ • L 2 , as a ↓ 0. These are defined as the dual norms (via the Riesz isomorphism) of the norms

H 1 0 (Λ) ∋ u → a Λ |∇u| 2 dξ + u 2 dξ 1/2 .
Then for s ≤ t we have (see e.g. [34, Theorem 2.6 and Lemma 2.7 (i),(ii)]) for a := 1 n (1.9)

Z t 2 H 1 n = Z s 2 H 1 n + 2 t s Φ(S r ), n(1 - 1 n ∆) -1 Z r -nZ r dr,
where for f, g : Λ → R measurable we set

f, g := Λ f g dξ if |f g| ∈ L 1 (Λ)
. We shall use (1.9) in a crucial way several times below.

Estimates for S t H and bounded absorption

In this section we construct bounded random sets that absorb trajectories with respect to the weak norm • H . For these estimates only the standard assumptions (A1)-(A3) are needed.

Theorem 2.1. Let β ∈ (0, ∞), with β ≤ a 2 , if p = 1.
Then there exists a function p

(β) 1 : R × Ω → R + such that p (β) 1 (t, •) is F-measurable for each t ∈ R, p (β) 1 (•, ω) is continuous and of at most polynomial growth for |t| → ∞, ω ∈ Ω and for all x ∈ H, ω ∈ Ω 0 (cf. Remark 1.7 (i)), s ∈ R: Z(t 2 , s, x, ω) 2 H ≤ Z(t 1 , s, x, ω) 2 H -β t 2 t 1 Z(r, s, x, ω) 2 2 dr + t 2 t 1 p (β) 1 (r, ω)dr, (2.1)
for all s ≤ t 1 ≤ t 2 .
Proof. We fix x, ω, s and set Z r := Z(r, s, x, ω), S r := S(r, s, ω)x for r ≥ s. All constants appearing in the proof below are, however, independent of x, ω and s! Since for

s ≤ t 1 ≤ t 2 Z t 2 2 H = Z t 1 2 H -2 t 2 t 1 Z r , Φ(S r ) dr,
we have for dr-a.e. r ∈ [s, ∞) by (A2)

d dr Z r 2 H = -2 Z r , Φ(S r ) = -2 S r -QW r , Φ(S r ) = -2 S r , Φ(S r ) + 2 QW r , Φ(S r ) ≤ -2a Λ |S r | p+1 dξ + 2 Λ (|QW r Φ(S r )| + c) dξ.
By Young's inequality, for arbitrary ǫ > 0 and some

C ǫ (= C ǫ (p)), C 1 , C 2 ∈ R we have by (A3) Λ |QW r Φ(S r )|dξ ≤ Λ C ǫ |QW r | p+1 + ǫ|Φ(S r )| p+1 p dξ ≤ ǫC 1 S r p+1 p+1 + C ǫ QW r p+1 p+1 + ǫC 2 |Λ|, where |Λ| := Λ dξ. Thus by choosing ǫ = a C 1 we obtain for dr-a.e. r ∈ [t 1 , t 2 ] d dr Z r 2 H ≤ -a S r p+1 p+1 + C ǫ QW r p+1 p+1 + 2|Λ|(c + C 3 ).
where

C 3 := aC 2 C 1 . Now, if p > 1, then for each β > 0 we can find a C β such that for all y ∈ R one has a|y| p+1 ≥ 2β|y| 2 -C β . If p = 1, then we have the same, provided β ∈ (0, a 2 ]. We obtain a S r p+1 p+1 ≥ 2β S r 2 2 -|Λ|C β = 2β Z r + QW r 2 2 -|Λ|C β ≥ β Z r 2 2 -2β QW r 2 2 -|Λ|C β . Hence for p (β) 1 (r, ω) := 2β QW r 2 2 + |Λ|C β + C ǫ QW r p+1 p+1 + 2|Λ|(c + C 3 ) , if ω ∈ Ω 0 0 , otherwise we obtain for dr-a.e. r ∈ [t 1 , t 2 ] d dr Z r 2 H ≤ -β Z r 2 2 + p (β) 1 (r, ω).
and the assertion follows.

Corollary 2.2. Let β ∈ (0, ∞), with β ≤ a 2 if p = 1 and let t ∈ R. Then there exists an F-measurable function q (β,t) 1

: Ω → R, such that for all x ∈ H, ω ∈ Ω 0 and s ≤ t

(2.2) Z(t, s, x, ω) 2 H ≤ q (β,t) 1 (ω) + e -β e 2 (t-s) Z(s, s, x, ω) 2 H . Proof. Since the embedding L 2 ֒→ H is continuous, there is a constant c > 0 such that v H ≤ c v 2 , for all v ∈ L 2 . Hence by Theorem 2.1 d dr ( Z r 2
H ) ≤ - β c 2 Z r 2 H + p (β) 1 (r, ω) dr-a.e. on [s, t].
Hence by Gronwall's Lemma the assertion follows with q β,t

1 (ω) := t -∞ e -β c 2 (t-r) p 1 (r, ω)dr.
Corollary 2.3 (Bounded absorption). Let t ∈ R. Then there is an F-measurable function q

(t) 1 : Ω → R such that for each ̺ > 0 there is an s(̺) ≤ t such that for all ω ∈ Ω 0 , x ∈ H with x H ≤ ̺ Z(t, s, x, ω) ∈ BH (0, q (t)
1 (ω)), for all s ≤ s(̺) i.e. there exists a bounded random set absorbing (Z t ) at time t.

Proof. Let β := a 2 . By Corollary 2.2, we have for β :

= β c 2 Z t 2 H ≤ e -β(t-s) Z s 2 H + q (β,t) 1 ≤2e -β(t-s) ( x 2 H + QW s 2
H ) + q (β,t) 1 ≤2̺ 2 e -β(t-s) + 2e -β(t-s) QW s 2 H + q (β,t) 1
, for all t ≥ s. Hence the result follows with

q (t) 1 := 1 + q (β,t) 1 + 2 sup s≤t (e -β(t-s) QW s 2
H )
and s(̺) ≤ t chosen so that 2̺ 2 e -β(t-s) ≤ 1 for all s ≤ s(̺).

We will need the following auxiliary estimate.

Corollary 2.4.

There is an F-measurable function q : Ω → R + such that for each ̺ > 0 there exists s(̺) ≤ -1 such that for all ω ∈ Ω 0 , x ∈ H with x H ≤ ̺ 0 -1 S(r, s, ω)x 2 2 dr ≤ q(ω) for all s ≤ s(̺).

Proof. Using (2.1) in Theorem 2.1 with t 1 = -1, t 2 = 0 and then using Corollary 2.3 for t = -1 yields for β = a 2 and s ≤ s(̺), where s(̺) ≤ -1 is as in Corollary 2.3,

β 0 -1 S(r, s, ω)x 2 2 dr ≤ 2 Z(-1, s, x, ω) 2 H + 2 0 -1 p (β) 1 (r, ω)dr + 2β 0 -1 QW r (ω) 2 2 dr ≤ βq(ω),
where q(ω) := 2 β q (-1) 1

(ω) + 2 β 0 -1 p (β) 1 (r, ω)dr + 2 0 -1 QW r (ω) 2 2 dr.
3 Estimate for S t 2 and compact absorption

In this section we utilize the stronger assumptions from Hypothesis 1.1 and 1.4 in order to obtain absorption with respect to the strong norm • 2 . By the compact embedding L 2 (Λ) ֒→ H this will prove the required compactness properties.

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

αλ 1 2 ] if p = 1.
Then there is a function p

(α) 2 : R × Ω → R such that p (α) 2 (t, •) is F-measurable for each t ∈ R, p (α) 2 (•, ω) is continuous and of at most polynomial growth for |t| → ∞, ω ∈ Ω and for all x ∈ L 2 (Λ), ω ∈ Ω 0 (cf. Remark 1.7 (i)), s ∈ R: Z(t 2 , s, x, ω) 2 2 ≤ Z(t 1 , s, x, ω) 2 2 -α t 2 t 1 Z(r, s, x, ω) 2 2 dr + t 2 t 1 p (α) 2 (r, ω)dr (3.1) for all s ≤ t 1 ≤ t 2 .
In particular, t → Z t is strongly right continuous in L 2 (Λ).

Proof. Again we fix x, ω, s and use the abbreviation Z r := Z(r, s, x, ω), S r := S(r, s, ω)x for r ∈ [s, ∞). But all constants appearing in the proof below are independent of x, ω and s. Case 1: Assume Hypothesis 1.1.

Let t 1 ≥ s such that Z t 1 ∈ L 2 (Λ) and t 2 ≥ t 1 . (1.9) implies

Z t 2 2 H 1 n = Z t 1 2 H 1 n + 2 t 2 t 1 Φ(S r ), n(1 -1 n ∆) -1 S r -nS r dr (3.2) -2 t 2 t 1 Φ(S r ), ∆(1 -1 n ∆) -1 QW r dr.
We now recall a formula given in [34, Lemma 5.1 (ii)]. Let q, q ′ > 1 such that

1 q + 1 q ′ = 1, f ∈ L q (Λ), g ∈ L q ′ (Λ) and p n (ξ, d ξ) the kernel corresponding to (1 -1 n ∆) -1 (cf. [34, Lemma 5.1 (i)]). Using the symmetry of (1 -1 n ∆) -1 in L 2 (Λ) we obtain f, g -(1 -1 n ∆) -1 g = 1 2 Λ Λ (f ( ξ) -f (ξ))(g( ξ) -g(ξ))p n (ξ, d ξ)dξ + Λ (1 -(1 -1 n ∆) -1 1)f gdξ.
Using this and proceeding analogously to the calculation following formula (5.6) in [START_REF] Röckner | Non-monotone stochastic generalized porous media equations[END_REF] yields for dr-a.e. r ∈ [s, ∞)

Φ(S r ), n(1 -1 n ∆) -1 S r -nS r = -n Φ(S r ), S r -(1 -1 n ∆) -1 S r = - n 2 Λ Λ [Φ(S r ( ξ)) -Φ(S r (ξ))][S r ( ξ) -S r (ξ)]p n (ξ, d ξ)dξ -n Λ (1 -(1 - 1 n ∆) -1 1)Φ(S r )S r dξ ≤ - n 2 Λ Λ (ζ(S r ( ξ)) -ζ(S r (ξ))) 2 p n (ξ, d ξ)dξ -n Λ (1 -(1 - 1 n ∆) -1 )ζ(S r ) 2 dξ = -n ζ(S r ), (1 -(1 - 1 n ∆) -1 )ζ(S r ) = -E (n) (ζ(S r ), ζ(S r )),
where (

E (n) , D(E (n) )) is the closed coercive form on L 2 (Λ) with D(E (n) ) = H 1 0 (Λ) and gener- ator n(1 -(1 -1 n ∆) -1 ) = ∆(1 -1 n ∆) -1
. We obtain from (3.2):

Z t 2 2 H 1 n + 2 t 2 t 1 E (n) (ζ(S r ), ζ(S r ))dr (3.3) ≤ Z t 1 2 H 1 n -2 t 2 t 1 Φ(S r ), ∆(1 -1 n ∆) -1 QW r dr.
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 QW t ∈ C 2 0 here. Using Young's inequality and (A3), for all ǫ > 0 and some C ǫ , C 1 , C 2 > 0 we obtain for dr-a.e. r ∈ [s, ∞)

| Φ(S r ), ∆(1 - 1 n ∆) -1 QW r | = | Φ(S r ), (1 - 1 n ∆) -1 ∆QW r | ≤ ǫ Λ |Φ(S r )| p+1 p dξ + C ǫ Λ |((1 - ∆ n ) -1 ∆QW r )| p+1 dξ ≤ ǫC 1 S r p+1 p+1 + C ǫ ∆QW r p+1 p+1 + C 2 .
Hence

Z t 2 2 H 1 n + 2 t 2 t 1 E (n) (ζ(S r ), ζ(S r ))dr (3.4) ≤ Z t 1 2 2 + 2 t 2 t 1 ǫC 1 S r p+1 p+1 + C ǫ ∆QW r p+1 p+1 + C 2 dr < ∞.
We note that by (1.8) the right hand side of (3.4) is indeed finite. Since

E (n) (ζ(S ), ζ(S r )) is increasing in n, we conclude that sup n∈N E (n) (ζ(S r ), ζ(S r )) < ∞ for dr-a.e. r ∈ [t 1 , ∞). By (A2)' and (A3) we know that for some c 1 , c 2 ≥ 0 ζ(s) 2 ≤ Φ(s)s ≤ c 1 |s| p+1 + c 2 |s|.
Since S r ∈ L p+1 (Λ) this implies ζ(S r ) ∈ L 2 (Λ) for dr-a.e. r ∈ [t 1 , ∞). We now recall the following result from the theory of Dirichlet forms: Let (E, D(E)) be the closed coercive form on L 2 (Λ) given by E(f, g) = Λ ∇f, ∇g R d dξ for f, g ∈ D(E) = H 1 0 (Λ). From [28, Chap. I, Theorem 2.13] we know for

f ∈ L 2 (Λ), that f ∈ D(E) = H 1 0 (Λ) iff sup n∈N E (n) (f, f ) < ∞. Moreover, lim n→∞ E (n) (f, g) = E(f, g) = Λ ∇f, ∇g R d dξ for all f, g ∈ D(E). Hence we obtain for dr-a.e. r ∈ [t 1 , ∞) that ζ(S r ) ∈ D(E) = H 1 0 (Λ) and that lim n→∞ E (n) (ζ(S r ), ζ(S r )) = E(ζ(S r ), ζ(S r )) = Λ |∇ζ(S r )| 2 R d dξ.
Using Fatou's lemma and taking n → ∞ in (3.4) yields

Z t 2 2 2 + 2 t 2 t 1 Λ |∇ζ(S r )| 2 R d dξ dr (3.5) ≤ Z t 1 2 2 + 2ǫC 1 t 2 t 1 S r p+1 p+1 dr + t 2 t 1 (C ǫ ∆QW r p+1 p+1 + C 2 )dr.
Since Z s = x -QW s ∈ L 2 (Λ), for all t 1 ≥ s we obtain Z t 1 ∈ L 2 (Λ) and thus (3.5) holds for all t 2 ≥ t 1 ≥ s.

Choosing ǫ = aλ 1 2C 1 , applying Poincaré's inequality and using the fact that if p > 1 for each α > 0 we can find Cα ≥ 0 such that for all y ∈ R one has aλ 1 |y| p+1 ≥ 2α|y| 2 -Cα ; the same is true for p = 1, if α ∈ (0, αλ 1 2 ]. We obtain from (A2)' that

Z t 2 2 2 ≤ Z t 1 2 2 -2λ 1 t 2 t 1 ζ(S r ) 2 2 dr + aλ 1 t 2 t 1 S r p+1 p+1 dr + t 2 t 1 (C ǫ ∆QW r p+1 p+1 + C 2 )dr ≤ Z t 1 2 2 -aλ 1 t 2 t 1 S r p+1 p+1 dr + t 2 t 1 (C ǫ ∆QW r p+1 p+1 + C 2 + c)dr ≤ Z t 1 2 2 -2α t 2 t 1 S r 2 2 dr + t 2 t 1 (C ǫ ∆QW r p+1 p+1 + C 2 + c + Cα )dr. Now Z r 2 2 = S r -QW r 2 2 ≤ 2 S r 2 2 + QW r 2 2 , whence Z t 2 2 2 ≤ Z t 1 2 2 -α t 2 t 1 Z r 2 2 dr + t 2 t 1 p α 2 (r, ω)dr, (3.6)
for α > 0 arbitrary and

p (α) 2 (r, ω) := C ǫ ∆QW r p+1 p+1 + C 2 + c + Cα + 2α QW r 2 2 , if ω ∈ Ω 0 0 , else.
To obtain right continuity of Z t in L 2 (Λ) first note that by (3.5) applied for t 1 = s and continuity of Z t in H we obtain weak continuity in L 2 (Λ). Now for t n ↓ t by (3.5) applied to t 1 = t we obtain lim sup

n→∞ Z tn 2 2 ≤ Z t 2 2 ;
which implies the right continuity of Z t in L 2 (Λ).

Case 2: Assume Hypothesis 1.4. Let ζ be as defined in Lemma 1.6 and again let t 1 ≥ s such that Z t 1 ∈ L 2 (Λ) and t 2 ≥ t 1 . In order to prove (3.1) in the case QW t ∈ C 1 0 (Λ) we need to be more careful when bounding the second term on the right hand side of (3.3). For this we need the regularity result proved in Lemma 3.3 below, which implies that for every ǫ > 0 there exist constants

C ǫ , Cǫ (= C ǫ (p), Cǫ (p)) such that for dr-a.e. r ∈ [s, ∞) -Φ(S r ), ∆(1 - 1 n ∆) -1 QW r = ∇Φ(S r ), ∇(1 - 1 n ∆) -1 QW r ≤ ǫ ∇Φ(S r ) p+1 p p+1 p + C ǫ ∇(1 - 1 n ∆) -1 QW r p+1 p+1 (3.6') ≤ ǫ ∇Φ(S r ) p+1 p p+1 p + Cǫ ∇QW r p+1 p+1 .
Now using Lemma 3.3 and (3.6') with ε = 1 in (3.3) yields for some constants c, C ∈ R

Z t 2 2 H 1 n + 2 t 2 t 1 E (n) (ζ(S r ), ζ(S r ))dr ≤ Z t 1 2 H 1 n + 2 t 2 t 1 ∇Φ(S r ) p+1 p p+1 p + C1 ∇QW r p+1 p+1 dr ≤ c Z t 1 2 2 + C t 2 t 1 ( ∇QW r p+1 p+1 + 1)dr < ∞ .
Now we can proceed as after (3.4)

to deduce ζ(S r ) ∈ D(E) = H 1 0 (Λ) and lim n→∞ E (n) (ζ(S r ), ζ(S r )) = Λ |∇ζ(S r )| 2 R d dξ, for dr-a.e. r ∈ [s, ∞). Since Φ ′ (r) > 0, ζ(s) = s 0 Φ ′ (r)dr is C 1 (R) with continuous inverse ζ -1 . Thus Φ(x) = x 0 Φ ′ (r)dr = x 0 Φ ′ (r) Φ ′ (r)dr = x 0 ζ ′ (r) Φ ′ (r)dr = ζ(x) 0 Φ ′ (ζ -1 (r))dr = F (ζ(x)), where F (s) := s 0 Φ ′ (ζ -1 (r))dr. Since F ∈ C 1 (R), ζ(S r ) ∈ H 1 0 (Λ) for dr-a.e. r ∈ [s, ∞) and F ′ (ζ(S r ))∇ζ(S r ) = Φ ′ (S r )∇ζ(S r ) ∈ L 1 (Λ) (by (1.4)), we have Φ(S r ) = F (ζ(S r )) ∈ H 1,1 0 (Λ) for dr-a.e. r ∈ [s, ∞) with (3.7) ∇Φ(S r ) = Φ ′ (S r )∇ζ(S r ) ∈ L 1 (Λ).
By (A2)' and (1.6) there are some constants

C 1 , C 2 such that ζ ′ (r) 2 p+1 p-1 ≤ C 1 ζ(r) 2 + C 2 .
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, ∞) ∇Φ(S r )

p+1 p p+1 p = Λ |∇Φ(S r )| p+1 p dξ = Λ | Φ ′ (S r )∇ζ(S r )| p+1 p dξ = Λ |ζ ′ (S r )∇ζ(S r )| p+1 p dξ ≤ ∇ζ(S r ) 2 2 + C Λ |ζ ′ (S r )| 2 p+1 p-1 dξ (3.10) ≤ ∇ζ(S r ) 2 2 + C ζ(S r ) 2 2 + C ≤ C ∇ζ(S r ) 2 2 + C.
We can now go on with bounding the second term on the right hand side of (3.3) as follows: (3.6') and (3.10) imply that for dr-a.e. r ∈ [s, ∞)

Φ(S r ), ∆(1 - 1 n ∆) -1 QW r ≤ ε ∇Φ(S r ) p+1 p p+1 p + Cε ∇QW r p+1 p+1
≤ εC 1 ∇ζ(S r ) 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 be generalized to noise

QW t ∈ H 2,p+1 0 (Λ) (QW t ∈ H 1,p+1 0 (Λ) resp.). Lemma 3.3. Assume Hypothesis 1.4 and let x ∈ L 2 (Λ), s ∈ R and ω ∈ Ω. Then Φ(S(•, s, ω)x) ∈ L p+1 p loc ([s, ∞); H 1, p+1 p 0
) and there exist constants c > 0, C ∈ R, independent of x, s and ω, such that

Z(t 2 , s, x, ω) 2 2 + c t 2 t 1 ∇Φ(S(r, s, ω)x) p+1 p p+1 p dr ≤ Z(t 1 , s, x, ω) 2 2 + C t 2 t 1 ( ∇QW r (ω) p+1 p+1 + 1)dr, ∀t ≥ t 1 ≥ s.
Proof. We use the Galerkin approximation and the notation used in the proof of unique existence of a solution to (1.7) in [START_REF] Prévôt | A concise course on stochastic partial differential equations[END_REF]Theorem 4.2.4]). Let {e i |i ∈ N} be the orthonormal basis of H consisting of eigenfunctions of ∆ on L 2 (Λ) with Dirichlet boundary conditions.

Then e i ∈ C ∞ (Λ) ∩ H 1 0 (Λ) ⊆ V . Furthermore, let H n = span{e 1 , .
.., e n } and define P n :

V * → H n ⊆ C ∞ (Λ) ∩ H 1 0 (Λ) by P n y := n i=1 V * y, e i V e i .
Note that via the embedding

L 2 (Λ) ⊆ H ⊆ V * , P n|L 2 (Λ) : L 2 (Λ) → H n is just the orthogonal projection in L 2 (Λ) onto H n . Let t 1 ≥ s such that Z t 1 ∈ L 2 (Λ), let Z n t denote the solution of Z n t = P n Z t 1 + t t 1 P n A ω (r, Z n r )dr, ∀t ≥ t 1
and let S n t := Z n t + QW t . By the chain rule, for all t 2 ≥ t 1

Z n t 2 2 2 = P n Z t 1 2 2 + 2 t 2 t 1 A ω (r, Z n r ), Z n r dr (3.13) = P n Z t 1 2 2 + 2 t 2 t 1 ∆Φ(S n r ), S n r dr -2 t 2 t 1 ∆Φ(S n r ), QW r dr.
By the same argument as for (3.7) and with ζ as defined in Lemma 1.6 we get

∆Φ(S n r ), S n r = -∇Φ(S n r ), ∇S n r = - Φ ′ (S n r )∇ζ(S n r ), ∇S n r = -∇ζ(S n r ) 2
and using Young's inequality

-∆Φ(S n r ), QW r = ∇Φ(S n r ), ∇QW r ≤ ε ∇Φ(S n r ) p+1 p p+1 p + C ε ∇QW r p+1 p+1 ,
for all ε > 0 and some C ε ∈ R. By (3.13) this yields

Z n t 2 2 2 ≤ P n Z t 1 2 2 -2 t 2 t 1 ∇ζ(S n r ) 2 2 dr (3.14) + 2ε t 2 t 1 ∇Φ(S n r ) p+1 p p+1 p dr + 2C ε t 2 t 1 ∇QW r p+1 p+1 dr.
By the same argument as for (3.10) we realize

∇Φ(S n r ) p+1 p p+1 p ≤ C 1 ∇ζ(S n r ) 2 2 + C 2 , for some constants C 1 , C 2 . Using this in (3.14), with ε = 1 2C 1 yields for some c > 0, C ∈ R Z n t 2 2 2 + c t 2 t 1 ∇Φ(S n r ) p+1 p p+1 p dr ≤ Z t 1 2 2 + C t 2 t 1 ( ∇QW r p+1 p+1 + 1)dr. (3.15)
Both C 1 , C 2 and c, C are independent of x, s and ω.

Hence we obtain the existence of a φ ∈ L

p+1 p ([t 1 , t 2 ]; H 1, p+1 p 0 ) such that (selecting a subse- quence if necessary) Φ(S n r ) ⇀ φ, in L p+1 p ([t 1 , t 2 ]; H 1, p+1 p 0
) and thus in

L p+1 p ([t 1 , t 2 ]; L p+1 p (Λ))
. By the proof of unique existence of a solution we also know that (again selecting a subsequence if necessary)

∆Φ(S n r ) ⇀ ∆Φ(S r ), in L p+1 p ([t 1 , t 2 ]; V * ) and by definition of ∆Φ : V → V * this is equivalent to Φ(S n r ) ⇀ Φ(S r ), in L p+1 p ([t 1 , t 2 ]; L p+1 p (Λ)). Hence φ = Φ(S r ). An analogous argument applied to Z n t 2 yields Z n t 2 ⇀ Z t 2 in L 2 (Λ). Letting n → ∞ in (3.15) we arrive at Z t 2 2 2 + c t 2 t 1 ∇Φ(S r ) p+1 p p+1 p dr ≤ Z t 1 2 2 + C t 2 t 1 ( ∇QW r p+1 p+1 + 1)dr. (3.16) Since Z s = x -QW s ∈ L 2 (Λ)
, for all t 1 ≥ s we obtain Z t 1 ∈ L 2 (Λ) and thus (3.16) holds for all t 2 ≥ t 1 ≥ s.

Corollary 3.4 (Compact absorption).

There is an F-measurable function κ : Ω → R + such that for each ̺ > 0 there exists s(̺) ≤ -1 such that for all x ∈ H with x H ≤ ̺ and all ω ∈ Ω 0 S(0, s, ω)x 2 ≤ κ(ω), for all s ≤ s(̺).

t 2 = 0 ≥ t 1 ≥ s implies Z 0 2 2 ≤ Z t 1 2 2 -α 0 t 1 Z r 2 2 + p (α)
2 (r, ω) dr.

Integrating over t 1 ∈ [-1, 0] yields Z 0 2 2 ≤ 0 -1 Z r 2 2 + |p (α) 2 (r, ω)| dr ≤ 0 -1 (2 S r 2 2 + 2 QW r 2 2 + |p (α) 2 (r, ω)|)dr.
Hence using Corollary 2.4 and recalling that Z 0 = S(0, s, ω)x we obtain the assertion.

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 the embedding L 2 (Λ) ֒→ H is compact, for each ω ∈ Ω the set

K(ω) := B L 2 (0, κ(ω))
H is nonempty and compact in H.

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 [START_REF] Crauel | Random probability measures on Polish spaces[END_REF]Proposition 2.4]

:= {ω ∈ Ω|O ∩ K(ω) = ∅} is measurable. But O ∩ K(ω) =O ∩ B L 2 (0, κ(ω)) H = O ∩ BL 2 (0, κ(ω)) =O ∩ L 2 (Λ) ∩ BL 2 (0, κ(ω)). For C ⊆ L 2 (Λ) and x ∈ L 2 (Λ) let d L 2 (x, C) := inf y∈C x -y 2 . If O ∩ L 2 (Λ) = ∅, then C O = ∅ is measurable and if O ∩ L 2 (Λ) = ∅, then C O = {ω ∈ Ω|d L 2 0, O ∩ L 2 (Λ) ≤ κ(ω)}
is measurable as κ is.

Let B be a bounded subset of H. Then B ⊂ BH (0, ̺), for some ̺ > 0. By Corollary 3.4 there exists a t B := -s(̺) ≥ 1 such that for all x ∈ B, t ≥ t B and ω ∈ Ω 0 ϕ(t, θ -t ω)(x) = S(t, 0, θ -t ω)x = S(0, -t, ω)x ≤ κ(ω).

Hence for all t ≥ t B , ω ∈ Ω 0 , ϕ(t, θ -t ω)(B) ⊂ K(ω), i.e. the random compact set K absorbs all deterministic bounded sets. Now we may apply Proposition 1.11 to get the existence of a global compact attractor A, given by:

A(ω) = B⊂H, B bounded Ω B (ω) H ,
where Ω B (ω) := T ≥0 t≥T ϕ(t, θ -t ω)B denotes the Ω-limit set of B. ), supported by A. Hence using [START_REF] Crauel | Markov measures for random dynamical systems[END_REF] there exists an invariant measure for the Markovian semigroup defined by P t f (x) = E[f (S(t, 0, x))] and it is given by

µ(B) = Ω µ ω (B)P (dω),
where B ⊆ H is a Borel set. If the invariant measure µ for P t is unique, then the invariant Markov measure µ • for ϕ is unique and given by

µ ω = lim t→∞ ϕ(t, θ -t ω)µ.

Attraction by a single point

So far we obtained the existence of the random attractor A for (0.0), but we did not deduce any information about its finer structure. Under a stronger monotonicity condition which was first introduced in [START_REF] Da Prato | Strong solutions of stochastic generalized porous media equations: existence, uniqueness and ergodicity[END_REF] we will now prove that A consists of a single random point. While we had to restrict to noise of regularity at least H 1,p+1 0 (Λ) before, we can now allow more general noise. Let Q be a Hilbert-Schmidt operator from L 2 (Λ) → H and W t be a cylindrical Brownian Motion on L 2 (Λ). Then QW t is an R := QQ * -Wiener process on H. Let e k be an orthonormal basis of eigenvectors of R with corresponding eigenvalues µ k . Assume further ∞ k=1 √ µ k e k V < ∞. Then QW t defines an almost surely continuous process in V . Now the associated RDS to (0.0) can be defined as before.

Define Φ : R → R to be a continuous function such that there exist some constants c ≥ 0, p ∈ (1, ∞), η > 0 such that

|Φ(s)| ≤ c(1 + |s| p ) (5.17) (s -t)(Φ(s) -Φ(t)) ≥ η|s -t| p+1 , s, t ∈ R.
It has been shown in [START_REF] Da Prato | Strong solutions of stochastic generalized porous media equations: existence, uniqueness and ergodicity[END_REF] that (5.17) holds if Φ ∈ C 1 (R), Φ(0) = 0 and if there exist constants κ, η > 0 such that

(5.18) (p + 1) 2 4 η|s| p-1 ≤ Φ ′ (s) ≤ κ(1 + |s| p-1 ), s ∈ R.
This, for example is true for Φ(s) = s|s| p-1 . By Remark 1.3 it is easy to see that (5.18) implies the weaker monotonicity assumption (A1)'. Also note that (5.17) implies the coercivity property (A2). Thus (A1)-(A3) are satisfied and we can define Z t , S t and the RDS ϕ as before (cf. (1.9)).

Remark 5.1. i. Obviously, for Φ as in Example 1.5 the conditions in (5.17) do not hold.

ii. Let Φ(r) := r 0 e -1 |s| ds. Then Φ ∈ C 1 (R), 0 < Φ ′ (r) ≤ 1 and Hypothesis 1.4 (hence also Hypothesis 1.1) holds with p = 1. But obviously (5.17) above does not hold.

Theorem 5.2. Assume (5.17). Then

S(t, s 1 , ω)x -S(t, s 2 , ω)y 2 H ≤ S(s 2 , s 1 , ω)x -y 1-p H + ηλ p+1 2 1 (p -1)(t -s 2 ) -2 p-1 ≤ ηλ p+1 2 1 (p -1)(t -s 2 ) -2 p-1 , for s 1 ≤ s 2 < t,
= S(s, s 1 , ω)x -S(s, s 2 , ω)y 2 H + 2 t s V * ∆Φ(S(r, s 1 , ω)x) -∆Φ(S(r, s 2 , ω)y), S(r, s 1 , ω)x -S(r, s 2 , ω)y V dr = S(s, s 1 , ω)x -S(s, s 2 , ω)y 2 H (5.19) -2 t s V * Φ(S(r, s 1 , ω)x) -Φ(S(r, s 2 , ω)y), S(r, s 1 , ω)x -S(r, s 2 , ω)y V dr ≤ S(s, s 1 , ω)x -S(s, s 2 , ω)y 2 H -2η t s S(r, s 1 , ω)x -S(r, s 2 , ω)y p+1 p+1 dr ≤ S(s, s 1 , ω)x -S(s, s 2 , ω)y 2 H -η t s S(r, s 1 , ω)x -S(r, s 2 , ω)y p+1 H dr,
where for notational convenience we have set η := 2ηλ p+1 2

1 . Thus, formally S(t, s 1 , ω)x -S(t, s 2 , ω)y 2

H is a subsolution of the ordinary differential equation h

′ (t) = -ηh(t) p+1 2 , ∀t ≥ s 2 (5.20) h(s 2 ) = S(s 2 , s 1 , ω)x -y 2 H . Let h ǫ (t) = ( S(s 2 , s 1 , ω)x -y H + ǫ) 1-p + η 2 (p -1)(t -s 2 ) -2 p-1 , t ≥ s 2 .
h ǫ is a solution of (5.20) with h ǫ (s 2 ) = ( S(s 2 , s 1 , ω)x-y H +ǫ)

Φ ǫ (t) = h ǫ (t) -S(t, s 1 , ω)x -S(t, s 2 , ω)y 2 H ≥ Φ ǫ (s) -η t s (h ǫ (r) p+1 2 -S(r, s 1 , ω)x -S(r, s 2 , ω) 2 H p+1 2 )dr ≥ Φ ǫ (s) -η p + 1 2 c p-1 2 ǫ t s Φ ǫ (r)dr.
Using the Gronwall Lemma we obtain

Φ ǫ (τ ǫ ) ≥ Φ ǫ (s 2 )e -η( p+1 2 )c p-1 2 ǫ (τǫ-s 2 ) > 0.
This contradiction proves τ ǫ = ∞ and since this is true for all ǫ > 0 we conclude:

S(t, s 1 , ω)x -S(t, s 2 , ω)y 2 H ≤ ( S(s 2 , s 1 , ω)x -y H ) 1-p + η 2 (p -1)(t -s 2 ) -2 p-1 ≤ S(s 2 , s 1 , ω)x -y 2 H ∧ η 2 (p -1)(t -s 2 ) -2 p-1 ≤ η 2 (p -1)(t -s 2 ) -2 p-1
, for each t > s 2 .

Theorem 5.3. Assume (5.17). The random dynamical system given by ϕ(t, ω)x = S(t, 0, ω)x has a compact global attractor A(ω) consisting of one point

A(ω) = {η 0 (ω)}.
Proof. Since η 0 (ω) is measurable, A(ω) is a random compact set. We need to check invariance and attraction for A(ω). Let t > 0. Then for any x ∈ H, by continuity of x → S(t, 0, ω)x and (1. It is easy to see that the convergence lim s→-∞ S(t, s, ω)x = η t (ω) implies the existence and uniqueness of an invariant measure for the associated Markovian semigroup, defined by P t f (x) := E[f (S(t, 0, •)x)] (cf. [START_REF] Da Prato | Strong solutions of stochastic generalized porous media equations: existence, uniqueness and ergodicity[END_REF]). This invariant measure is given by µ = P • η -1 0 . In fact we can deduce much more. Since evidently η 0 is measurable with respect to F -(as defined in [START_REF] Crauel | Markov measures for random dynamical systems[END_REF]), by [START_REF] Crauel | Markov measures for random dynamical systems[END_REF] µ ω := lim t→∞ ϕ(t, θ -t ω)µ exists P-a.s. and defines an invariant measure for the random dynamical system ϕ (for more details on invariant random measures cf. [START_REF] Crauel | Attractors for random dynamical systems[END_REF]). Moreover by [START_REF] Crauel | Random probability measures on Polish spaces[END_REF]Theorem 2.12] every invariant measure for ϕ is supported by A = {η 0 }, i.e. µ ω ({η 0 (ω)}) = 1 for P-a.a. ω. Hence we have proved the following Corollary 5.4. There exists a unique invariant random measure µ • ∈ P Ω (H) for the random dynamical system ϕ and it is given by µ ω = δ η 0 (ω) , P-a.s. .

Concluding remarks on computational approaches

The porous medium equation considered here is a model case for a general type of equations that include more details of the permeable medium and that has important applications to the simulation of oil reservoirs. We refer to [START_REF] Aarnes | Mixed multiscale finite element methods for stochastic porous media flows[END_REF] for such an application and for an up-todate finite element method that can be used for solving the deterministic version of (1.1). One of the major difficulties here is to account for the spatial variations (represented by the functions ϕ j in the operator Q) by introducing different scales in the finite element subspace. For the quasilinear steady state equation suitable finite element approximations have been set up, cf. [START_REF] Matthies | Stochastic finite elements: computational approaches to stochastic partial differential equations[END_REF], [START_REF] Matthies | Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations[END_REF] and the references therein.

It seems, however, that computational methods for random attractors in infinite dimensional systems (except for the case of a singleton) are well beyond today's computational capabilities.

There are a few approaches to approximate random attractors in stochastic ordinary differential equations [START_REF] Julitz | Approximation of random attractors and random invariant manifolds with subdivision algorithm[END_REF], [START_REF] Keller | Numerical approximation of random attractors[END_REF], [START_REF] Tearne | Boundary approximation of deterministic and random attractors[END_REF]. These are based on the subdivision and box covering techniques developed over the last years by Dellnitz and coworkers (see [START_REF] Dellnitz | Set oriented numerical methods for dynamical systems[END_REF] for a survey). However, these methods are essentially still limited to lower dimensions. In order to proceed to high-dimensional or even infinite-dimensional cases (see e.g. [START_REF] Yang | Random attractors for the stochastic Kuramoto-Shivahinsky equation[END_REF]) one will need reduction principles as they are well established in the theory of inertial manifolds for deterministic PDEs. The corresponding properties of squeezing and flattening (cf. [START_REF] Foias | Some analytic and geometric properties of the solutions of the Navier-Stokes equations[END_REF], [START_REF] Ma | Necessary and sufficient conditions for the existence of global attractors for semigroups and applications[END_REF]) have been generalized to random dynamical systems in [START_REF] Kloeden | Flattening, squeezing and the existence of random attractors[END_REF]. It is also shown in [START_REF] Kloeden | Flattening, squeezing and the existence of random attractors[END_REF] that squeezing is a stronger condition than flattening, but that the latter one is sufficient to establish the existence of a compact random attractor. The determining modes occuring in these properties should form the basis of a reduced space to which numerical methods apply.

  where for nonempty sets A, B ∈ 2 H we set d(A, B) = sup x∈A inf y∈B d(x, y) and d(x, B) = d({x}, B). A measurable set-valued map is also called a random set.

Remark 3 . 5 .

 35 This is analogous to [15, Lemma 5.5, p. 380]. Proof. (3.1) in Theorem 3.1 with

Remark 4 . 2 .

 42 By[START_REF] Crauel | Attractors for random dynamical systems[END_REF] Proposition 4.5] the existence of a random attractor as constructed in the proof of Theorem 4.1 implies the existence of an invariant Markov measure µ • ∈ P Ω (H) for ϕ (in the sense of[START_REF] Crauel | Attractors for random dynamical systems[END_REF] Definition 4.1]

  8'),(1.8") ϕ(t, ω)A(ω) = S(t, 0, ω) lim st, θ t ω)x = {η 0 (θ t ω)} = A(θ t ω).Since the convergence in Theorem 5.2 is uniform with respect to x ∈ H, for any bounded set B ⊆ H we have (again using (1.8"))d(ϕ(t, θ -t ω)B, A(ω)) = sup x∈B S(t, 0, θ -t ω)xη 0 (ω) H = sup x∈B S(0, -t, ω)xη 0 (ω) H → 0,for t → ∞. Hence A(ω) attracts all deterministic bounded sets.

  and • H is its dual norm. Here |•| R d denotes Euclidian norm on R d and below •, • R d shall denote the corresponding inner product. By • p we will denote the L p -norm. Further, for r ∈ N, p ≥ 1 let H r,p 0 (Λ) denote the usual Sobolev space of order r in L p (Λ) with Dirichlet boundary conditions.

  , it is enough to check that for each open set O ⊂ H, C O

  2 , which suggests S(t, s 1 , ω)x-S(t, s 2 , ω)y 2 H ≤ h ǫ (t). This will be proved next.Let Φ ǫ (t) := h ǫ (t) -S(t, s 1 , ω)x -S(t, s 2 , ω)y 2 H and τ ǫ = inf {t ≥ s 2 | 0 ≥ Φ ǫ (t)}. Using 0 < Φ ǫ (s 2) and continuity of Φ ǫ we realize τ ǫ > s 2 . Further note that by definition we haveh ǫ (t) ≥ S(t, s 1 , ω)x -S(t, s 2 , ω)x 2 H on [s 2 , τ ǫ ] and that h ǫ (t) ≤ ( S(s 2 , s 1 , ω)xy H + ǫ) 2 =: c ǫ .Assume τ ǫ < ∞. Then Φ ǫ (τ ǫ ) ≤ 0 and for all s 2 ≤ s ≤ t ≤ τ ǫ , by the mean value theorem and (5.19):

* Supported by the DFG through SFB-701. † Supported by DFG-Internationales Graduiertenkolleg Stochastics and Real World Models, the SFB-701 and the BiBoS-Research Center.

‡ Supported by the DFG through SFB-701 and IRTG 1132 as well as the BIBOS-Research Center.

1 2000


M Said Seddiki

Ye-Qiong Song
email: ye-qiong.song@loria.fr

Mounir Frikha
email: m.frikha@supcom.rnu.tn

Dynamic node allocation in Network Virtualization

Keywords: Network Virtualization, Resource allocation, Game theory, Queuing theory

Network virtualization is a new technology that provides a transparent abstraction of the networking resources. The most important challenge in network virtualization is the allocation of the physical substrate network among the pool of the active virtual networks (VNs). Our work in progress aims at allocating the node resources in a virtualized networking infrastructure. We believe that this allocation should be dynamic to lead to higher performance and better utilization of the physical resources. In this paper, we propose two models for dynamic node allocation for multiple VNs. The first model uses game theory and market-based approach in order to better allocate the physical node. The second one proposes a dynamic weighted round robin (WRR) approach where each VN receives a fraction of the physical node according to an estimation of its current number of waiting packets and its weight. Both models use a distributed approach to minimize the packet delays inside the physical router and to fairly allocate the nodes between different VN.

I. INTRODUCTION

In the recent years, network virtualization has been introduced to provide a solution for the ossification issue of the Internet [START_REF] Chowdhury | A Survey of Network Virtualization[END_REF]. This technology splits the role of Internet service provider ISP into two different roles [START_REF] Chowdhury | Network virtualization: State of the art and research challenges[END_REF]; the service provider (SP) and the infrastructure provider (InP). The InP is responsible for deploying and managing the physical network substrate. The service provider leases resources from multiple InPs and creates VNs by deploying customized protocols. Allocating physical resource for multiple SPs is a difficult task for the InP. The resource allocation approaches are classified into static and dynamic [START_REF] Haider | Challenges in Resource Allocation in Network Virtualization[END_REF]. The dynamic approach offers a higher utilization by reallocating the resource to various VN instantiations. This work aims at maximizing the node throughput and to minimize the packet processing delay. Game theory offers an efficient tool to model, analyze, and solve resource allocation problems involving multiple rational decision makers that strategically interact with each other in intelligent way. Queuing theory also provides an interesting mathematical tool to model and analyze the physical router where a packet needs to be processed by multiple pipe-lined resources from the input ports to the output ports. In this paper, we study the resource allocation problem for multiple VNs with the goal of maximizing the number of VNs provisioned. We focus on how the VNs share a physical node in a fair and efficient way. We proposed two models using game theory and queuing theory to fairly allocate the node resources and to find out bottleneck nodes in order to prevent physical node downtime and performance degradation. This paper is organized as follows. Section II describes the related work and the literature review. The two models for dynamic node allocation are proposed in section III. Results of the proposed models are presented in terms of performance in section IV. Section V concludes the paper and suggests the future work.

II. RELATED WORK

The most critical issue in Network virtualization is the Virtual Network Embedding (VNE) problem [START_REF] Fischer | Virtual Network Embedding: A Survey[END_REF]. It is a three steps process that deals with the efficient mapping of the virtual network and the optimal constrained allocation of the shared physical resources in the substrate network [START_REF] Wang | Network Virtualization: Technologies, Perspectives, and Frontiers[END_REF]. The first step is the resource discovery which is usually performed by the InP to share the current state of the monitored physical network with multiple SPs. The second one is virtual network mapping which is performed by the SPs to match their requests with the available network resources [START_REF] Hsu | Virtual network mapping algorithm in the cloud infrastructure[END_REF]. The third one, which is the step addressed in this work, concerns the virtual network allocation. It is performed by the InP upon the reception of all the allocation requests from the SPs. Several research projects on network virtualization such as CABO [START_REF] Feamster | How to lease the internet in your spare time[END_REF] and 4WARD [START_REF] Niebert | The way 4ward to the creation of a future internet[END_REF] addressed the VNE issue. Many researchers have also extensively studied the resource allocation problem in network virtualization. The allocation is performed by InP once it receives a VN request. The InP should find the optimal resource allocation for both nodes and links in order to avoid deterioration of the Quality of service (QoS) levels for the active VNs. This problem has been addressed using mainly heuristic approaches because it is considered as an NP-hard problem [START_REF] Belbekkouche | Resource Discovery and Allocation in Network Virtualization[END_REF]. The VNs are deployed on top of a physical infrastructure that can be managed by one or more InPs. Each VN is a virtual topology of a set of virtual nodes connected together by a set of virtual links. A portion of the shared physical link and node is assigned to each VN. Razzaq and Rathore [START_REF] Razzaq | An approach towards resource ef?cient virtual network embedding[END_REF] proposed a technique to use the physical resources in an efficient way. The authors suggested mapping the vertices of the VN as closely as possible in the substrate network. Then, all the virtual edges are assigned to the shortest paths that satisfy their demands. The proposed technique optimizes the resource utilization by accommodating more VNs than the approach of greedy node while it increases the VN embedding time. Houidi et al. [START_REF] Houidi | A distributed virtual network mapping algorithm[END_REF] proposed heuristic optimization algorithms to address the resource allocation where each VN request is divided among several InPs. The problem is solved as a mixed integer program in order to increase the acceptance ratio compared to existing heuristic solutions. The approach does not take into consideration the migration of a VN and offers a limited flexibility for handling the VN requests. Botero et al. [START_REF] Botero | Optimal mapping of virtual networks with hidden hops[END_REF] proposed a heuristic algorithm to solve the VN embedding problem based on the optimization theory. The algorithm takes into account the CPU demanded by the hidden hops when a virtual link is mapped to fairly allocate the resources for multiple VNs. It also offers the possibility to accept VNs that do not ask for any demand. The goal of the proposed approach is to leave as much resource as possible in the substrate network resources while mapping the virtual nodes and links with explicit demands. Butt et al. [START_REF] Butt | Topology-awareness and re-optimization mechanism for virtual network embedding[END_REF] proposed a mechanism for the selection among node and link resources based on their importance in the substrate network to prioritize some VNs over others. The mechanism also offers the re-optimization of the embedded VN requests. They concluded that the proposed mechanism improves the performance of the existing algorithms in terms of acceptance ratio and revenue-cost ratio. He et al. [START_REF] He | Davinci: Dynamically Adaptive Virtual Networks for a Customized Internet[END_REF] proposed a framework for Dynamically Adaptive Virtual Networks for a Customized Internet. The framework supports the periodic reassignment of bandwidth among multiple VNs. The proposed approach allows the use of multiple paths to reach a single node. Each path is selected to avoid the creation of bottlenecks. This approach can cause packetreordering problem. In [START_REF] Cheng | Virtual network embedding through topology-aware node ranking[END_REF], the authors formulated a Markov random walk model to compute topology-aware resource ranking of nodes in a virtualized network environment. Based on the rank of each node, they proposed two new algorithms called RW-MaxMatch and RWBFS. The first one sorts virtual nodes in non-increasing order according to their rank values. The second algorithm constructs a breadth-first search tree where the root node is the virtual node with the largest rank value. The authors claimed that the proposed approach outperform the existing approaches in terms of the long-term average revenue and the acceptance ratio.

In this work, we address the problem of dynamic node allocation to satisfy the resource requirements of each VN while optimizing the use of the CPU cycles, the memory access, and the size of the buffer in the substrate node.

III. DYNAMIC NODE ALLOCATION MODELS

In this section, we develop two models for dynamic node allocation in a virtualized network infrastructure using game theory and queuing theory. We model the substrate network as a undirected graph and denote it by G p =(N p ,L p ) where N P is the set of substrate nodes and L p is the set of substrate links. The substrate node is associated with the CPU and memory capacity. The substrate link l p (i, j) ∈ L p between two substrate nodes i and j is associated with the bandwidth capacity. Each VN, using the shortest algorithm, k-shortest paths, or multicommodity flow algorithms [START_REF] Chowdhury | Virtual network embedding with coordinated node and link mapping[END_REF] performs the VN mapping. Once the mapping is done, each VN has to compete with other VNs in order to receive a fraction of each physical node. We consider there are n virtual networks, each with a budget, that are competing on m physical nodes deployed and managed by multiple infrastructure providers. Through the proposed models, we aim at providing a fair allocation scheme between isolated concurrent VNs while they are sharing the same physical node.

A. Dynamic node allocation modeling through game theory

The first dynamic node allocation model is based on proportional share mechanism where each VN divides its budget among the multiple nodes and receives a fraction of each node proportional to its bid. We propose an approach similar to the work proposed by Feldman et al. [START_REF] Feldman | Allocation Market for Computational Resources[END_REF] that was presented for allocating shared computational resources in computer networks. However, we extend the game using a prediction of the next bids in order to maximize the utilization of the physical node resources. The delay decreases when the utility function increases. The objectives of the proposed game are to maximize the VN's utility rather than focusing on the uniqueness of the Nash equilibrium. We assign to each VN a fixed budget G i . This budget depends on requested QoS in order to satisfy its end users. The higher is the requested QoS, the greater is the budget for the VN. For a given VN i the strategy in the game is to submit a bidding vector (B i,j ) j=1,....,m for different physical node and then receives a fraction of each node that depends on its bid. The VN also assigns a weight for the requested physical node. The weights reflect the preference of the VN to each physical node. The total bids for VN i are less than or equal to its initial budget G i . We define the price P j of a node j as the sum of all the bids submitted by all the active VNs for that node. Thus, the fraction of the physical node j received by a virtual network VN i is given by:

F i,j = B i,j P j (1)
The utility function is represented as a function of the fraction F i,1 , F i,2 , ...F i,m , which it receives from each physical node and a vector of weight w i,1 , w i,2 , ...w i,m assigned to each node. It defines the player's preferences on the physical routers. Thus the utility function is a linear function defined as:

U i (F i,1 , ..., F i,m ) = w i,1 F i,1 + ... + w i,m F i,m (2)
By maximizing its utility function, every VN decreases the delay for a packet to be processed by the physical nodes. In our reasoning, we assume that the VN is selfish trying to maximize its utility function and enhancing its performance by allocating the maximum of the node resource. In order to reach the Nash equilibrium state of the best response of VN i is a solution of the following optimization problem:

     M aximize U i = ( Bi,1 P1 , Bi,2 P2 , ... Bi,m Pm ) Subject to l j=1 B i,j = G i and B i,j ≥ 0 (3)
In our approach, we assume that a VN will try to maximize its utility function by predicting the next bids. Each VN uses the past performance of the resource and its price in order to find the best response in the game and maximize its utility function. The key of the prediction is to find the bottlenecked physical node, which is the physical node where a packet for a given VN spends the maximum delay to be processed. Each VN has a bottlenecked resource at a given time. The main idea of the prediction is to minimize the packet delay in the bottlenecked resource and keep the same performance of the other resources. At each time of the game, the bottlenecked physical node changes. The next bid at T + 1 instant for the VN i when the physical node j is considered as bottlenecked is equal to:

B T +1 i,j = P T j * F req i,j (4)
At T +1 instant , the virtual network VN i tries to maximize his bid B T i,j for the rest of non-bottlenecked nodes as a solution of the following optimization problem:

     (B T +1 i,j ) j =j = argmax((B T i,j ) j =j ) Subject to m j=1 B i,j ≤ G i and B i,j ≥ 0 (5)

B. Dynamic node allocation using queuing theory

This work aims at developing multiple schedulers for multiple physical nodes based on WRR algorithm. This scheduler is able to predict the requirement for different flows of packets for a given VN that needs to be processed by different physical nodes. Each VN is served on the basis of the weight assigned to the related queue. We aim at providing a higher isolation for different instance of VNs by allocating the physical node in a fair, adaptive and efficient way. We suppose that a virtual network VN i has a fixed global weight Gw i that depends on the QoS requested to achieve its performance. We also suppose that the processing capacity of a single node N j with a WRR scheduler is C j . If all the concurrent VNs are sharing this node, then each of them receives a fraction of that capacity. The minimum guaranteed service rate given by assigned weight for each VN is equal to:

C T i,j = Gw T i,j * C j N k=1 Gw T k,j (6)
We use the exponential moving average to estimate the current input rate for each VN in each physical node at T instant. The estimation of the input rate is given by the following equation:

λT i,j = (1 -α) * λT -1 i,j + α * λ T i,j (7)
The packets arrive to the first physical node N 1 according to a Poisson process with a rate λ equal to the sum of all the input rate of all the active VNs. The first physical node N 1 is considered as a queuing system with multiple queues and a single sever. The service rate is deterministic and the interarrival is exponential. In order to update the weights of each virtual network VN i in the resource R 1 , the dynamic WRR algorithm uses the estimation about the average number of packet in each queue at T instant according the estimation of the input packet arrival. Let A T i,j be the average number of waiting packets for a given virtual network VN i at T instant in order to be processed by the node N 1 . According to Pollaczek-Khinchin formula [START_REF] Pochec | Modelling with queues: an empirical study[END_REF] it is defined as:

A T i,1 = ( λT i,1 ) 2 2 C T i,1 * (C T i,1 -λT i,1 ) (8)
We define D T i,1 as the average delay for packet for a given virtual network VN i spent in the physical node N 1 at T instant. This delay is equal to the average waiting delay plus the average service delay of the node. The average packet delay is given by the following equation:

D T i,1 = 1 -( λT i,1 / 2 * C T i,1 ) C T i,1 -λT i,1 (9)
For the first physical node N 1 , the new weight of a virtual network VN i at T + 1 depends of its global weight Gw i and the estimation of its average number of waiting packet at T instant. At T + 1 instant each virtual network VN i receives a new weight in the node N 1 equal to :

w T +1 i,1 = ( Gw i N k=1 Gw k + A T i,1 N k=1 A T k,1 ) C 1 2 (10)
For the other physical nodes N j =1 , each virtual network VN i receives a new weight according to its global weight Gw i and the number of packets that have been already processed by the previous node N j-1 . Thus, at T + 1 instant every virtual network VN i receives a new weight equals to :

w T +1 i,j =1 = ( Gw i N k=1 Gw k + C T i,j-1 N k=1 C T k,j-1 ) C j =1 2 (11)

IV. EXPERIMENTAL RESULTS

In this section, we implement the proposed node allocation models using Matlab and present the experimental results. In order to discuss the performance results of the node allocation, we suppose that 3 VNs are sharing 3 physical nodes for a given infrastructure provider. We try to simulate the fraction received for each VN during a time period. The virtual networks (VN 1 , VN 2 , VN 3 ) are competing for the nodes (N 1 , N 2 , N 3 ). We assign three budgets (G 1 , G 2 , G 3 ) for the VNs. We assume that G 1 ≥ G 2 ≥ G 3 . The budget assigned for the VN depends on the level of the quality of service requested. All the VNs have a Poisson arrivals at rate λ 1 , λ 2 , λ 3 with fixed length packets. We implement the delays for processing a packet in each node. We suppose there is a random input packet flow for each physical node. We run our algorithm and we observe the packet delays in every node and how much fraction received by the VNs at T and T + 1 instants. Each VN predicts the next bids in order to find the best response during the game at T + 1 instant.

Figure 2 illustrates the fraction received by the three concurrent VNs in order to get a fraction of the node N 1 . Figure 3 shows the delay for packets of the virtual network of VN 1 inside the node N 1 . According to both figures, the physical node N 1 was considered as a bottlenecked node for the Virtual network VN 1 between T= 22ms and T= 25ms. Thus, VN 1 increased its bid in order to receive a higher fraction and decrease the packet delay inside the node N 1 .

For the second model of node allocation, we assign three global weights (Gw 1 , Gw 2 , Gw 3 ) for the three virtual networks. We assume that Gw 1 > Gw 2 > Gw 3 , and the weights assigned to the VN depend on the level of the requested QoS where Gw 1 = 50, Gw 2 = 30, and Gw 3 = 20. The queuing system is supposed to be stable such as the sum of the arrival rates of all the VNs is always less than the node capacity. We update dynamically the weight of each VN i and we compare the results obtained by the static and dynamic weighted round robin when the arrival rate is the same and the exponential moving average parameter α is equal to 0,5. The performances of our simulation are expressed in terms of packet delays, the number of waiting packets in different queues for different VNs and the efficient usage of the physical nodes.

Figure 4 shows the fraction received by the virtual network VN 1 of the physical node N 1 , that has a processing capacity of 2000 packets/s, with the proposed dynamic WRR algorithm and the static one. Figure 5 illustrates the average packet delay of the packets of the virtual network VN 1 inside the physical node N 1 . We can clearly conclude that our approach achieves better results than the static approach in terms of minimizing the packet delay inside a physical node.

V. CONCLUSION

In this paper, we applied game theory and queuing theory concepts to solve the problem that occurs where a physical node is allocated dynamically to a set of VNs. Our main contribution is to use a prediction of the current workload in order to share the node among multiple instances of VNs and avoid service degradation. We have shown that both models offer higher utilization of physical node and better managing the satisfaction of virtual networks by minimizing the packet delays inside the physical node.

The future steps consist in exploring other distributions of packet inter-arrival rates at each node that could be more realistic for Internet-like traffic [START_REF] Adas | Traffic models in broadband networks[END_REF]. We will also propose an admission control for each model in order to bind the limit for creation requests for a virtual network and to prevent it from deteriorating the performance of the active VNs.

Figure 1 .

 1 Figure 1. The virtualized networking environment

Figure 2 .Figure 3 .

 23 Figure 2. The fractions of the node N 1 received by the three concurent VNs during a given time interval

Figure 4 .

 4 Figure 4. The fractions of the node N 1 received by the virtual network VN 1

Figure 5 .

 5 Figure 5. The average delay for the packets of VN 1 inside the node N 1


Michal Malatek

Basile Dufay
email: basile.dufay@unicaen.fr

Sébastien Saez

Christophe Dolabdjian

Improvement of the off-diagonal magnetoimpedance sensor white noise

Keywords: Off-diagonal magnetoimpedance, amorphous ribbon, magnetic field sensor, noise

This work deals with the analysis of noise sources of the off-diagonal magneto-impedance sensor with stressannealed amorphous ribbon. This experimental analysis shows that the equivalent magnetic noise spectral density is mostly due to the electronic conditioning circuitry that operates the sensor. The electronic noise was then reduced with employment of an improved excitation-detection technique for our sensor and an equivalent magnetic noise spectral density below 1 pT/ √ Hz in white noise region has been reached. Indeed, the equivalent magnetic white noise level of the output signal decreased from 5 pT/ √ Hz to 560 fT/ √ Hz. Nevertheless, this noise level is still limited by the electronic conditioning, leaving room for further improvement.

Introduction

The giant magnetoimpedance (GMI) effect is a phenomena observed in ferromagnetic conductors driven by high frequency current and exposed to external magnetic field. Basically, the ac permeability, and so the impedance, of such material changes when exposed to an external magnetic field variation [START_REF] Knobel | Giant magnetoimpedance[END_REF]. In the last few decades it has attracted considerable attention due to its strong potential in high sensitivity magnetometry.

Diagonal (or classical) GMI effect is observed as an impedance change of the magnetic materials related to an external magnetic field variation. Another approach to improving sensitivity is the choice of a two port network configuration, in which the GMI magnetic material is associated with a pick-up coil [START_REF] Sasada | Orthogonal fluxgate mechanism operated with dc biased excitation[END_REF][START_REF] Dufay | Improved gmi sensors using strongly-coupled thin pick-up coils[END_REF]. In such a case, a change of the apparent (off-diagonal) impedance, Z 12 = V 2 /I 1 (see Fig. 1), occurs versus an external magnetic field variation. Sensitivity of the off-diagonal GMI device to the external magnetic field can be magnified with a large number of turns of the pick-up coil. It was also proven that parameters like linearity and temperature stability are also improved [4] compared to the classical (diagonal) GMI devices. Different names, like Inverse Wiedemann effect [START_REF] Kraus | Magnetic field sensor based on asymmetric inverse wiedemann effect[END_REF] or Orthogonal fluxgate sensor in fundamental frequency [START_REF] Sasada | Orthogonal fluxgate mechanism operated with dc biased excitation[END_REF][START_REF] Plotkin | Compensation of the thermal drift in the sensitivity of fundamental-mode orthogonal fluxgates[END_REF], were sometimes used.

In this paper, following up on the work from ref. [START_REF] Malátek | Off-diagonal gmi sensor with stress-annealed amorphous ribbon[END_REF], main attention is paid to the analysis of noise sources and to the improvement of the sensor white noise level.Regarding the output noise spectral density, it usually can be separated in a low frequency excess noise, named 1/f noise, and a white noise floor [START_REF] Dufay | Impact of electronic conditioning on the noise performance of a two-port network giant magnetoimpedance magnetometer[END_REF][START_REF] Ding | Equivalent magnetic noise limit of low-cost gmi magnetometer[END_REF]. The white noise is mostly expected as a limit given by the electronic noise while the intrinsic noise of the sensing element is expected to be more of the 1/f behavior. However, exact differentiation of noise sources and their spectrum is a difficult task.

Investigation of GMI sensors noise performance is not a novelty. Several researches reported on noise level achieved with GMI devices [START_REF] Yabukami | A design of highly sensitive gmi sensor[END_REF]. However, a conceptual improvement of noise properties of GMI sensing element was first given by Melo et al. [START_REF] Melo | Optimization of the magnetic noise and sensitivity of giant magnetoimpedance sensors[END_REF]. Later, a detailed description and a model of noise contributors of classical GMI sensors were presented by Ding et al. [START_REF] Ding | Equivalent magnetic noise limit of low-cost gmi magnetometer[END_REF]. In the last Hz for an off-diagonal GMI sensor based on a 45 mm long and 120 µm Co-rich magnetic wire [START_REF] Butta | Reduction of noise in fundamental mode orthogonal fluxgates by optimization of excitation current[END_REF].

The paper is organized as follows. Section 2 describes the sensor operation, the experimental setup and its noise model, based on previous development [START_REF] Dufay | Impact of electronic conditioning on the noise performance of a two-port network giant magnetoimpedance magnetometer[END_REF]. The electronic conditioning for low-noise measurement performance is also pointed out there. Then, results of an improved set-up are given in section 3, which is followed by a conclusion.

Classical set-up

This section deals with the characterization of the sensitivity and noise performances of the measuring chain in a classical set-up illustrated on Fig. 2. The sensor is made of 11 cm long, 1.2 mm wide and 17 µm thick ribbon of Co 67 Fe 4 Si 8 B 14 Cr 7 produced by planar flow casting. The ribbon, annealed under tensile stress of 50 MPa for 10 minutes at 390 °C, is placed on non-magnetic plate and is electrically connected to a coaxial cable. A pick-up coil of 290 turns is wound on fiberglass tube of 9 mm in diameter, in which the ribbon is inserted.

Taking into account all electronic noise sources appearing in the measuring chain, and considering that the sensing device is fully described by means of classical two port network parameter [START_REF] Dufay | Characterization of an optimized off-diagonal gmibased magnetometer[END_REF] and its associated impedance matrix (Fig. 1b), the expected equivalent magnetic white noise spectral density of the sensor, at the magnetic field working point, B 0 , is given by [START_REF] Dufay | Impact of electronic conditioning on the noise performance of a two-port network giant magnetoimpedance magnetometer[END_REF][START_REF] Ding | Equivalent magnetic noise limit of low-cost gmi magnetometer[END_REF]

b 2 n (f ) B=B0 ≈ 2 R1+|Z11(f0)| eg1 ∂|Z 21 (f 0 )| ∂B 2 |Z21(f0)| R1+|Z11(f0)| 2 eg1 Dy 2 + e 2 nR1 (f 0 ) + |Z 22 (f 0 )| 2 i 2 n-LIA (f 0 ) + e 2 n-LIA (f 0 ) (1)
where R 1 is the generator output impedance, Z ij are the impedance matrix parameters of the sensing element at the static working point B 0 , e g1 is the amplitude of the excitation generator, i n-LIA and e n-LIA are the current and voltage noise of the Lock-In Amplifier (LIA), and D y , is the generator dynamic range. This equation shows how each element contributes to the equivalent magnetic noise level of the sensor. Notice that noises coming from the lead battery power fluctuations and the intrinsic magnetic sensor noise are neglected here. All noise measurements were accomplished with sensor placed into the GREYC magnetic-shielded room excluding all noise contribution from the surrounding environment. The AC current signal, i 1 , Figure 2: Sensing head and associated conditioning electronic set-ups. R is a DC bias resistance. The lock-in amplifier is synchronized to the excitation frequency f 0 .

was driven by a waveform generator HP33120A at frequency, f 0 (f 0 ≈ 500 Hz). The DC bias current came from 12 V lead battery. In addition, the sensor was exposed to an external low frequency sine wave magnetic field, h ext , induced by Helmholtz coils. This field, of nT-range magnitude at frequency, f 1 , around 10 Hz, was used as reference signal for sensitivity calibration. The output signal, v 2 , of the off-diagonal GMI sensing element is therefore a classical AM modulated signal with high frequency carrier at f 0 and modulating sine wave signal at frequency f 1 . The modulating signal is extracted thanks to a synchronous demodulation accomplished with high-frequency SR-844 lock-in amplifier (Fig. 2). Output noise spectral density of the measuring chain was recorded by a low frequency spectral analyzer HP3562A.

Firstly, the effect of the AC amplitude current excitation was investigated. In this experiment, the AC current amplitude, I 1 , was increased up to 50 mA RMS (sinewave) and was then set back to a value of 10 mA RMS . This sequence is shown in Fig. 3, where correlation between sensitivity and AC amplitude is depicted.

As it can be seen, for three different DC bias current values, the sensitivity increases linearly with the amplitude of the AC excitation up to a certain point. At the largest excitation of 50 mA RMS , the sensitivity was not measurable for DC bias current of 1 mA (solid squares) and dropped down by 25% for DC bias current of 5 mA (solid circles). Afterward, when the amplitude of excitation current was set back to 10 mA RMS , the sensitivity of the sensor was lower than the original one for these two DC bias currents (empty marks at 10 mA value on x-axis). Nevertheless, it appears that the sensor re- trieves its original behavior after being exposed to a strong external magnetic field pulse (labeled as magnetic reset in Fig. 3). In the case of DC bias current of 10 mA, the sensor works properly even at 50 mA RMS and keeps linearity between sensitivity and excitation current amplitude. The sensitivities at 10 mA RMS excitation, before and after the highamplitude excitation, are nearly the same in that case. In all the following we make sure to avoid anomalies occuring at high excitation current.

The output noise spectral densities for the different amplitudes of AC excitation are depicted on Fig. 4a.

These experiments highlight the weakness of the measurement set-up, which is the dynamic reserve of the lock-in amplifier. Each record of the Fig. 4a was acquired with lock-in amplifier gain set to the most suitable value (i.e. a maximal gain with nonsaturated/non-overloaded input filters). Thus, gain and dynamic range of the amplifier were not the same for the different excitation parameters of the sensor. Looking at the output voltage noise spectral densities (Fig. 4a), one can see that the noise floor reaches value around 30 nV/ √ Hz for lock-in amplifier ranges of 3 mV and 10 mV, whereas it jumps up to about 100 nV/ √ Hz when the measurement range is 30 mV, 100 mV, 300 mV or 1 V. Obviously, the white noise level in our experiments is determined by the lock-in amplifier dynamic range and hides the other electronic noise sources as well as the intrinsic noise of the sensor. Figure 4b shows the equivalent magnetic noise spectral density expressed in units of T/ √ Hz after divided by the sensitivity. To summarize, we observe, for used ribbon sensor in the classical electronic set-up, the following points

• best noise performance was achieved for AC current excitation amplitudes around 40 mA RMS ,

• the measured white noise level of 4 pT/ √ Hz is limited by the lock-in amplifier noise and its dynamic reserve,

• a further increase of the sensitivity must not necessarily reduce the equivalent magnetic noise spectral density of the measuring chain since at excitation current of 50 mA RMS the sensitivity rises proportionally (Fig. 3), but the noise background grows even more (Fig. 4b).

Similar behaviors were also recently reported in [START_REF] Butta | Reduction of noise in fundamental mode orthogonal fluxgates by optimization of excitation current[END_REF].

Improved set-up

Previous section highlights the problematic part of the signal conditioning chain due to the dynamic of the demodulation stage. In order to operate at a lower measurement range, with lower white noise level, of the lock-in amplifier, an improved set-up was realized by means of differential low noise preamplifier and additional generator. The schematic of this set-up is shown on Fig. 5. The phase and amplitude of the second generator is adjusted in order to be subtracted from the pick-up voltage, v 2 , of the sensor. By this way, an AM signal with suppressed carrier is obtained at the output of the preamplifier allowing to reduce the lock-in required dynamic range, still having the same amplitude of modulated signal. Improvement of the white noise level is shown on Fig. 6, where both records (in the original and the improved configurations) are compared. Here, the sensor is driven by 10 mA RMS + 10 mA DC current. The capacitor, C ap , in parallel to the pick-up coil, tunes the sensor into the resonance according to the excitation frequency, which increases the sensor sensitivity and so the SNR [START_REF] Malátek | Off-diagonal gmi sensor with stress-annealed amorphous ribbon[END_REF]. Similarly as Eq. ( 1), the expected equivalent magnetic white noise level of the sensor, at the magnetic field working operation, B 0 , is given by

b 2 n (f ) B=B0 ≈ 2 R1+|Z11(f0)| Eg1 ∂|Z 21 (f 0 )| ∂B 2 |Z21(f0)| R1+|Z11(f0)| 2 2 Eg1 Dy 2 + e 2 nR1 (f 0 ) +e nR2 + |Z 22 (f 0 )| 2 i 2 n-buf f (f 0 ) +e 2 n-buf f (f 0 ) + e 2 n-LIA G 2 (2)
where G, i n-buf f and e n-buf f are the gain, the current noise source and voltage noise source of 200 400 600 800 1000 1200 0,5

Equivalent magnetic white noise spectral density (pT/ÖHz) Excitation frequency (kHz)

AC amplitude 20mA 25mA 30mA

Figure 7: Equivalent magnetic white noise levels versus excitation current frequency, for several excitation current amplitudes (measured with the improved measurement set-up described on Fig. 5).

Results are obtained with the new pick-up coil, which increase the filling factor, tuned with a parallel capacitor.

the preamplifier, respectively. Several noise measurements were accomplished at various excitation frequencies to find the best regime of operation of the sensor. Experimentally, the best noise performances, with a white noise level of 0.7 pT/ √ Hz, were achieved at f 0 = 360 kHz, for an excitation current amplitude of 30 mA RMS and a 10 mA DC bias current.

Following these results, a new pick-up coil, with same number of turns as the previous one and smaller diameter (6.7 mm), was used in order to compare the effect of a higher filling factor. Fig. 7 shows the measured equivalent magnetic white noise levels, in units of pT/ √ Hz, versus the frequency of excitation. Capacitance, C ap , was tuned accordingly to each excitation frequency. Employing the new pick-up coil, we achieve minimum white noise level of around 0.6 pT/ √ Hz at an excitation frequency of 1.27 MHz.

The output voltage noise spectral density of the sensor operated under these optimal conditions is given on Fig. 8a (gray empty triangles), which is also compared to the noise spectral density from generator (red line). The system noise was evaluated by means of additional generator (replacing the sensor). The amplitude and phase of the generator output were adjusted in the way to match with the signal in the second input of the preamplifier. It clearly appears that the generator noise mostly contributes to the total noise level of the measur-ing chain. The black minus marks in Fig. 8 show the noise floor of the detection stage, which was acquired when the amplifier's input was terminated with 50 Ω impedance.

Fig. 8b shows the equivalent magnetic noise spectral densities associated to the Fig. 8a after dividing by the sensitivity of the measuring chain. Although the sensor, with the new pick-up coil, reaches an equivalent magnetic white noise level below 0.6 pT/ √ Hz at frequencies higher than 300 Hz (gray triangles in Fig. 8b), the performances at very low frequencies are worse than those with the original pick-up coil (green squares in Fig. 8b). Strong correlation (between the generator and sensor noise spectral densities with new coil in its best performance) shows that, in this particular case, the generator noise is responsible for the noise behavior in the low frequency range (below 300 Hz).

Conclusion

Off-diagonal magnetoimpedance sensor employing amorphous ribbon with creep-induced anisotropy was examined in order to investigate its magnetic noise performances. Based on detailed experimental procedures, a successful approach to sub-pT resolution was achieved with an improved operation set-up of the sensor. A reduction of white noise level from about 5 pT/ √ Hz [START_REF] Malátek | Off-diagonal gmi sensor with stress-annealed amorphous ribbon[END_REF] to 560 fT/ √ Hz has been achieved for frequencies higher than 300 Hz. Our experiments also proved the fact that resolution of magnetoimpedance sensors is mainly limited by electronic devices that operate the sensors. Hence, the intrinsic magnetic noise level of magnetoimpedance sensor is difficult to observe without deep characterization of full measurement of the operation set-up.

The experiments also point out the irreversible transformation of the magnetic structure in the ribbons occurring with high amplitude excitation. At certain amplitude of AC current, the magnetic ordering in the ribbon changes from circumferentially oriented magnetization moments into a domain structure with left-and right-hand oriented magnetic domains and the 1/f noise component of the sensor dramatically increases which may be a consequence of domain walls (domain wall movements) presence in the structure. The point, at which the magnetic structure changes, depends on the excitation current amplitude, on the DC bias current and also on the external static magnetic field working point. Further work on this area has to be done to improve the low frequency equivalent magnetic noise of GMI-based sensor. The best performances achieved with the original coil and the one with the new coil is given by the green full squares and empty gray triangles, respectively.White noise floor is limited by the detection stage noise level (minus black marks). The noise level of the generator dynamic range is given by the red line.

Figure 1 :

 1 Figure 1: (a) Sensing element (an amorphous ribbon with associated pickup coil). i 1 and v 2 are the AC excitation current and the induced pick-up voltage, respectively. (b) Associated electrical two port network model.

Figure 3 :

 3 Figure 3: Sensitivity versus excitation current amplitude for three different values of the DC bias current. The excitation frequency was set to 490 kHz. Arrows indicate the direction of AC amplitude development within the experiment. The magnetic reset consists of a strong external applied magnetic field pulse.

Figure 4 :

 4 Figure 4: (a) Output voltage noise spectral densities (µV/ √ Hz) of the basic measurement set-up for a 10 mA DC bias current and different amplitudes of sine wave excitation current at f 0 = 490 kHz. Input range of the lock-in amplifier is written in square brackets. (b) Equivalent magnetic noise spectral densities of the sensor (pT/ √ Hz) after dividing by the sensitivity.

Figure 5 :

 5 Figure 5: (a) Sensor head operation in the improved signal conditioning set-up. The resistor R allows to set a DC bias current flowing through the ribbon. The additional generator on input B of the preamplifier is used in order to suppress the carrier of the AM signal from the sensing element. The lock-in amplifier is synchronized to the excitation frequency, f 0 . The capacitor, C ap , in parallel to the pick-up coil, tunes the sensor at the resonance, according to the excitation frequency.

Figure 6 :

 6 Figure 6: Output voltage noise spectral density for the two used set-up considering an excitation current amplitude of 10 mA RMS and a static bias current of 10 mA in both cases.

Figure 8 :

 8 Figure 8: Voltage (a) and equivalent magnetic (b) noise spectral density versus experimental condition.The best performances achieved with the original coil and the one with the new coil is given by the green full squares and empty gray triangles, respectively.White noise floor is limited by the detection stage noise level (minus black marks). The noise level of the generator dynamic range is given by the red line.

Acknowledgments

Authors would like to thank to Luděk Kraus for his valuable remarks and for preparation of the ferromagnetic material that was used in the sensor's core.