Large deviations for invariant measures of stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term

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www.elsevier.com/locate/anihpb

Large deviations for invariant measures of stochastic reaction–diffusion systems with multiplicative noise

and non-Lipschitz reaction term

Sandra Cerrai

^a,^∗

, Michael Röckner

^b

aDip. di Matematica per le Decisioni, Università di Firenze, Via C. Lombroso 6/17, 50134 Firenze, Italy bFakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

Received 15 May 2003; accepted 30 March 2004 Available online 11 September 2004

Abstract

In this paper we prove a large deviations principle for the invariant measures of a class of reaction–diffusion systems in bounded domains ofR^d,d1, perturbed by a noise of multiplicative type. We consider reaction terms which are not Lipschitz- continuous and diffusion coefficients in front of the noise which are not bounded and may be degenerate. This covers for example the case of Ginzburg–Landau systems with unbounded and possibly degenerate multiplicative noise.

Résumé

Dans cet article on prouve un principe de grandes déviations pour les mesures invariantes de systèmes de réaction–diffusion stochastiques dans des domaines bornés deR^d,d1, perturbés par un bruit multiplicatif. On considère des termes de réaction qui ne sont pas lipschitziens et des coefficients de diffusion qui ne sont pas bornés et peuvent être dégénérés. Ceci s’applique par exemple au cas de systèmes de Ginzburg–Landau avec bruit multiplicatif non borné et éventuellement dégénéré.

MSC: 60F10; 60H15

Keywords: Large deviations principle; Stochastic partial differential equations; Invariant measures; Multiplicative noise

* Corresponding author.

E-mail addresses: sandra.cerrai@dmd.unifi.it (S. Cerrai), roeckner@mathematik.uni-bielefeld.de (M. Röckner).

doi:10.1016/j.anihpb.2004.03.001

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1. Introduction

In this paper we are dealing with the long-term behavior of the stochastic reaction–diffusion system









∂u_i

∂t (t, ξ )=Aiu_i(t, ξ )+f_i(ξ, u₁(t, ξ ), . . . , u_r(t, ξ )) +ε_r

j=1g_ij(ξ, u₁(t, ξ ), . . . , u_r(t, ξ ))Q_j^∂w_∂t^j(t, ξ ), t0, ξ∈O, u_i(0, ξ )=x_i(ξ ), ξ∈O, Bⁱu_i(t, ξ )=0, t0, ξ∈∂O, 1ir,

(1.1)

withε >0. HereOis a bounded open set ofR^d, withd1, having aC^∞boundary. For eachi=1, . . . , r Ai(ξ, D)=

d h,k=1

∂

∂ξ_h

a_hkⁱ (ξ ) ∂

∂ξ_k

−αi, ξ∈O. (1.2)

The constantsα_i are strictly positive, the coefficientsaⁱ_hkare taken inC^∞(O)and the matricesaⁱ(ξ ):= [aⁱ_hk(ξ )]hk

are non-negative and symmetric, for eachξ∈O, and fulfill a uniform ellipticity condition, that is inf

ξ∈Oaⁱ(ξ )h, hλ_i|h|², h∈R^d,

for some positive constantsλ_i. Finally, the operators Bⁱ act on ∂O and are assumed either of Dirichlet or of co-normal type.

The mappingf:=(f₁, . . . , f_r):O×R^r→R^ris only locally Lipschitz-continuous and has polynomial growth.

The mappingg:= [gij]:O×R^r →L(R^r)is Lipschitz-continuous, without any assumption of boundedness and non-degeneracy.

The linear operatorsQj are bounded onL²(O)and may be taken to be equal to the identity operator in case of space dimensiond=1. The noisy perturbations ∂wj/∂t are independent cylindrical Wiener processes on a stochastic basis(Ω,F,Ft,P).

For example, in the case of space dimensiond=1 andr=2, we can deal with systems of the following type









∂u₁

∂t =_∂ξ^∂ a1∂u₁

∂ξ

−α1u1−c1u^2k₁ ⁺¹+f1(u1, u2)+

g1(u1, u2),^∂w_∂t ,

∂u₂

∂t =_∂ξ^∂ a₂^∂u_∂ξ²

−α₂u₂−c₂u^2k₂ ⁺¹+f₂(u₁, u₂)+

g₂(u₁, u₂),^∂w_∂t ,

u_i(0, ξ )=x_i(ξ ), ξ∈O, u_i(t, ξ )=η_iu_i(t, ξ )+(1−η_i)^∂u_∂ξⁱ(t, ξ ), ξ ∈∂O,

whereaiare positive functions inC¹(O),ηi∈ {0,1},αiandciare positive constants (in factαi can be taken zero in the case of Dirichlet boundary conditions, that is ifηi=1),f=(f₁, f₂):R²→R²is aC¹function having linear growth, withf (0)=0 andDf (0)diagonal, andg=(g₁, g₂):R²→L(R²)is any Lipschitz continuous function such thatg(0)either vanishes or is diagonal invertible and such that

g(σ )

L(R²)c 1+ |σ|^γ

, σ∈R², with

2k+1> (1+6γ )∨2.

In particular, ifgis bounded in the reaction term we can take any power 2k+13.

In [2] it is proved that for any ε > 0 and p 1 system (1.1) admits a unique global solution u^x_ε ∈ L^p(Ω;C([0, T];E)), whereE is the space of continuous functions onO with values in R^r, and for each initial datumx∈Eanda >0 the family of probability measures{L(u^x_ε(t)}ta is tight in(E,B(E)). In particular,

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due to the Krylov–Bogoliubov theorem this yields the existence of a sequence{tn} ↑ +∞(possibly depending onε) such that the sequence of probability measures defined by

ν_ε,n(Γ ):= 1 tn

t_n

0

P u⁰_ε(s)∈Γ

ds, Γ ∈B(E), (1.3)

converges weakly to some measureν_ε, which is invariant for system (1.1).

In the earlier paper [3] we have proved that the process{u^x_ε}ε>0is governed by a large deviation principle in C([0, T];E), for anyT >0. Our aim here is to prove that the family of invariant measures{νε}ε>0 defined as the weak limits of the sequences of measures as in (1.3) obeys a large deviation principle inE, asεgoes to zero (precise hypotheses on the coefficients are specified in Section 2 below to which the reader is referred to).

Clearly the first step in the proof of large deviations estimates is defining an appropriate action functionalV having compact level sets. The hardest part here is not to findV (see (5.1) below for its initial definition and, in particular, [9] and [14]) but to find a good characterization of it, in order to prove that its level sets are compact and, maybe more importantly, to get a better intuition about its meaning. So, we spend a great deal of effort to prove that (as in [8] and [14]), the action functionalV, also called quasi-potential, has the following form

V (x)=min

I_−∞(z); z∈C (−∞,0];E

, z(0)=x, lim

t→−∞z(t)

E=0

. (1.4)

HereI_−∞(z)is the minimum energy required to producezas a solution of the skeleton equation corresponding to (1.1) fort∈(−∞,0], i.e. replace ∂w/∂t by a deterministic functionϕ∈L²(−∞,0;L²(O,R^r)) so that the corresponding solutionz(ϕ)equalszand the energy (:= |ϕ|²_L2(−∞,0;L²(O,R^r))) is minimal (cf. Section 3 and the beginning of Section 5 below).

By compactness the infimum in (1.4) is indeed achieved by somez₀, which exhibits more regularity in the space variables than just being inE(cf. Lemma 3.5 which in turn is essential for the proof of Proposition 5.4, but also for the proof of Lemma 7.1 which yields upper bounds). We would like to mention at this point that proving (1.4) requires considerable new input, since we consider space dimensiond1, so the coefficient in front of the noise (in contrast to the one-dimensional case considered in [9] and [14]) can no longer be invertible. In addition truly degenerate multiplicative noise is included in our framework.

Once we have shown that the mappingV:E→ [0,+∞]is lower semi-continuous, with compact level sets, we prove that the family of probability measures{ν_ε}ε>0obeys a principle of large deviations with action functionalV (cf. [9,10] and [14] for the formulation), i.e.

1. lower bounds (cf. Section 6 below): for anyδ, γ >0 andx¯∈Ethere existsε0>0 such that νε x∈E: |x− ¯x|E< δ

exp

−V (x)¯ +γ ε²

, εε0;

2. upper bounds (cf. Section 7 below): for anys, δ, γ >0 there existsε0>0 such that νε x∈E;distE x, K(s)

δ

<exp

−s−γ ε²

, εε₀, whereK(s):= {x∈E: V (x)s}is the level set ofV.

In accordance to the general ideas about the large deviations for the invariant measures{ν_ε}ε>0(as beautifully explained in the introduction of [9]) we have the following interpretation. Due to the definition ofνε for any set A⊂Ethe numberνε(A)is the mean expected time the processuεspends inA. Moreover, by the large deviation results in [3] points inK(s), for smalls, are of course more likely to be visited byuε. So, according to statement 2 above, the mass ofν_εwill concentrate asε→0 at points inEwhich are minimum points ofV. In our case

V (x)=0⇔x=0

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(cf. (5.2) below), soνεwill converge to the Dirac measure at the zero function inE, i.e. the only stationary solution of Eq. (1.1) forε=0.

In the framework considered in the present paper the skeleton equation associated with system (1.1) is not null controllable, as in the case considered by Sowers in [14]. Then the proof of lower bounds turns out to be more complicate than in [14]. In fact, a crucial role is played by Lemma 6.2, whose proof is not immediate, as we are dealing with non-Lipschitz reaction term, unboundedGand any space dimensiond1. To this purpose we note that for the proof of Lemma 6.2 we also benefited from some ideas of I. Daw (see [6]).

Concerning the upper bounds, we have distinguished the case of bounded and unboundedG. WhenGis bounded we can use exponential estimates for the solutionuε proved in [3] and generalize some arguments of Sowers to our more delicate situation. In the case of unboundedG this is not anymore possible. Hence we need to prove estimate (3.13) in Theorem 3.4 below, i.e. an estimate on the solution of the skeleton equation which is uniform with respect to the initial datum. This allows us to prove Theorem 7.5, wheregonly satisfies the growth condition in Hypothesis 6, without using the exponential tail estimates (7.2) for the solution of (1.1) which are only known to hold for boundedg. Thus, Theorem 3.4 turns into a key step, since here we have not succeeded in applying a localization argument as we did in [3].

Finally, let us mention that our general strategy mainly follows R. Sowers [14], but our more general situation requires various new techniques. These, in particular, becomes necessary because of the following.

1. Unlike in [13] and [14] (see also [6] and [11]), where global Lipschitz assumptions were imposed, here the functionsfi in (1.1) are only locally Lipschitz and of polynomial growth (see Hypothesis 3 and Remark 2.4 below).

2. g= [g_ij]in (1.1) is not assumed to be globally bounded (as e.g. done in [13] and [6]) and just assumed to be globally Lipschitz (see Hypothesis 2, but also Hypothesis 6 for the proof of upper bounds). Moreover,gmay be degenerate. This means that we can consider for exampleg_ij(u)=λ_iju_j, withλ_ij∈R.

3. We consider systems ofr coupled stochastic reaction–diffusion equations, ruling out the maximum principle and hence comparison techniques commonly used in caser=1.

4. Unlike in [14], where space dimensiond=1 is considered, we can allow arbitrary space dimension, i.e. for E=C(O;R^r)we can allowO to be a bounded open subset of R^d, for arbitraryd 1 (cf. Hypothesis 2 below).

2. Assumptions and preliminaries

LetObe a bounded open set ofR^d, withd1, having aC^∞boundary. In what follows we shall denote by Hthe Hilbert spaceL²(O;R^r),r1, endowed with the usual scalar product·,·H and the corresponding norm

| · |H. The norm inL^p(O;R^r),p∈ [1,∞],p=2, shall be denoted by| · |p.

For any 1p∞andm∈N, byW^m,p(O)we shall denote the space of functionsf ∈L^p(O)such that the weak derivativesD^αf exist inL^p(O), for each 0|α|m.W^m,p(O)is a Banach space, endowed with the norm

|f|W^m,p(O):=

|α|m

|D^αf|L^p(O).

Moreover, ifs >0 is not integer, we defineW^s,p(O)as the space of functionsf ∈W^[^s^]^,p(O)such that

|f|W^s,p(O):= |f|W^[s],p(O)+

|α|=[s]

O×O

|D^αf (ξ )−D^αf (η)|^p

|ξ−η|^d⁺^(s^−[^s^]^)p dξ dη <∞.

Next, we recall that for anys∈Randp∈(1,∞)the Bessel potential spaceH^s,p(R^d)is defined by H^s,p(R^d):=

f ∈S(R^d):|f|H^s,p(R^d):=F⁻¹ 1+ |ξ|²s/2

Ff

L^p(R^d)<∞ ,

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whereS(R^d) is the space of tempered distributions onR^d andF the Fourier transform. The Bessel potential spaces onOare defined by restriction as

H^s,p(O):=

f =g_|_∂O;g∈H^s,p(R^d) , with

|f|H^s,p(O):= inf

f=g_|_∂O|g|H^s,p(R^d).

We note that fork∈Nwe haveH^k,p(O)=W^k,p(O)(for all definitions and detailed proofs see [15]).

Finally, we shall denote byW^s,p(O;R^r)andH^s,p(O;R^r)the space of R^r-valued functions such that each component belongs toW^s,p(O)andH^s,p(O), respectively.

In what follows we shall denote byAthe realization inHof the differential operatorA=(A1, . . . ,Ar)defined in (1.2), endowed with the boundary conditionsB=(B¹, . . . ,B^r), where for eachi=1, . . . , r

Bⁱu=u, or Bⁱu= aⁱν,∇u, (2.1)

(hereνis the normal vector at∂O). As proved e.g. in [15, Chapter 5] we have D(A)=

u∈H^2,2(O;R^r): Bu=0 on∂O

=:H_B^2,2(O;R^r) and the following optimal regularity result holds

u∈D(A), Au∈H^l,2(O;R^r), l∈N⁺⇒u∈H^l⁺^2,2(O;R^r). (2.2) We recall that for any integerk2 thekth power of the operatorAis defined by

D(A^k):=

u∈D(A^k⁻¹): A^k⁻¹u∈D(A)

, A^ku:=A(A^k⁻¹u).

Analogously, we can define thekth power ofAby setting A^ku:=A(A^k⁻¹u)= A1(A^k₁⁻¹u₁), . . . ,Ar(A^kr⁻¹u_r)

, u∈H^2k,2(O;R^r).

Thanks to (2.2) it is immediate to show that for any fixed integerk D(A^k)=H_B^2k,2

k (O;R^r):=

u∈H^2k,2(O;R^r): Bu= · · · =B(A^k⁻¹u)=0

, (2.3)

so that the operatorA^kis the realization inHof the differential operatorA^kendowed with the boundary conditions Bk:=

B,BA, . . . ,BA^k⁻¹ .

Notice thatAgenerates an analytic semigroupe^{t A}in eachL^p(O;R^r), with 1p∞, which is self-adjoint onHand of negative type. Thus, as−Ais a positive self-adjoint operator onH, for any 0αβ andθ∈ [0,1] we have

D (−A)^α

, D (−A)^β

θ= D (−A)^α

, D (−A)^β

θ,2=D (−A)⁽¹⁻^{θ )α}⁺^θβ

, (2.4)

where in general, given any two Banach spacesXandY,[X, Y]θ denotes their complex interpolation space and (X, Y )_θ,2denotes their real interpolation space (for a proof see [15, Theorem 1.18.10]).

By complex interpolation arguments it is possible to characterize the domain of the fractional powers of−A.

Proposition 2.1. Letm_i:=(1+2 ordBⁱ)/4. Then, for anyγ >0 andi=1, . . . , r we have D (−A)^γ

=H_B^{2γ ,2}

γ (O;R^r), whereBγ:=(B_γ¹, . . . ,B^r_γ)with

Bⁱγ:=

∅ ifγ∈ [0, m_i],

{Bⁱ,BⁱAi, . . . ,BⁱA^k_i} ifγ∈(k+mi, k+1+mi], k∈N∪ {0}. (2.5)

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Proof. Due to (2.4) we have D((−A)^γ)=

H, D (−A)^[^γ^]+¹

[γ]+1γ

and then from (2.3) we obtain D((−A)^γ)=

H, H_B²⁽^[^γ^]+^1),2

[γ]+1 (O;R^r)

[γ]+1γ . (2.6)

It is not difficult to prove that for any integerkthe operatorA^k endowed with the boundary conditionBk is regular elliptic (for the definition and all details see [15, Section 5.2.1]). Thus, as proved in [4, Lemma 11] from (2.6) we obtain

D (−A)^γ

=H_B^{2γ ,2}

γ (O;R^r), where

Bⁱ_γ:=

BⁱA^j_i;0j[γ], ord(BⁱA^j_i) <2γ−1 2

.

Hence, by easy computations we can check that the boundary conditionsBⁱγ above coincide with the boundary conditionsBⁱγ in (2.5). 2

Remark 2.2. It is immediate to check that ifγ∈Nthe boundary conditionsBγ introduced in (2.3) coincide with the boundary conditionsBγ introduced in the proposition above.

In what follows we shall set

E:=D(A)^C(Ô;R^r⁾=D(A₁)^C(Ô;R⁾× · · · ×D(A_r)^C(Ô;R⁾.

Each set D(A_i)^C(^O;R⁾ coincides with C(O;R) orC₀(O;R), ifBi is respectively a co-normal or a Dirichlet boundary condition. In any case, with this definition of the spaceE, endowed with the sup-norm| · |E and the duality·,·E:=E·,·E, the part ofe^{t A}inE(which we will still denote bye^{t A}) is strongly continuous. Moreover for anyδ_x∈∂|x|E:= {x∈E, x, xE= |x|E, |x|E=1}we have

Ax, δ_xE−α|x|E, x∈D(A), (2.7)

whereα:=min_i₌_1,...,rα_i.

As recalled also in [2] and [3],e^{t A}has a smoothing effect. In fact, for anyt >0, 1qp∞andε0 the semigroupe^{t A}mapsL^q(O;R^r)intoW^ε,p(O;R^r)and

|e^{t A}x|W^ε,p(O;R^r)ce⁻^αt(t∧1)⁻⁽^ε²⁺^d(p−q)^2pq ⁾|x|q, x∈L^q(O;R^r). (2.8) Moreover,e^{t A}mapsEintoC^θ(O;R^r), for anyθ0, and

|e^{t A}x|_Cθ(O;R^r)ce⁻^αt(t∧1)⁻^θ²|x|E, x∈E. (2.9) We also notice thate^{t A} is compact onL^p(O;R^r), for all 1p∞andt >0, and the spectrum{−α_n}is independent ofp.

Our first hypothesis concerns the eigenvalues ofA.

Hypothesis 1. The complete orthonormal system ofH which diagonalizesAis equi-bounded in the sup-norm.

Next, we assume thatQ:=(Q1, . . . , Qr):H→H is a bounded linear operator which satisfies the following conditions.

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Hypothesis 2.Qis non-negative and diagonal with respect to the complete orthonormal basis which diagonal- izesA, with eigenvalues{λ_n}. Moreover, ifd2,

there exists





<∞ ifd=2 < 2d

d−2 ifd >2 such thatQ:=

_∞

k=1

λ_n 1/

<∞. (2.10)

Remark 2.3. Hypothesis 1 is satisfied e.g. by the Laplace operator on[0, T]^d endowed with Dirichlet boundary conditions. But there are several important cases in which it is not satisfied and it is only possible to say that

|ek|∞ck^γ,

for someγ0. In this more general situation one has to assume that the summability condition (2.10) imposed on the eigenvalues ofQis satisfied for some smaller constant. In other words one has to color the noise more.

In Hypotheses 3 and 4 below we give conditions on the coefficientsf andg.

Hypothesis 3. The mappingg:O×R^r→L(R^r)is continuous. Moreover the mappingg(ξ,·):R^r →L(R^r)is Lipschitz-continuous, uniformly with respect toξ∈O, that is

sup

ξ∈O

sup

σ,ρ∈R^r σ=ρ

g(ξ, σ )−g(ξ, ρ)_L(R^r)

|σ−ρ| <∞.

In what follows for anyx, y:O→R^r we set G(x)y

(ξ ):=g ξ, x(ξ )

y(ξ ), ξ∈O.

Next, settingf:=(f₁, . . . , fr), for anyx:O→R^r we define F (x)(ξ ):=f ξ, x(ξ )

, ξ∈O.

Hypothesis 4.

(1) The mappingF:E→Eis locally Lipschitz-continuous and there existsm1 such that F (x)

Ec 1+ |x|^m_E

, x∈E. (2.11)

Moreover,F (0)=0.

(2) For anyx, h∈E

F (x+h)−F (x), δ_h

E0, (2.12)

for someδh∈∂|h|E:= {h∈E; |h|E=1,h, hE= |h|E}. (3) There exista >0 andc0 such that for eachx, h∈E

F (x+h)−F (x), δh

E−a|h|^mE+c 1+ |x|^mE

, (2.13)

for someδ_h∈∂|h|E. Remark 2.4. Assume that

fi(ξ, σ1, . . . , σr):=ki(ξ, σi)+hi(ξ, σ1, . . . , σr), i=1, . . . , r,

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wherehi:O×R^r→Ris a continuous function such thathi(ξ,·):R^r→Ris locally Lipschitz-continuous with linear growth, uniformly with respect toξ∈O, and

k_i(ξ, σ_i):= −c(ξ )σ_i²ⁿ⁺¹+ 2n k=0

c_k(ξ )σ_i^k,

wherec(ξ )andc_k(ξ )are continuous functions,c(ξ )ε >0,ξ ∈Oandc₀(ξ )= −h_i(ξ,0).

Under these assumptions the functionf satisfies conditions (1) and (3) in Hypothesis 4 (see also [1, Chapter 6], [2] and [3, Remark 2.1] for more general examples of functionsf fulfilling Hypothesis 4 and for all details).

The next set of conditions assure the compactness of level sets for the quasi-potential associated with system (1.1).

Hypothesis 5. EitherG(0)=0 or there exists a continuous increasing functionc(t)such that for anyt0 Q

G(0)

e^t^[^A⁺^F⁽⁰⁾^]h

Hc(t)|Qe^{t A}h|H, h∈H. (2.14)

In the case (2.14) is verified, the following conditions hold.

(1) If{−αn}and{λn}are respectively the eigenvalues ofAandQ, then 1

cα_n⁻^δλncα_n⁻^δ, (2.15)

for somec >0 and someδsuch that δ0, ifd=1, δ >d−2

4 , ifd2. (2.16)

(2) The mappingsf andgare of classC^∞onO×R^r.

(3) If δ is the constant in (2.15) andBγ is the boundary operator introduced in (2.5), then for anyγ δ and u, v∈H^{2γ ,2}(O;R^r)we have

Bγu_|_∂O=0⇒BγF (u)_|_∂O=0,

Bγu_|_∂O=Bγv_|_∂O=0⇒Bγ(G(u)v)_|_∂O=0. (2.17)

Moreover, ifu, v, w∈H^2δ,2(O;R^r)we have Bδu_|_∂O =Bδv_|_∂O=0⇒Bδ(F(u)v)_|_∂O=0,

Bδu_|_∂O =Bδv_|_∂O=Bδw_|_∂O=0⇒Bδ([G(u)v]w)_|_∂O=0. (2.18) Remark 2.5.

1. We note that the assumption (2.14) is fulfilled when there exist two diagonalr×rmatricesD₁andD₂, with D₁invertible, such that

g(ξ,0)=D₁, Dσf (ξ,0)=D₂, ξ∈O.

In particular, when instead of a system a single equation is considered, condition (2.14) is always fulfilled if bothg(ξ,0)andD_σf (ξ,0)do not depend onξ.

2. Condition (2.15) means that RangeQ=D((−A)^δ).

3. We assumef andgto beC^∞(O×R^r)only for simplicity. In fact we needf andgto be of classC^k(O×R^r), for someklarge enough, depending on the constantδ introduced in (2.15) (for example, in the caseBⁱ=I it is sufficient to takek <2δ+1/2, see also next remark).

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4. If we have

α_n∼n^2/d, n∈N

(this happens for example in the case of the Laplace operator in strongly regular open sets, both with Dirichlet and with Neumann boundary conditions, see [5, Theorem 1.9.6]), then if (2.16) holds, there exists someρwhich fulfills condition (2.10).

5. WhenBⁱ=I, for eachi=1, . . . , r, condition (2.17) is verified for example by functionsf andgsuch that

D^j_σf (ξ,0)=0, D^j_σg(ξ,0)=0, ξ∈O, (2.19)

for anyj=1, . . . ,2k, wherek∈ [δ−5/4, δ−1/4)(notice that in this casemi=1/4, for eachi). In the same setting, condition (2.18) holds forf andgfulfilling (2.19) for anyj =1, . . . ,2k+1, withkas above.

For the proof of upper bounds in the case of unboundedgwe need the following condition on its growth.

Hypothesis 6. There existsγ∈ [0,1]such that sup

ξ∈O

g(ξ, σ )

L(R^r)c 1+ |σ|^γ

, σ∈R^r, (2.20)

and m >

1+(2+d)γ

1−d(−2) 2

₋1

∨2, (2.21)

whereandmare the constants introduced respectively in (2.10) and (2.13).

Remark 2.6. Condition (2.21) ond,m,andγsays how the space dimension, the dissipativity ofF, the regularity ofQand the growth ofGare related to one another, in order to have upper bounds.

In the case of space dimensiond=1 and white noise (which meansQ=I and hence= +∞) the relation betweenm(the dissipativity ofF) andγ(the growth ofG) is

m > (1+6γ )∨2,

so that in the case ofGhaving linear growth (that isγ=1) we have to assumem >7. If instead of a white noise we take a coloured noise with Hilbert–Schmidt covarianceQ(that is=2) we have

m > (1+3γ )∨2

which becomesm >4 in the case ofγ=1.

In general, from (2.10) we have that the bigger the space dimensiond becomes, the smallerhas to be chosen (and hence the more regularQhas to be taken). Due to (2.21) this means that if we want to allow the same growth ofgwith increasing dimensions, we have to take reaction termsF with stronger and stronger dissipativity, that is, larger and largerm.

3. The skeleton equation

With the notations introduced in the previous section system (1.1) can written more concisely as du(t)=

Au(t)+F u(t)

dt+G u(t)

Q dw(t), u(0)=x. (3.1)

In this section we prove some results for the skeleton equation associated with the system above.

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For any−∞t₁< t₂+∞andϕ∈L²(t₁, t₂;H )we denote byz(ϕ)any solution belonging toC([t₁, t₂];E) of the deterministic problem

z(t)=Az(t)+F z(t)

+G z(t)

Qϕ(t), z(t₁)=x. (3.2)

In several cases, when we need to stress thatz(ϕ)starts fromx at timet₁, we shall writez^x_t

1(ϕ). As shown in [3, Theorem 4.1], for anyr0 andt₁< t₂there exists a constantcr,t₂−t₁>0 such that for anyx∈E

sup

|ϕ|_L2(t1,t2;H )r

z^x(ϕ)

C([t₁,t₂];E)c_r,t₂₋_t₁ 1+ |x|E

.

In fact, by proceeding as in [2, proofs of Proposition 6.1 and Theorem 6.2], it is possible to get the following stronger result.

Theorem 3.1. Under Hypotheses 1–4, for anyr0 there exists a constantcr >0 such that for anyT ∈Rand x∈E

sup

|ϕ|_L2(T ,∞;H )r

z_T^x(ϕ)

C([T ,∞);E)cr 1+ |x|E

. (3.3)

Moreover, there existsθ∈(0,1)andc_r∈(0,+∞)such that for anyt > T andx∈E sup

|ϕ|_L2(T ,∞;H )r

z_T^x(ϕ)(t)

C^θ(O;R^r)cr 1+ |x|^m_E 1+(t−T )⁻^θ²

. (3.4)

Proof. For any fixedϕ∈L²(T ,∞;H ),z∈C([T ,∞);E)andλ0 we define γ_ϕ,λ^T (z)(t):=

t T

e^(t⁻^s)(A⁻^λ)G z(s)

Qϕ(s) ds, tT ,

(and we setγ_ϕ^T(z):=γ_ϕ,0^T (z)). Clearly,γ_ϕ,λ^T (z)is the unique mild solution of the problem dv

dt(t)=(A−λ)v(t)+G z(t)

Qϕ(t), tT , v(T )=0.

Thanks to the same arguments used in [2, proofs of Theorem 4.2 and Proposition 4.5, Remark 4.6], due to (2.9) we can fix someθ∈(0,1)such that for anyλ0 andT ∈R

sup

tT

γ_ϕ,λ^T (z)(t)

C^θ(O;R^r)c(λ) 1+ |z|C([T ,∞);E)

|ϕ|L²(T ,∞;H ), (3.5)

for a constantc(λ)decreasing to zero asλgoes to infinity.

Now, if we setγ^T(t):=γ_ϕ,λ^T (z_T^x(ϕ))(t)andu(t):=z^x_T(ϕ)(t)−γ^T(t), fortT, we have u(t)=(A−λ)u(t)+F u(t)+γ^T(t)

+λz^x_T(ϕ)(t), u(T )=x.

We recall here that if a mappingu:[0, T] →Eis differentiable at some pointt0then d

dt

−u(t₀)

E=min

u(t₀), x

E, x∈∂u(t₀)

E

see for example [1, Proposition A.1.3]. Hence, ifδ_{u(t )}is the element of∂|u(t)|Eintroduced in (2.13), due to (2.11) we have

d dt

−u(t)

E

Au(t), δ_{u(t )}

E+

F u(t)+γ^T(t)

−F γ^T(t) , δ_{u(t )}

E

+

F γ^T(t)

+λz^x_T(ϕ)(t), δ_{u(t )}

E−a|u(t)|^m_E+c 1+γ^T(t)^m

E+λz_T^x(ϕ)(t)

E

.

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Then, recalling thatz^x_T(ϕ)=u+γ_ϕ,λ^T (z^x_T(ϕ)), by a comparison argument (see for example [2, proof of Lemma 5.4]) for anytT we obtain

z^x_T(ϕ)(t)

E|x|E+c

1+sup

rT

γ_ϕ,λ^T z^x_T(ϕ) (r)

E+λ^m¹z^x_T(ϕ)(t)^m¹

E

.

Thanks to (3.5) and to the Young inequality, this implies that if|ϕ|_L²_{(T ,}_∞;_{H )}r sup

tT

z^x_T(ϕ)(t)

E|x|E+1 4sup

tT

z^x_T(ϕ)(t)

E+c(λ) 1+z^x_T(ϕ)

C([T ,∞);E)

r+λ^m−1¹ .

Now, as limλ→∞c(λ)=0, we can findλ¯such thatc(¯λ)r1/4 and then sup

tT

z^x_T(ϕ)(t)

Ec_r 1+ |x|E

, for some positive constantc_r.

Finally, in order to obtain (3.4), we remark that thanks to (2.9), (2.11) and (3.3) for anytT we easily have

t T

e^(t⁻^s)AF z^x_T(ϕ)(s) ds

C^θ(O;R^r)

c

t T

e⁻^α(t⁻^s) (t−s)∧1₋^θ₂

1+z^x_T(ϕ)(s)^m

E

dscr 1+ |x|^mE

. (3.6)

Then, as

z_T^x(ϕ)(t)=e^(t⁻^{T )A}x+ t T

e^(t⁻^s)AF z_T^x(ϕ)(s)

ds+γ_ϕ^T z^x_T(ϕ) (t),

from (3.3), (3.5) and (2.9) for anyt > T we get z^x_T(ϕ)(t)

C^θ(O;R^r)c e⁻^α(t⁻^{T )} (t−T )∧1₋^θ

2|x|E+c_r 1+ |x|^m_E , which easily implies (3.4) 2

The next proposition shows that if we start fromx=0 at timeT, thenz⁰_T(ϕ)decreases to zero inC([T ,∞);E), asϕdecreases to zero inL²(T ,∞;H ).

Proposition 3.2. Under Hypotheses 1–4, for anyT ∈Rwe have

|ϕ|_L2(T ,lim∞;H )→0

z⁰_T(ϕ)

C([T ,∞);E)=0. (3.7)

Proof. As in the proof of Theorem 3.1, if we setu(t):=z⁰_T(ϕ)(t)−γ_ϕ^T(z⁰_T(ϕ))(t), we have u(t)=Au(t)+F u(t)+γ_ϕ^T z⁰_T(ϕ)

(t)

, u(T )=0, so that, with the notations of Theorem 3.1,

d dt

−u(t)

E

Au(t), δ_{u(t )}

E+

F u(t)+γ_ϕ^T z⁰_T(ϕ) (t)

−F γ_ϕ^T z⁰_T(ϕ) (t)

, δ_{u(t )}

E

+

F γ_ϕ^T z_T⁰(ϕ) (t)

, δ_{u(t )}

E−au(t)^m

E+F γ_ϕ^T z_T⁰(ϕ)(t)

E.

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Recalling thatu(t):=z⁰_T(ϕ)(t)−γ_ϕ^T(z⁰_T(ϕ))(t), by comparison this yields sup

tT

z⁰_T(ϕ)(t)

Esup

tT

u(t)

E+γ_ϕ^T z⁰_T(ϕ)(t)

E

csup

tT

γ_ϕ^T z⁰_T(ϕ) (t)

E+F γ_ϕ^T z⁰_T(ϕ)(t)

E^m¹ .

Now, thanks to (3.5) and (3.3), if|ϕ|L²(T ,∞;H )rwe have sup

tT

γ_ϕ^T z⁰_T(ϕ)(t)

Ec_r|ϕ|_L²_{(T ,}_∞;_{H )}, and then, asF (0)=0, we can conclude. 2

Now we show that under the growth conditions of Hypothesis 6 it is possible to give estimates of|z^x(ϕ)(t)|E

which are uniform with respect to the initial datumx. To this purpose we need a preliminary result on the convo- lutionγ_ϕ⁰(z).

Lemma 3.3. Let us assume Hypotheses 1–4 and 6. Then, if andγ are the constants introduced in (2.10) and (2.20), respectively, for anyq1 such that

q

γ 1, (2+d)γ

q <1−d(−2)

2 (3.8)

there exists some continuous increasing functioncq(t)vanishing att=0 such that for anyz∈L^q(0,+∞;E)and ϕ∈L²(0,+∞;H )

γ_ϕ⁰(z)(t)

Ec_q(t) 1+ |z|^γ_Lq(0,t;E)

|ϕ|L²(0,t;H ), t0.

Proof. For anyβ∈(0,1)andt0 we have γ_ϕ⁰(z)(t)=sinπβ

π t 0

(t−s)^β⁻¹e^(t⁻^s)Avβ(s) ds,

where

vβ(s):=

s 0

(s−σ )⁻^βe^(s⁻^{σ )A}G(z(σ ))Qϕ(σ ) dσ.

Thanks to (2.8), for anyβ∈(0,1),ε >0 andp1 such that(β−1−ε/2)p/(p−1) >−1 we have γ_ϕ⁰(z)(t)

W^ε,p(O;R^r)sinπβ π

t 0

(t−s)^β⁻¹⁻^ε²v_β(s)

pds

sinπβ π

t 0

vβ(s)^p

pds ¹

pt 0

(t−s)^(β⁻¹⁻^ε²⁾

p p−1 ds

^p−1

p

.

Hence, ifεp > d, that is, if

β > (2+d)/2p, (3.9)