Mean attractors and invarant measures of locally monotone and generally coercive spdes driven by superlinear noise

MEAN ATTRACTORS AND INVARIANT MEASURES OF LOCALLY MONOTONE AND GENERALLY COERCIVE SPDES DRIVEN BY SUPERLINEAR NOISE RENHAI WANG, TOM´ AS CARABALLO, AND NGUYEN HUY TUAN Abstract. We study the global solvability, mean attractors and invariant measures for an abstract locally monotone and generally coercive SPDEs driven by infinite-dimensional superlinear noise defined in a dual space of intersection of finitely many Banach spaces.The main feature of this abstract system is that it covers a larger class of fundamental models which are included or not included previously. Under an extended locally monotone variational setting, we establish the global well-posedness, Itˆo’s energy equality and existence of mean random attractors in some high-order Bochner spaces. The existence, uniqueness, support, (high-order and exponential) moment estimates, ergodicity, (pointwise and Wasserstein-type) exponentially mixing and asymptotic stability of invariant measures and evolution systems of measures are discussed for autonomous and nonautonomous stochastic equations. A stopping time technique is used to prove the convergence of solutions in probability in order to overcome the difficulty caused by the local monotonicity and superlinear growth of the coefficients. Our abstract results and unified methods are expected to be applied to various types of SPDEs like 2D Navier-Stokes equations, 2D MHD equations, 2D magnetic B´enard problem, Burgers type equations, 3D Leray α-model, convective Brinkman-Forchheimer equations, fractional (s, p)-Laplacian equations with monotone nonlinearities of polynomial growth of arbitrary order, and others. 1. Introduction 1.1. Statement of problems. Let (H, h·,·iH) be a separable Hilbert space identified with its dual space H∗by Riesz’s representation theorem. For i= 1,2, . . . , m ∈N, let (Vi,k·kVi) be a reflexive Banach space continuously and densely embedded into H. Let V:= Ti=1,2,...,m Viwith the norm kvkV=Pm i=1 kvkVi, and V∗h·,·iVand V∗ ih·,·iVibe the duality products of V, Viand their dual spaces V∗, V ∗ i. Then V∗= Pm i=1 V∗ i,f∈V∗if and only if f=Pm i=1 fiwith fi∈V∗ i, and V∗hf, viV=Pm i=1 V∗ ihfi, viVifor v∈V. The norm of V∗is defined by kfkV∗= inf f=Pm i=1 fiPm i=1 kfikV∗ i. Assume that Vis separable. Then we get the variational triples: Vi⊆H≡H∗⊆V∗ iand V⊆H≡H∗⊆V∗.Given a separable Hilbert space (U, h·,·iU), we denote by L2(U, H) the space of Hilbert-Schmidt operators from Uto Hwith the norm k·kL2(U,H)and inner product h·,·iL2(U,H). Let Wbe a two-sided U-valued cylindrical Wiener process defined on the complete filtered probability space (Ω,F,{Ft}t∈R,P). In this work we are concerned with the global solvability and long-time dynamics of the following stochastic evolution equation defined in V∗:      dX(t) = m X i=1 Ai(t, X(t))dt +B(t, X(t))dW(t), t > τ, X(τ) = X0, (1.1) where the progressively measurable1evolution operators Ai: [τ, τ +T]×Ω×Vi→V∗ iand B: [τ, τ +T]×Ω×V→ L2(U, H), T > 0, satisfy some locally monotone, generally coercive and superlinear growth conditions: there exist constants L0>0, ϑi>0, θi>0αi>1, βi∈[1, αi), κ⩾0, $⩾0, and Ft-adapted nonnegative processes φ1, φ3, φ5∈L1([τ, τ +T]×Ω, dt×P;R+), φ2, φ4, φ6∈L∞([τ, τ +T]×Ω, dt×P;R+) and ηi∈L αi αi−βi([τ, τ + T]×Ω; dt ×P;R+) such that A:= Pm i=1 Aiand Bsatisfy, for all (t, ω)∈[τ, τ +T]×Ω and v, v1, v2∈V, 2010 Mathematics Subject Classification. 35B40, 35B41, 37L30. Key words and phrases. Stochastic evolution equation; Variational approach; Random attractors; Invariant measure; Ergodicity; Mixing; Nonlinear noise. 1Airestricted to [τ, τ +T]×Ω×Viis B([τ, τ +T])×Ft×B(Vi)-measurable (Bdenotes the corresponding Borel σ-algebra). 1 2 RENHAI WANG, TOM ´ AS CARABALLO, AND NGUYEN HUY TUAN (C1) (Hemicontinuity) The map s→V∗hA(t, ω, v1+sv2), viVis continuous from Rto R; (C2) (Local Monotonicity) 2V∗hA(t, ω, v1)−A(t, ω, v2), v1−v2iV+kB(t, ω, v1)−B(t, ω, v2)k2 L2(U,H) ⩽− m X i=1 2ϑikv1−v2kαi Vi+φ1(t, ω) + κ m X i=1 ρi(v2)kv1−v2k2 H, where ρi:Vi→[0,+∞) is a measurable, hemicontinuous and locally bounded function satisfying m X i=1 ρi(v)⩽L01 + m X i=1 kvkαi Vi(1 + kvk$ H); (1.2) (C3) (General Coercivity) 2V∗hA(t, ω, v), viV+kB(t, ω, v)k2 L2(U,H)⩽− m X i=1 2θikvkαi Vi+φ2(t, ω)kvk2 H+φ3(t, ω); (C4) (Superlinear Growth) kB(t, ω, v)k2 L2(U,H)⩽ m X i=1 ηi(t, ω)kvkβi Vi+φ4(t, ω)kvk2 H+φ5(t, ω), m X i=1 kAi(t, ω, v)k αi αi−1 V∗ i ⩽φ6(t, ω)1 + m X i=1 kvkαi Vi(1 + kvk$ H). 1.2. Global well-posedness of (1.1).Under our settings, we demonstrate the global existence, uniqueness and Itˆo’s formula of solutions to (1.1) in some high-order Bochner spaces. Definition 1.1. (Probabilistically strong solutions) We say a continuous H-valued Ft-adapted stochastic process {X(t)}t∈[τ,τ+T]is a solution to (1.1) if its dt ×P-equivalent class ¯ Xsatisfies ¯ X∈ \ i=1,2,...,m Lαi([τ, τ +T]×Ω, dt ×P;Vi)!\L2([τ, τ +T]×Ω, dt ×P;H), and P-a.s., the following equation holds true in V∗: X(t) = X0+Zt τ A(s, ¯ X(s))ds +Zt τ B(s, ¯ X(s))dW(s), t ∈[τ, τ +T]. Theorem 1.2. (Well-posedness, Itˆo’s formula, mean energy equality) Suppose (C1)-(C4) hold true for φ3, φ5∈L`([τ, τ +T]×Ω, dt ×P;R+)and ηi∈L `αi αi−βi([τ, τ +T]×Ω, dt ×P;R+)with some `⩾$ 2+ 1. Then, for any X0∈L2`(Ω,Fτ,P;H), problem (1.1) has a unique solution {X(t)}t∈[τ,τ+T]according to Definition 1.1 such that X∈C([τ, τ +T], L2(Ω,P;H)), and satisfies Itˆo’s formula kX(t)k2` H=kX0k2` H+ 2`(`−1) Zt τkX(s)k2`−4 Hk(B(s, X(s)))∗X(s)k2 Uds + 2`Zt τkX(s)k2`−2 HhX(s), B(s, X(s))dW (s)iH +`Zt τkX(s)k2`−2 H2V∗hA(s, X(s)), X(s)iV+kB(s, X(s))k2 L2(U,H)ds, t ⩾τ, P-a.s.,(1.3) the mean energy equality d dtE[kX(t)k2` H]=2`(`−1)E[kX(t)k2`−4 Hk(B(t, X(t)))∗X(t)k2 U] +`EhkX(t)k2`−2 H2V∗hA(t, X(t)), X(t)iV+kB(t, X(t))k2 L2(U,H)i, t ⩾τ, (1.4) and the uniform estimate Ehsup t∈[τ,τ+T]kX(t)k2` Hi+EZτ+T τkX(s)k2`−2 H m X i=1 kX(s)kαi Vids⩽C(τ, T, X0).(1.5) MEAN ATTRACTORS AND INVARIANT MEASURES OF SPDES 3 The global well-posedness results in Theorem 1.2 permit us to investigate the behavior of solutions to (1.1). In the literature, the behavior of solutions to stochastic systems has been discussed in several different directions: pathwise random attractors (behavior in almost sure, see [3, 4, 6, 23, 27, 43, 44, 47, 56, 61]), mean random attractors (behavior in mean, see [25, 50, 52]), invariant measures (behavior in distribution, see [7, 17, 22, 28, 29, 32, 33]), large or moderate deviation principle (behavior in small probability, and the law of large numbers or central limit theorems [28, 29]. In the present paper we study mean random attractors and invariant measures of (1.1). To do this, we shall need the following remark. Remark 1.3. Assume that there exists i1∈[1, m]∩Nsuch that αi1⩾2. Then by (C2)-(C3) we deduce 2V∗hA(t, ω, v1)−A(t, ω, v2), v1−v2iV+kB(t, ω, v1)−B(t, ω, v2)k2 L2(U,H) ⩽− m X i=1 ϑikv1−v2kαi Vi+−λi1ϑi1+φ1(t, ω) + κ m X i=1 ρi(v2)kv1−v2k2 H+ϑi1Cαi1, and 2V∗hA(t, ω, v), viV+kB(t, ω, v)k2 L2(U,H)⩽− m X i=1 θikvkαi Vi+ (φ2(t, ω)−λi1θi1)kvk2 H+φ3(t, ω) + θi1Cαi1, where Cαi1= 0 if αi1= 2 and Cαi1=α−1 i1(αi1−2)(αi1/2)2/(2−αi1)if αi1>2, and λi1>0is the best embedding constant such that pλi1kvkH⩽kvkVi1. 1.3. Mean random attractors of (1.1).A basic but very restrictive condition to investigate almost sure behavior of solutions to SPDEs by pathwise random attractors is that SPDEs should be converted into pathwise systems via an Ornstein-Uhlenbeck process, see [6, 8, 9, 10, 11, 12, 14, 15, 23, 31, 42, 48, 49, 54, 55, 58, 62]. In general, such a transformation can be achieved for SPDEs driven by additive or linear multiplicative noise. For SPDEs like (1.1) with nonlinear noise, it seems that there are no methods available in the literature to achieve such a conversion. Then, we alternatively study the mean (not pathwise) random dynamics of (1.1), and prove that the mean random dynamical system (RDS) generated by the solution operators has a unique mean random attractor in L2`(Ω,F,P;H) over (Ω,F,{Ft}t∈R,P) in the sense of Wang [50, 52], see Theorem 3.2. If `= 1, then similar results can be found in [25, 50, 51] for stochastic parabolic equations. If `= 2, the reader is referred to Wang [52] for the existence of mean random attractors of stochastic Navier-Stokes equations. We notice that a new concept of mean random invariant manifolds for mean RDSs was recently proposed by Wang [53]. 1.4. Invariant measures of (1.1): autonomous case. An important and universal conclusion in the theory of pathwise RDSs is that all invariant measures are supported by pathwise random attractors. Since the existence of a pathwise RDS for SPDEs like (1.1) with nonlinear noise is still unknown, we are currently unable to discuss the relationship between pathwise random attractors and invariant measures of (1.1). Although we can establish the existence of mean random attractors of (1.1), we still do not know the relationship between mean random attractors and invariant measures of (1.1). In this paper we also discuss the existence and some properties of invariant probability measures of (1.1) with nonlinear noise. Owing to the local monotonicity and superlinear growth of Aiand B, it is difficult to prove the Feller property of the transition operators which will be used to prove the existence of invariant probability measures of (1.1). A stopping time technique is used to overcome this difficult by proving the continuous dependence on initial data of the solutions in probability, see Lemma 4.2. For a global monotonous and linear growth case, the reader is referred to [51, 57]. For the autonomous case, we prove the existence and regularity of invariant measures of (1.1) when Aiand Bare independent of sample and time. 4 RENHAI WANG, TOM ´ AS CARABALLO, AND NGUYEN HUY TUAN Theorem 1.4. (Existence and support) Let assumptions in Theorem 1.2 hold true. (i) If there exist i0, i1∈[1, m]∩Nsuch that Vi0,→His compact, αi1⩾2and θi1> λ−1 i1φ2, then the transition semigroup (P0,t)t⩾0for (1.1) has an invariant probability measure on H. (ii) If there exists i1∈[1, m]∩Nsuch that αi1⩾2and θi1> λ−1 i1φ2, then each invariant probability measure of (P0,t)t⩾0on His supported by V=Ti=1,2,...,m Vi. Next, we take a stationary solution of (1.1) to look at the moment estimates of invariant measures of (P0,t)t⩾0, which are useful to discuss the uniqueness, ergodicity and mixing of invariant measures of (1.1). Theorem 1.5. (High-order and exponential moment estimates) Let assumptions in Theorem 1.2 hold. (i) If there exists i1∈[1, m]∩Nsuch that αi1= 2 and θi1>2λ−1 i1ℵ−1 `[φ2+ 2(`−1)φ4], where ℵ`= 2 if `= 1 and ℵ`= 1 if ` > 1, then every invariant measure ηof (P0,t)t⩾0on Hsatisfies ZHkxk2`−2 H m X i=1 kxkαi Viη(dx)<∞,∀`⩾1.(1.6) (ii) If there exists i1∈[1, m]∩Nsuch that αi1>2, then every invariant measure ηof (P0,t)t⩾0on H satisfies ZHkxk2`−2 H m X i=1 kxkαi Viη(dx)<∞,∀`⩾1.(1.7) (iii) If B(v)≡B, and there exists i1∈[1, m]∩Nsuch that αi1⩾2and θi1> λ−1 i1φ2, then every invariant measure ηof (P0,t)t⩾0on Hsatisfies, for any ∈h0,λi1θi1−φ2 2kBk2 L2(U,H)i, ZH ekxk2 H m X i=1 θikxkαi Viη(dx)⩽(φ3+θi1Cαi1)ZH ekxk2 Hη(dx)⩽2(φ3+θi1Cαi1)e(2θ−1 i1(φ3+θi1Cαi1))2/αi1. Under some specified assumptions, we then further look at the uniqueness, ergodicity, strong mixing and exponential mixing of invariant measures of (P0,t)t⩾0on H. Theorem 1.6. (Uniqueness, ergodicity and mixing) Let assumptions in Theorem 1.2 hold. (i) If κ= 0 and there exists i1∈[1, m]∩Nsuch that αi1= 2,ϑi1> λ−1 i1φ1and θi1> λ−1 i1φ2, then every invariant measure ηof (P0,t)t⩾0on His unique, ergodic, strongly mixing, and exponentially mixing in the sense that for any ϕ∈Lipb(H)and X0∈H, (P0,tϕ)(X0)−ZH ϕ(x)η(dx)⩽c1 + kX0kHe−1 2(λi1ϑi1−φ1)t,(1.8) where c > 0is a constant independent of X0and t. Furthermore, ηis also exponentially mixing under the Wasserstein metric of P(H), that is, for any ϕ∈Lipb(H)and µ∈ P(H)satisfying RHkxk2 Hµ(x)<∞, dP W(Q0,tµ, η)⩽c1 + ZHkxk2 Hµ(dx)1/2 e−1 2(λi1ϑi1−φ1)t,(1.9) where c > 0is a constant independent of t,Q0,t is the adjoint operator of P0,t. (ii) If $= 0,κ6= 0,B(v)≡Band there exists i1∈[1, m]∩Nsuch that αi1= 2,ϑi1> λ−1 i1[φ1+ κL0(1+θφ3)] and θi1> λ−1 i1[φ2+2(1+κθL0)kBk2 L2(U,H)], where θ:= max i=1,2,...,m{θ−1 i}, then every invariant measure ηof (P0,t)t⩾0on His unique, ergodic, strongly mixing, and exponentially mixing in the sense that for any ϕ∈Lipb(H)and X0∈H, (P0,tϕ)(X0)−ZH ϕ(x)η(dx)⩽C(X0)e1 2[−λi1ϑi1+φ1+κL0(1+θφ3)]t,(1.10) where C(X0)>0is a constant independent of t. Furthermore, ηis also exponentially mixing under the Wasserstein metric of P(H), that is, for any ϕ∈Lipb(H)and µ∈ P(H)satisfying RHekxk2 Hµ(dx)<∞, dP W(Q0,tµ, η)⩽Cµe1 2[−λi1ϑi1+φ1+κL0(1+θφ3)]t,(1.11) MEAN ATTRACTORS AND INVARIANT MEASURES OF SPDES 5 where Cµ>0is a constant independent of t. Let B(X(t)) be replaced by B(X(t)) in (1.1) for ∈[0,1/√2]. Let ˙ P(H) be the collection of all invariant measures of (1.1) for ∈[0,1/√2]. Then we discuss the limiting stability of invariant measures taken from ˙ P(H). In particular, we show that the union of ˙ P(H) over [0,1/√2] is tight on H, and the limit of every sequence of invariant measures taken from Sε∈[0,1/√2] ˙ P(H) must be an invariant measure of a limiting system of (1.1). Theorem 1.7. (Limiting stability) Let assumptions in Theorem 1.2 hold. (i) For every δ > 0,T > 0,0∈[0,1/√2] and bounded set B⊆H, we have lim →0 sup t∈[0,T ] sup X0∈B P(kX(t, 0, X0)−X0(t, 0, X0)kH⩾δ)=0. (ii) If there exists i0, i1∈[1, m]∩Nsuch that Vi0,→His compact, αi1⩾2and θi1> λ−1 i1φ2, then Sε∈[0,1/√2] ˙ P(H)is tight on H. If, in addition, ηn∈˙ Pn(H)and n→0with 0, n∈[0,1/√2], then there exists a subsequence nkand η0∈˙ P0(H)such that ηnk→η0weakly. 1.5. Invariant measures of (1.1): nonautonomous case. In many applications to physics and other fields of science, the evolution equations are often driven by stochastic and nonautonomous forcing simultaneously. In such a case the classical concept of invariant measures for time homogeneous transition semigroups does not work for nonautonomous stochastic equations like (1.1). To close the gap, a new concept called evolution systems of probability measures of time inhomogeneous transition operator (Pτ,t)t⩾τwas introduced and studied in [18, 19, 28, 29]. By an evolution system of probability measures of (Pτ,t)t⩾τwe mean a family of probability measures {ηt}t∈Ron Hsatisfying the invariance RHPτ,tϕ(x)ητ(dx) = RHϕ(x)ηt(dx) for any t⩾τ∈Rand continuous bounded function ϕon H. As far as we know, there are not many results on the investigation of evolution system of probability measures for nonautonomous SPDEs, and the quoted results are all concerned with SPDEs with linear drift terms and additive noise. In this paper we study the existence, uniqueness, global exponentially mixing, forward strongly mixing and backward strongly mixing of evolution system of probability measures for a class of abstract SPDEs with nonlinear drift and diffusion terms. Theorem 1.8. Let assumptions in Theorem 1.2 hold. If κ= 0 and there exists i1∈[1, m]∩Nsuch that αi1= 2,ϑi1> λ−1 i1φ1,θi1> λ−1 i1φ2,λi1(ϑi1−θi1) + φ2−φ1⩾0and Rτ −∞ e[λi1θi1−φ2]sφ3(s)ds < ∞for any τ∈R, then (Pτ,t)t⩾τhas a unique evolution system of probability measures {ηt}t∈Ron Hsuch that ZHkxk2 Hηt(dx)⩽Zt −∞ e[λi1θi1−φ2](s−t)φ3(s)ds, ∀t∈R.(1.12) In addition, {ηt}t∈Ris exponentially mixing in the sense that for any ϕ∈Lipb(H),X0∈Hand t⩾τ∈R, |(Pτ,tϕ)(X0)−ZH ϕ(x)ηt(dx)|⩽2kϕkLipe1 2[−λi1ϑi1+φ1](t−τ)kX0kH +e1 2[−λi1ϑi1+φ1]te1 2[λi1(ϑi1−θi1)+φ2−φ1]τZτ −∞ e[λi1θi1−φ2]sφ3(s)ds1/2.(1.13) Furthermore, {ηt}t∈Ris also exponentially mixing under the Wasserstein metric of P(H), that is, for any ϕ∈Lipb(H),t⩾τ∈Rand {µt}t∈R⊆ P(H)satisfying RHkxk2 Hµt(dx)⩽Rt −∞ e[λi1θi1−φ2]sφ3(s)ds, dP W(Qτ,tµτ, ηt)⩽2e1 2[−λi1ϑi1+φ1]te1 2[λi1(ϑi1−θi1)+φ2−φ1]τZτ −∞ e[λi1θi1−φ2]sφ3(s)ds1/2 .(1.14) Remark 1.9. (i) Unlike the autonomous case, we prove the existence of evolution system of probability measures without using compact Sobolev embeddings. (ii) Every evolution system of probability measures 6 RENHAI WANG, TOM ´ AS CARABALLO, AND NGUYEN HUY TUAN satisfying (1.12) must be unique and exponentially mixing in the sense of (1.13)-(1.14). (iii) By (1.13) we find that for every ϕ∈Cb(H)and X0∈H, lim t→+∞|(Pτ,tϕ)(X0)−ZH ϕ(x)ηt(dx)|= 0, τ ∈R lim τ→−∞|(Pτ,tϕ)(X0)−ZH ϕ(x)ηt(dx)|= 0, t ∈R. (iv) The restrictive condition κ= 0 means that operators Aand Bhave to be global monotone. We plan a future work to prove the existence of evolution systems of probability measures of (1.1) in the case κ6= 0. 1.6. Remarks on models. If m= 1, ϑ=κ=$= 0, ηi≡0 and φ1=φ2=φ4=φ6≡c, then above framework reduces to the standard or locally monotone variational framework [13, 26, 34, 40]. A significant advantage of the present variational framework is κ6= 0, $6= 0, m6= 1, ϑ6= 0 and ηi6= 0, which permits us to possibly study the global well-posedness and long-term dynamics for a wide class of SPDEs covering several important models included or not included before. A typical example within the present framework but not the clcial setting of is the tamed Navier-Stokes equation in dimension N= 2,3: ∂u ∂t −µ∆u+ (u· ∇)u+β|u|r−1u+∇p= 0, where µ, β > 0 and r⩾1. It is known that the 2D Navier-Stokes equation (β= 0) satisfies the framework of (1.1) with m= 1. However, we can show that the tamed Navier-Stokes equation satisfies the present setting for N= 2,3. Let Pbe the Helmholtz-Hodge projection, and consider A1(u) = Pµ∆u−(u·∇)uand A2(u) = −βP|u|r−1u. It can be proved that A1satisfies (C2)-(C4) only when N= 2. While, currently, it is impossible to prove that A1satisfies (C2) and (C4) simultaneously if N= 3 due to the term (u·∇)u. Nevertheless, by using the dissipative effect of β|u|r−1uto carefully control (u·∇)u, we can prove that A=A1+A2satisfies (C2) and (C4) together. Table 1. Asatisfies (C2) and (C4) (see Proposition 7.11) 2V∗hA(v1)−A(v2),v1−v2iV⩽ m P i=1  Ai(t, ω, v)  αi/(αi−1) V∗ i+ρi(v)⩽ N= 2,3−µkv1−v2k2 V−βγkv1−v2kr+1 Lr+1 c(1 + kf(t)k2 V∗)1 + kvk2 V+kvkr+1 Lr+1 1 + kvk 2(r−3) r−1 H r > 3 +h24 r−3r−3 r−1µ 1−r r−3(β(r−1)) 2 3−ri v1−v2  2 H N= 2, r⩾1 −µkv1−v2k2 V−2βγkv1−v2kr+1 Lr+1 + 27 64µ3kv2k2 Hkv2k2 Vkv1−v2k2 H c(1 + kf(t)k2 V∗)1 + kvk2 V+kvkr+1 Lr+1 1 + kvk2 H N=r= 3, βµ ⩾1−µkv1−v2k2 V−2βγkv1−v2kr+1 Lr+1 c(1 + kf(t)k2 V∗)1 + kvk2 V+kvk4 L4 1.7. Applications of abstract results. As applications of our abstract results for problem (1.1), we show that the stochastic tamed Navier-Stokes equation (see Example (7.2)) and a stochastic fractional (s, p)-Laplacian equation with polynomial growth nonlinearities of arbitrary order for all s∈(0,1) and p∈[2,∞) (see Example (7.9)), do fall within our abstract variational setting. Then by directly applying our abstract results in Theorems 1.2-1.8, we obtain the global well-posedness and long-time dynamics results for the two typical examples under some specified conditions. Our abstract results are expected to be applied to many different types of SPDEs. MEAN ATTRACTORS AND INVARIANT MEASURES OF SPDES 7 1.8. Outline of paper. In the next section we establish the well-posedness and Itˆo’s formula of (1.1). In Section 3 we prove the existence a uniqueness of a mean random attractor for (1.1). In Section 4 we discuss the existence and properties of invariant measures and evolution systems of measures of (1.1) in autonomous and nonautonomous cases. In the last section we illustrate our abstract results with two typical models. 2. Well-posedness of (1.1) in high-order Bochner spaces: Existence, uniquness and Itˆ o’s formula In this section we prove our first main result on the global well-posedness of problem (1.1) under assumptions (C1)-(C4). The proofs are based on the Galerkin approximation and the theory of monotone operators, see e.g., [1, 5, 38, 39, 45, 46, 59, 60] for deterministic case, and [26, 34, 40] for stochastic case. 2.1. Approximate systems and functional spaces. Note that if f∈Hand v∈V, then V∗hf, viV=V∗ i hfi, viVi=hf, viH. Since the separable Banach space Vis continuously and densely embedded into H, there exists an orthonormal basis {ei, i ∈N}of Hsuch that {ei, i ∈N} ⊆ V, and the span{ei, i ∈N} is dense in V. For n∈N, let Pn:V∗→Hn:= span{e1, e2· ··, en}be the projection given by Pnf=Pn i=1 V∗hf, eiiVeifor f∈V∗. This is crucial to treat the operator A=Pm i=1 Ai. Note that Pn|H defines an orthogonal projection onto Hnin H, and hence V∗hPnA(t, ω, u), viV=hPnA(t, ω, u), viH= V∗hA(t, ω, u), viVfor all u∈Vand v∈Hn. Let {gi, i ∈N}be an orthonormal basis of U. Let e Pn:U→ Un:= span{g1, g2···, gn}be an orthogonal projection, and write Wn(t) := e PnW(t) = Pn i=1hW(t), giiUgi. Then we consider a finite-dimensional stochastic system on Hn: (dXn(t) = PnA(t, Xn(t))dt +PnB(t, Xn(t))dWn(t), t > τ ∈R, Xn(τ) = PnX0.(2.1) According to the classical results for the solvability of finite-dimensional SDEs, see e.g., [26], we can prove that problem (2.1) has a unique continuous strong solution. In what follows, we will derive a priori estimates of solutions to (2.1) in the spaces Ki=Lαi([τ, τ +T]×Ω, dt ×P;Vi), K∗ i=L αi αi−1([τ, τ + T]×Ω, dt ×P;V∗ i), i= 1,2, . . . , m,J=L2([τ, τ +T]×Ω, dt ×P;L2(U, H)) and S=L∞([τ, τ + T], dt;L2(Ω,P;H)). In addition, the letter c > 0 denotes a generic constant which may change its value in different places. 2.2. A priori estimates. The main difficulty of deriving a priori estimates of solutions to (2.1) in the spaces above is how to estimate the nonlinear diffusion term B(t, ω, v) with a superlinear growth rate in v. This difficulty can be surmounted by using the dissipativeness of the operator Ato carefully control the superlinear growth of B(t, ω, v). To simplify calculations, we derive the following two formulations which are just direct consequences of (C4). Proposition 2.1. Let (C4) be valid. If βi< αi, then for any ε, ε1, ε2, r1, K > 0and r2> r1, KkB(t, ω, v)k2 L2(U,H)⩽ε m X i=1 θikvkαi Vi+Kφ4(t, ω)kvk2 H+c m X i=1 η αi αi−βi i(t, ω) + Kφ5(t, ω),(2.2a) Kkvkr1 HkB(t, ω, v)k2 L2(U,H)⩽ε1kvkr1 H m X i=1 θikvkαi Vi+ε2kvkr2 H+Kφ4(t, ω)kvk2+r1 H +cφ r2 r2−r1 5(t, ω) + c m X i=1 η αir2 (αi−βi)(r2−r1) i(t, ω).(2.2b) Lemma 2.2. Let (C1)-(C4) hold. Then for any X0∈L2`(Ω,Fτ,P;H), we have the following conclusions. 8 RENHAI WANG, TOM ´ AS CARABALLO, AND NGUYEN HUY TUAN (i) If φ3, φ5∈L`([τ, τ +T]×Ω, dt ×P;R+)and ηi∈L `αi αi−βi([τ, τ +T]×Ω, dt ×P;R+)hold for any `⩾1, then there exists a constant C(τ, T, X0)>0independent of nsuch that E"sup t∈[τ,τ+T]kXn(t)k2` H#+E"Zτ+T τkXn(s)k2`−2 H m X i=1 kXn(s)kαi Vids#⩽C(τ, T, X0),∀`⩾1. (ii) If φ3, φ5∈L`([τ, τ +T]×Ω, dt ×P;R+)and ηi∈L `αi αi−βi([τ, τ +T]×Ω, dt ×P;R+)hold for all `⩾$/2+1, then for every n∈N, there exists a constant C(τ, T, X0)>0independent of nsuch that kXnk2` L2`(Ω,P;L∞([τ,τ+T,dt;H])) + m X i=1 hkXnkαi Ki+kAi(·, Xn)k αi αi−1 K∗ ii+kB(·, Xn)k2 J⩽C(τ, T, X0). Proof. (i) Applying the finite-dimensional Itˆo formula to (2.1) we find, P-a.s., kXn(t)k2 H=kPnX0k2 H+Zt τ2V∗hPnA(s, Xn(s)), Xn(s)iV +kPnB(s, Xn(s)) e Pnk2 L2(U,H)ds + 2 Zt τhXn(s), PnB(s, Xn(s))dWn(s)iHds. (2.3) Note that by Parseval’s identity we have k(PnB(s, Xn(s)) e Pn)∗Xn(s)kL2(U,R)=k(PnB(s, Xn(s)) e Pn)∗Xn(s)kU⩽ kPnB(s, Xn(s)) e PnkL2(U,H)kXn(s)kH⩽kB(s, Xn(s))kL2(U,H)kXn(s)kH. For any `⩾1, applying the finite-dimensional Itˆo formula to (2.3) again, we infer from (2.2b) with K=`(2`−1), ε1=ε2=`/2, r1= 2`−2 and r2= 2`that P-a.s., kXn(t)k2` H=kPnX0k2` H+ 2`(`−1) Zt τkXn(s)k2`−4 Hk(PnB(s, Xn(s)) e Pn)∗Xn(s)k2 L2(U,R)ds +`Zt τkXn(s)k2`−2 H2V∗hPnA(s, Xn(s)), Xn(s)iV+kPnB(s, Xn(s)) e Pnk2 L2(U,H)ds +Mn(t) ⩽kPnX0k2` H−2`Zt τkXn(s)k2`−2 H m X i=1 θikXn(s)kαi Vids +`(2`−1) Zt τkXn(s)k2`−2 HkB(s, Xn(s))k2 L2(U,H)ds +cZt τ1 + φ2(s)kXn(s)k2` H+φ` 3(s)ds +Mn(t) ⩽kPnX0k2` H−3` 2Zt τkXn(s)k2`−2 H m X i=1 θikXn(s)kαi Vids +cZt τ1 + X i=2,4 φi(s)kXn(s)k2` H+ m X i=1 η `αi αi−βi i(s) + X i=3,5 φ` i(s)ds +Mn(t),(2.4) where Mn(t) := 2`Rt τkXn(s)k2`−2 HhXn(s), PnB(s, Xn(s))dWn(s)iH. For k∈N, we define a stopping time ςk n= inf t⩾τ:kXn(t)kH> k and m X i=1 Zt τkXn(s)kαi Vids > k, where inf ∅= +∞. Since we can control B(s, Xn(s) by Proposition 2.1, by the definition of ςk nwe know the quadratic variation of Mn(t∧ςk n): hMn(·∧ςk n)it⩽4`2Rt∧ςk n τkXn(s)k4`−4 Hk(PnB(s, Xn(s)) e Pn)∗Xn(s)k2 L2(U,R)⩽ 4`2Rt∧ςk n τkXn(s)k4`−2 HkB(s, Xn(s))k2 L2(U,H)ds ⩽C(τ, T, k), P-a.s.. Then Mn(t) is a real-valued continuous local martingale. By (2.4) we obtain, for all t∈[τ, τ +T], Esup τ⩽r⩽tkXn(r∧ςk n)k2` H+3` 2E"Zt∧ςk n τkXn(s)k2`−2 H m X i=1 θikXn(s)kαi Vids# ⩽2E[kPnX0k2` H] + cZt τ E"sup τ⩽r⩽s1 + X i=2,4 φi(r∧ςk n)kXn(r∧ςk n)k2` H#ds MEAN ATTRACTORS AND INVARIANT MEASURES OF SPDES 9 +cZτ+T τ E X i=3,5 φ` i(s) + m X i=1 η `αi αi−βi i(s) ds + 2Ehsup τ⩽r⩽t∧ςk n|Mn(r)|i.(2.5) By the BDG inequality for real-valued continuous local martingales, we infer from (2.2b) with ε1=`/4, ε2= 1/4, r1= 2`−2 and r2= 2`that the stochastic term in (2.5) is bounded by 2Ehsup τ⩽r⩽t∧ςk n |Mn(r)|i⩽12`E  Zt∧ςk n τ kXn(s)k4`−2 HkB(s, Xn(s))k2 L2(U,H)ds!1 2  ⩽12`E sup τ⩽s⩽t kXn(s∧ςk n)k` H Zt∧ςk n τ kXn(s)k2`−2 HkB(s, Xn(s))k2 L2(U,H)ds!1 2  ⩽1 2Esup τ⩽r⩽t kXn(r∧ςk n)k2` H+cE Zt∧ςk n τ kXn(s)k2`−2kB(s, Xn(s))k2 L2(U,H)ds# ⩽1 2Esup τ⩽r⩽t kXn(r∧ςk n)k2` H+` 4E"Zt∧ςk n τ kXn(s)k2`−2 H m X i=1 θikXn(s)kαi Vids# +cZt τ Esup τ⩽r⩽s1 + φ4(r∧ςk n)kXn(r∧ςk n)k2` Hds +cZτ+T τ E"φ` 5(s) + m X i=1 η `αi αi−βi i(s)#ds. (2.6) By (2.5)-(2.6) we see, for all `⩾1, Esup τ⩽r⩽tkXn(r∧ςk n)k2` H+E"Zt∧ςk n τkXn(s)k2`−2 H m X i=1 kXn(s)kαi Vids# ⩽ec+cPi=2,4kφikL∞([τ,τ+T]×Ω) E[kX0k2` H] + Zτ+T τ EX i=3,5 φ` i(s) + m X i=1 η `αi αi−βi i(s)ds.(2.7) By Markov’s inequality and (2.7) we find P{τ⩽ςk n< τ +T } ⩽Psupτ⩽r⩽τ+TkXn(r∧ςk n)kH⩾k⩽ Ck−2`<∞for any T ∈ N, where C > 0 is a constant independent of k. Let ΩT=T∞ m=1 S∞ k=m{ςk n< τ+T}. By the Borel-Cantelli lemma we know P(ΩT) = PT∞ m=1 S∞ k=m{τ⩽ςk n< τ +T }= 0. This implies, for every ω∈Ω\ΩT, there exists k0=k0(ω)>0 such that ςk n(ω)⩾τ+Tfor all k⩾k0. Taking Ω0=ST ∈NΩT, we have P(Ω0) = 0 and ςk n(ω)⩾τ+Tfor all ω∈Ω\Ω0, and thus ςk n→ ∞ as k→ ∞, P-a.s. Consequently, we complete the proof of (i) by letting k→ ∞ in (2.7). (ii) Note that condition (ii) implies condition (i). Then by (C4), (i) and Proposition 2.1, there exists a constant C(τ, T, X0)>0, independent of n, such that for all `⩾$/2 + 1, m X i=1 EZt τkAi(s, Xn(s))k αi αi−1 V∗ ids+EZt τkB(s, Xn(s))k2 L2(U,H)ds⩽C(τ, T, X0).(2.8) This yields (ii).  2.3. Proof of Theorem 1.2. A special attention in the proof of Theorem 1.2 is that the approximated solutions converge in both Sobolev and finitely many Banach spaces simultaneously. The idea of the proof is motivated by the classical works in [26, 34], where the SPDEs is defined in a single Banach space and the growth rate of the noise is linear. In Theorem 1.2 we consider a large class of SPDEs defined in a dual space of intersection of finitely many Banach spaces driven by superlinear noise in order to cover more models which do not fall within the variational framework. Our approaches are different from the semigroup method [17], and the regularization method [51] since the operator Ain there is restricted to be linear. Proof. Proof of Theorem 1.2. By Lemma 2.2 there exist a subsequence {nk}∞ k=1 of {n}∞ n=1 such that Xnk→¯ Xweakly star in L2(Ω,P;L1([τ, τ +T], dt;H])∗; (2.9a) Xnk→¯ Xweakly in L2(Ω,P;L2([τ, τ +T], dt;H]), K1,K2,. . . ,Km; (2.9b) 16 RENHAI WANG, TOM ´ AS CARABALLO, AND NGUYEN HUY TUAN ∨Zt τ m X i=1 kX(s, τ, X0)kαi Vids> R().(4.2) As before, by (1.1), (C2) and the product rule we find eφ1(τ−t∧ςn)−κPm i=1 Rt∧ςn τρi(X(r,τ,X0))drkX(t∧ςn, τ, Xn,0)−X(t∧ςn, τ, X0)k2 H ⩽kXn,0−X0k2 H+Mn(t∧ςn),(4.3) where Mn(t∧ςn) is given by Mn(t∧ςn) := 2 Zt∧ςn τ eφ1(τ−s)−κPm i=1 Rs τρi(X(r,τ,X0))dr ×hX(s, τ, Xn,0)−X(s, τ, X0),B(s, X(s, τ, Xn,0)) −B(s, X(s, τ, X0))dW(s)iH. By (4.2) and Proposition 2.1 we can prove that the expectation of the quadratic variation of M(t∧ςn) is finite. Then, taking the expectation in (4.3), we obtain Eheφ1(τ−t∧ςn)−κPm i=1 Rt∧ςn τρi(X(r,τ,X0))dr ×kX(t∧ςn, τ, Xn,0)−X(t∧ςn, τ, X0)k2 Hi⩽kXn,0−X0k2 H. By (1.2) and (4.2) there exists another constant C(τ, T, R()) to bound Pm i=1 Rt∧ςn τρi(X(r, τ, X0))dr ⩽ C(τ, T, R()), P-a.s.. This implies EhkX(t∧ςn, τ, Xn,0)−X(t∧ςn, τ, X0)k2 Hi⩽eφ1T+C(τ,T,R())kXn,0−X0k2 H.(4.4) Note that ςn(ω)⩾t, for all ω∈Ω n, by the definitions of ςnand Ω n. Then by (4.4), Chebyshev’s inequality and P(Ω \Ω n)< , we conclude that for every δ > 0, Pω∈Ω : kX(t, τ, Xn,0)−X(t, τ, X0)kH⩾δ =Pω∈Ω\Ω n:kX(t, τ, Xn,0)−X(t, τ, X0)kH⩾δ +Pω∈Ω n:kX(t, τ, Xn,0)−X(t, τ, X0)kH⩾δ ⩽+δ−2EhkX(t∧ςn, τ, Xn,0)−X(t∧ςn, τ, X0)k2 Hi ⩽+δ−2eφ1T+C(τ,T,R())kXn,0−X0k2 H.(4.5) Since is arbitrary and Xn,0→X0in H, we infer from (4.5) that lim n→∞ Pω∈Ω : kX(t, τ, Xn,0)− X(t, τ, X0)kH⩾δ= 0. Then there exists a subsequence {nk}∞ k=1 ⊆ {n}∞ n=1 such that lim k→∞kX(t, τ, Xnk,0)− X(t, τ, X0)kH= 0,P-a.s.. By a contradiction argument we deduce from the uniqueness of the solutions that the original sequence X(t, τ, Xn,0)→X(t, τ, X0) in Has n→ ∞,P-a.s.. This along with the continuity of ϕshows that ϕ(X(t, τ, Xn,0)) →ϕ(X(t, τ, X0)) in Ras n→ ∞,P-a.s.. Since ϕ(X(t, τ, Xn,0)) ⩽kϕk∞for all n∈N, by Lebesgue’s theorem we complete the proof of (i). (ii) By the argument of proving (i), we can follow [17, Theorem 9.14] to show that X(t, τ, X0) is an H-valued Markov process for t⩾τ. (iii) By (ii) and [17, Corollaries 9.15], we can similarly prove (iii).  Lemma 4.3. Let assumptions in Theorem 1.2 hold. If there exist i1∈[1, m]∩Nsuch that αi1⩾2and θi1> λ−1 i1φ2, then EkX(t)k2 H+ m X i=1 θiZt τ e−[λi1θi1−φ2](t−s)EkX(s)kαi Vids ⩽e−[λi1θi1−φ2](t−τ)kX0k2 H+Zt τ e−[λi1θi1−φ2](t−s)[θi1Cαi1+φ3(s)]ds, MEAN ATTRACTORS AND INVARIANT MEASURES OF SPDES 17 and m X i=1 θiZt τ EkX(s)kαi Vids ⩽kX0k2 H+Zt τ [θi1Cαi1+φ3(s)]ds. Proof. The proof is similar to that in Lemma 3.1, and so omitted.  5. Invariant measures of (1.1): autonomous case In this subsection we discuss the existence and several important properties including integrability, uniqueness, ergodicity, mixing and stability of invariant probability measures for equation (1.1) in the autonomous case. In this part we further assume that operators and functions in (C1)-(C4) are all independent of t. Then we know Pτ,t =P0,t−τand Pτ,t(X0,·) = P0,t−τ(X0,·) for any t⩾τ⩾0. 5.1. Proof of Theorem 1.4. Proof. Proof of Theorem 1.4. (i) (Existence) By Lemma 4.3, there exist i0, i1∈[1, m]∩Nsuch that k−1Zk 0 EhkX(s, 0, X0)kαi0 Vi0ids ⩽C0, k ∈N.(5.1) where C0:= θ−1 i0(kX0k2 H+θi1Cαi1+φ3). For ε > 0 and l∈N, we define Yε l=nv∈Vi0:kvkVi0⩽ (ε−1C022l)1/αi0oand Zε l=u∈H:ku−vkH⩽1 2lfor some v∈ Yε l. By Markov’s inequality we get Pω∈Ω : X(t, 0, X0)/∈ Zε l⩽Pω∈Ω : X(t, 0, X0)/∈ Yε l +Pω∈Ω : X(t, 0, X0)/∈ Zε land X(t, 0, X0)∈ Yε l ⩽Pnω∈Ω : kX(t, 0, X0)kVi0>(ε−1C022l) 1 αi0o ⩽ε C022lE(kX(t, 0, X0)kαi0 Vi0).(5.2) By the compact Sobolev embedding Vi0,→Hwe know Zε:= Tl∈NZε lis compact in H. Define a family of probability measures {ηk}k∈N⊆ P(H) by ηk=k−1Rk 0P0,t(X0,·)dt. By (5.1) and (5.2) we have ηk(H\Zε l) = k−1Zk 0 P{ω∈Ω : X(t, 0, X0)/∈ Z l}dt ⩽ε C022lkZk 0 E[kX(t, 0, X0)kαi0 Vi0]dt ⩽ε 22l.(5.3) It follows from (5.3) that ηk(H\Zε)⩽P∞ l=1 ηk(H\Zε l)⩽P∞ l=1 ε 22l< ε, and so ηk(Zε)>1−ε. Then the sequence {ηk}k∈Nis tight on H. By the Prokhorov’s theorem, there exists a probability measure η∈ P(H) such that, up to a subsequence, ηk→ηweakly as k→ ∞. Therefore by Lemma 4.2 and the classical Krylov-Bogolyubov method (see [17]), we know, for any t⩾0 and ϕ∈Cb(H), hϕ, ηi= lim k→∞k−1Zk 0ZH P0,r(X0, dy)(P0,tϕ)(y)dr = lim k→∞hP0,tϕ, ηki=hϕ, P∗ 0,tηi. This completes the proof of (i). (ii) (Regularity) It is sufficient to show that each invariant probability measure ηof (P0,t)t⩾0on H satisfies η(V) = 1. Given R > 0, we consider the ball BV R:= {x∈V:kxkV⩽R}in V. Denoting α:= min i=1,2,...,m αi, by Lemma 4.3 and θi1> λ−1 i1φ2we find that for every X0∈H, there exists k0(X0)∈N such that for all k⩾k0(X0), k−1Zk 0 E[kX(s, 0, X0)kα V]ds ⩽c+ck−1 m X i=1 Zk 0 EkX(s, 0, X0)kαi Vids ⩽c, (5.4) where c > 0 is a constant independent of kand X0. By Markov’s inequality and (5.4), for all k⩾k0(X0), k−1Zk 0 Pnω∈Ω : X(s, 0, X0)∈BV Rods ⩾1−cR−α.(5.5) 18 RENHAI WANG, TOM ´ AS CARABALLO, AND NGUYEN HUY TUAN By the invariance of ηand Fubini’s theorem we deduce that for all k⩾k0(X0), ZHk−1Zk 0 Pnω∈Ω : X(s, 0, X0)∈BV Rodsη(dX0) = η(BV R). This along with (5.5) and Fatou’s lemma implies η(BV R)⩾ZH lim inf k→∞ k−1Zk 0 Pnω∈Ω : X(s, 0, X0)∈BV Rodsη(dX0)⩾1−cR−α. Letting R→ ∞ in the above, we derive η(V)⩾1, and hence η(V) = 1.  5.2. Proof of Theorem 1.5. Proof. Proof of Theorem 1.5 We choose an H-valued F0-measurable random variable X0with law νsuch that ν=η. Noting that (P0,tϕ)(X0) = E[ϕ(X(t, 0, X0))] = hϕ, P0,t(X0,·)i=hP0,tϕ, νifor any ϕ∈Cb(X), we have P0,t(X0,·) = P∗ 0,tν. By the invariance of η=νwe have P0,t(X0,·) = η, which means that the law of X(t, 0, X0) does not change for all time t⩾0. Then by [17] we know X(t, 0, X0) is a stationary solution of (1.1). Then we are able to show (i)-(iii) of Theorem 1.5. (i) (High-order integrability for αi1= 2). By (3.3) with αi1= 2 we find E[kX(t)k2` H] + `1 2λi1θi1−φ2−2(`−1)φ4Zt 0 EkX(r)k2` Hdr +` 2Zt 0 EkX(r)k2`−2 H m X i=1 θikX(r)kαi Vidr ⩽EkX0k2` H+ct. Since θi1>2λ−1 i1(φ2+ 2(`−1)φ4) and ηis the law of the stationary solution X(t, 0, X0), we obtain RHkxk2` Hη(dx) + RHkxk2`−2 HPm i=1 kxkαi Viη(dx)⩽c. This implies (1.6). (ii) (High-order integrability for αi1>2). By (3.3) we obtain E[kX(t)k2` H] + ` 2Zt 0 EkX(r)k2`−2 H m X i=1 θikX(r)kαi Vidr ⩽EkX0k2` H+cZt 0 EkX(r)k2` Hdr +ct. (5.6) By the Sobolev embedding Vi,→Hwe know pλi1kxkH⩽kxkVifor i= 1,2, . . . , m. Since ηis the law of the stationary solution X(t, 0, X0), by (5.6) and Young’s inequality we have ZHkxk2` Hη(dx)⩽c1+c2ZHkxk2`−2 H m X i=1 kxkαi Viη(dx)⩽c3ZHkxk2` Hη(dx) + c4,(5.7) where c2, c3>0, 0 < c1⩽c4. Letting R > 0 and splitting RHkxk2` Hη(dx) into two parts, by (5.7) we deduce ZHkxk2` Hη(dx)⩽R2`+cR2−αi1ZHkxk2`−2 Hkxkαi1 Vi1η(dx)⩽R2`+cR2−αi1ZHkxk2` Hη(dx) + cR2−αi1. Since αi1>2, there exists R > 0 such that cR2−αi1=1 2, then we obtain the finiteness of RHkxk2` Hη(dx) immediately. This along with (5.7) completes the proof of (1.7). (iii) (Exponential integrability for αi1⩾2) By Itˆo’s formula and (1.3) with `= 1, for any  > 0, dekX(t)k2 H=ekX(t)k2 H2hX(t), BdW (t)iH+ 2kB∗X(t)k2 Udt +2V∗hA(X(t)), X(t)iV+kBk2 L2(U,H)dt. Taking the expectation in the above equality, and by a stopping time argument as in Lemma 2.2, we infer from (C3) and Remark 1.3 that d dtEhekX(t)k2 Hi=EhekX(t)k2 H2kB∗X(t)k2 U+ 2 V∗hA(X(t)), X(t)iV+kBk2 L2(U,H)i ⩽EhekX(t)k2 H2kBk2 L2(U,H)+φ2−λi1θi1kX(t)k2 H− m X i=1 θikX(t)kαi Vi+φ3+θi1Cαi1i. MEAN ATTRACTORS AND INVARIANT MEASURES OF SPDES 19 This implies for all ∈0,2−1kBk−2 L2(U,H)(λi1θi1−φ2), EekX(t)k2 H+EZt 0 ekX(s)k2 H m X i=1 θikX(s)kαi Vids⩽EekX(0)k2 H+(φ3+θi1Cαi1)EZt 0 ekX(s)k2 Hds. As before, we have RHekxk2 HPm i=1 θikxkαi Viη(dx)⩽(φ3+θi1Cαi1)RHekxk2 Hη(dx).Then, for all R > 0, ZH ekxk2 Hη(dx)⩽eR2+R−αi1ZH ekxk2 Hkxkαi1 Vi1η(dx)⩽eR2+θ−1 i1(φ3+θi1Cαi1)R−αi1ZH ekxk2 Hη(dx). Taking R= (2θ−1 i1(φ3+θi1Cαi1))1/αi1, we then complete the proof.  Remark 5.1. The finiteness of RHkxk2` Hη(dx)is unknown if αi∈(1,2) for any i∈[1, m]∩N. 5.3. Proof of Theorem 1.6. Proof. Proof of Theorem 1.6. (i) Let X(t, 0, X0) and X(t, 0, Y0) be two solutions to (1.1). Then by Remark 1.3 with αi1= 2, we have EkX(t, 0, X0)−X(t, 0, Y0)k2 H⩽e−(λi1ϑi1−φ1)tkX0−Y0k2 H. Let η∈ P(H) be an invariant probability measure of (P0,t)t⩾0. By Cauchy-Schwarz’s inequality we deduce, for any ϕ∈Lipb(H) and X0∈H, |(P0,tϕ)(X0)−hϕ, ηi|2=E[ϕ(X(t, 0, X0))] −ZH (P0,tϕ)(Y0)η(dY0) 2 =ZH E[ϕ(X(t, 0, X0)) −ϕ(X(t, 0, Y0))]η(dY0) 2 ⩽2kϕk2 LipbkX0k2 H+ZHkX0k2 Hη(dX0)e−(λi1ϑi1−φ1)t,(5.8) where RHkX0k2 Hη(dX0)<∞due to (i) of Theorem 1.5 with `= 1. Then (1.8) follows from (5.8). Given ϕ∈Cb(H), by Proposition 4.1, there exists ϕn∈Lipb(H) satisfying supx∈Xsupn∈Nϕn(x)<∞ such that ϕn(x)→ϕ(x) for any x∈H. By (1.8) we have (P0,tϕn)(X0)→ hϕn, ηias t→+∞. Then by Lebesgue’s theorem we deduce that (P0,tϕ)(X0)→ hϕ, ηias t→+∞, and hence ηis strongly mixing on Hfor (P0,t)t⩾0. Furthermore, for µ∈ P(H) satisfying RHkxk2 Hµ(dx)<∞, we have dP W(Q0,tµ, η) = sup ϕ∈Lipb(H),kϕkLipb ⩽1ZH E[ϕ(X(t, τ, X0))]µ(dX0)−ZH E[ϕ(X(t, τ, Y0))]η(dY0) = sup ϕ∈Lipb(H),kϕkLipb ⩽1ZHZH E[ϕ(X(t, τ, X0)) −ϕ(X(t, τ, Y0))]µ(dX0)η(dY0) ⩽√2ZHkX0k2 Hµ(dX0) + ZHkY0k2 Hη(dY0)1/2 e−1 2(λi1ϑi1−φ1)t. This gives (1.9). Let bη∈ P(H) be another invariant measure for (P0,t)t⩾0. As before, for any ϕ∈Lipb(H), |hϕ, ηi−hϕ, bηi|2=ZH (P0,tϕ)(X0)η(dX0)−ZH (P0,tϕ)(Y0)bη(dY0) 2 =ZHZH E[ϕ(X(t, 0, X0)) −ϕ(X(t, 0, Y0))]η(dX0)bη(dY0) 2 ⩽2kϕk2 Lipb e−(λi1ϑi1−φ1)tZHkX0k2 Hη(dX0) + ZHkY0k2 Hbη(dY0)→0 as t→+∞. Then, by Proposition 4.1 and Lebesgue’s theorem, we have hϕ, ηi=hϕ, bηifor all ϕ∈Cb(H). This further gives the uniqueness of η. From this and [17], we know that ηis ergodic on Hfor (P0,t)t⩾0. 20 RENHAI WANG, TOM ´ AS CARABALLO, AND NGUYEN HUY TUAN (ii) Let Z(t) := X(t, 0, X0)−X(t, 0, Y0) be the difference of two solutions to (1.1). By Remark 1.3 with αi1= 2 we find d dtkZ(t)k2 H⩽h−λi1ϑi1+φ1+κ m X i=1 ρi(X(t, 0, Y0))ikZ(t)k2 H. This along with (1.2) for $= 0 yields E[kZ(t)k2 H]⩽kZ(0)k2 HEhe[−λi1ϑi1+φ1]t+κRt 0Pm i=1 ρi(X(r,0,Y0))dri ⩽kZ(0)k2 HEe(−λi1ϑi1+φ1+κL0)t+κθL0Rt 0Pm i=1 θikX(s,0,Y0)kαi Vids,(5.9) where θ:= max i=1,2,...,m{θ−1 i}. To estimate the term eRt 0Pm i=1 θikX(s,0,Y0)kαi Vids in (5.9), we let Υ(t) := kX(t, 0, Y0)k2 H+Rt 0Pm i=1 θikX(s, 0, Y0)kαi Vids. Then by (1.3) with `= 1 for X(t, 0, Y0) we obtain dΥ(t) = m X i=1 θikX(t, 0, Y0)kαi Vidt + 2hX(t, 0, Y0), BdW (t)iH + 2 V∗hA(X(t, 0, Y0)), X(t, 0, Y0)iVdt +kBk2 L2(U,H)dt. Then using Itˆo’s formula to the above equation, we find, for any  > 0, deΥ(t)=eΥ(t)m X i=1 θikX(t, 0, Y0)kαi Vidt + 2 V∗hA(X(t, 0, Y0)), X(t, 0, Y0)iVdt +kBk2 L2(U,H)dt + 2hX(t, 0, Y0), BdW (t)iH+ 2kB∗(X(t, 0, Y0))k2 Udt.(5.10) Taking expectation in (5.10), as in Lemma 2.2, we infer from Remark 1.3 with αi1= 2 that d dtEeΥ(t)⩽φ3EeΥ(t)],∀⩽λi1θi1−φ2 2kBk2 L2(U,H) . This implies EeRt 0Pm i=1 θikX(s,0,Y0)kαi Vids⩽e(kY0k2 H+φ3t),∀⩽λi1θi1−φ2 2kBk2 L2(U,H) .(5.11) By the condition on θi1, we have κθL0⩽λi1θi1−φ2 2kBk2 L2(U,H) . This along with (5.9) and (5.11) for =κθL0 yields E[kX(t, 0, X0)−X(t, 0, Y0)k2 H]⩽kX0−Y0k2 He[−λi1ϑi1+φ1+κL0(1+θφ3)]t+κθL0kY0k2 H. Let η∈ P(H) be an invariant measure of (P0,t)t⩾0. As before, for any ϕ∈Lipb(H) and X0∈H, |P0,tϕ(X0)−hϕ, ηi|2=ZH E[ϕ(X(t, 0, X0)) −ϕ(X(t, 0, Y0))]η(dY0) 2 ⩽kϕk2 Lipb e[−λi1ϑi1+φ1+κL0(1+θφ3)]tZHkX0−Y0k2 HeκθL0kY0k2 Hη(dY0) ⩽2kϕk2 Lipb e[−λi1ϑi1+φ1+κL0(1+θφ3)]tekX0k2 HZH e(1+κθL0)kY0k2 Hη(dY0). Since 1 + κθL0⩽λi1θi1−φ2 2kBk2 L2(U,H) , by taking = 1 + κθL0in the above inequality and (iii) of Theorem 1.5, we find (1.10). This also implies that ηis strongly mixing. Furthermore, for any ϕ∈Lipb(H) and µ∈ P(H) satisfying RHekX0k2 Hµ(dX0)<∞, we have dP W(Q0,tµ, η) = sup ϕ∈Lipb(H),kϕkLipb ⩽1ZHZH E[ϕ(X(t, 0, X0)) −ϕ(X(t, 0, Y0))]µ(dX0)η(dY0) ⩽√2e1 2[−λi1ϑi1+φ1+κL0(1+θφ3)]tZH ekX0k2 Hµ(dX0)1/2ZH e1+κθL0kY0k2 Hη(dY0)1/2 . By taking = 1 + κθL0in the above inequality and (iii) of Theorem 1.5, we have (1.11). MEAN ATTRACTORS AND INVARIANT MEASURES OF SPDES 21 Let bηbe another invariant measure of (P0,t)t⩾0, then for any ϕ∈Lipb(H), as t→+∞, |hϕ, ηi−hϕ, bηi|2=ZHZH E[ϕ(X(t, 0, X0)) −ϕ(X(t, 0, Y0))]η(dX0)bη(dY0) 2 ⩽2kϕk2 Lipb e[−λi1ϑi1+φ1+κL0(1+θφ3)]tZH ekX0k2 Hη(dX0)ZH e1+κθL0kY0k2 Hbη(dY0)→0, This proves the uniqueness and ergodicity of η. 5.4. Proof of Theorem 1.7. Proof. Proof of Theorem 1.7. (i) By Theorem 1.2 we find, for any ∈[0,1/√2], Ehsup t∈[0,T ]kXε(t, X0)k2` Hi+EZT 0kXε(s, X0)k2`−2 H m X i=1 kXε(s, X0)kαi Vids +Ehsup t∈[0,T ]kXε0(t, X0)k2` Hi+EZT 0kXε0(s, X0)k2`−2 H m X i=1 kXε0(s, X0)kαi Vids⩽M2, where M2is a constant independent of . By Markov’s inequality we can prove that for every γ > 0, there exists a constant R(γ) such that for any ∈[0,1/√2], Pω∈Ω : sup t∈[0,T ] kX(t, X0)kH> R(γ)⩽γ 2and Pω∈Ω : sup t∈[0,T ] kX0(t, X0)kH> R(γ)⩽γ 2. Define Ωγ =ω∈Ω : supt∈[0,T ]kX(t, Xn,0)kH⩽R(γ) and supt∈[0,T ]kX0(t, X0)kH⩽R(γ). Define ς= inf t⩾0 : kX(t, X0)kH∨ kX0(t, X0)kH∨Zt 0 m X i=1 kX(s, X0)kαi Vids ∨Zt 0 m X i=1 kX0(s, X0)kαi Vids> R(γ). By (1.1) we have d(X(t)−X0(t)) = A(X(t)) −A(X0(t)) + (−0)B(X(t)) + 0[B(X(t)) −B(X0(t))]dW(t). As before, by (C2) and 0∈[0,1/√2] we have kX(t)−X0(t)k2 H=Zt 0h2V∗hA(X(s)) −A(X0(s)), X(s)−X0(s)iV +k(−0)B(X(s)) + 0[B(X(s)) −B(X0(s))]k2 L2(U,H)ids + 2 Zt 0X(s)−X0(s),(−0)B(X(s)) + 0[B(X(s)) −B(X0(s))]dW(s)H ⩽Zt 0hφ1+κ m X i=1 ρi(X0(s))ikX(s)−X0(s)k2 Hds + 2 Zt 0X(s)−X0(s),(−0)B(X(s)) + 0[B(X(s)) −B(X0(s))]dW(s)H + 2|−0|2Zt 0kB(X(s))k2 L2(U,H)ds. By the product rule we deduce e−Rt 0φ1+κPm i=1 ρi(X0(r))drkX(t)−X0(t)k2 H ⩽2|−0|2Zt 0 e−Rs 0φ1+κPm i=1 ρi(X0(r))drkB(X(s))k2 L2(U,H)ds +Mε(t),(5.12) where M(t) is a continuous real-valued local martingale given by Mε(t) := 2 Zt 0 e−Rs 0φ1+κPm i=1 ρi(X0(r))dr ×hX(s)−X0(s),(−0)B(X(s)) + 0[B(X(s)) −B(X0(s))]dW(s)H. 22 RENHAI WANG, TOM ´ AS CARABALLO, AND NGUYEN HUY TUAN By Proposition 2.1 we can prove that E[M(t∧ς)] = 0. Replacing tby t∧ςand taking the expectation in (5.12), we obtain Ee−Rt∧ς 0φ1+κPm i=1 ρi(X0(r))drkX(t∧ς)−X0(t∧ς)k2 H ⩽2|−0|2EZt∧ς 0 e−Rs 0φ1+κPm i=1 ρi(X0(r))drkB(s, X(s))k2 L2(U,H)ds ⩽2|−0|2EZt∧ς 0kB(s, X(s))k2 L2(U,H)ds⩽C(T, R(γ))|−0|2→0 as →0,(5.13) where C(T, R(γ)) >0 is a constant independent of . Note that there exists another constant C(T, R(γ)) independent of εsuch that Pm i=1 Rt 0ρi(X0(r))dr ⩽C(T, R(γ)), P-a.s.. This along with (5.13) implies limε→ε0EkX(t∧ς)−X0(t∧ς)k2 H= 0. Then by the arguments of (4.5) we know that X(t, X0)→ X0(t, X0) in Hin probability as →0, and thus we complete the proof of (i). (ii) We prove that the union S∈[0,1/√2] ˙ P(H) is tight on H, that is, for any δ > 0, ∈[0,1/√2] and η∈˙ P(H), there exists a compact set Zδindependent of such that η(Zδ)>1−δ. By ∈[0,1/√2], θi1> λ−1 i1φ2and Lemma 4.3, we find that for every X0∈H, there exists k0=k0(X0)∈Nsuch that for all k⩾k0, k−1Zk 0 EhkX(s, X0)kαi0 Vi0ids ⩽C0,(5.14) where C0>0 is a constant independent of X0,kand . For l∈N, we define Yδ l=v∈Vi0:kvkVi0⩽ (δ−1C022l)1/αi0and Zδ l=u∈H:ku−vkH⩽1 2lfor some v∈ Yδ land Zδ:= T∞ l=1 Zδ l. By the compactness of Vi0,→Hwe know that Zδis compact in H. It is sufficient to show η(Zδ)>1−δ. For n∈N, we set Xδ n:= Tn l=1 Zδ l. Then T∞ n=1 Xδ n=Zδand Xδ n+1 ⊆ Xδ n, and hence η(Zδ) = ηT∞ n=1 Xδ n= lim n→∞ηXδ n. Thus, there exists N=N(δ)∈Nsuch that 0 ⩽η(Xδ n)−η(Zδ)⩽δ/3 for all n⩾N. By the invariance of ηand Fubini’s theorem, ZHk−1Zk 0 Pnω∈Ω : X(s, X0)∈ Xδ Nodsη(dX0) = η(Xδ N). This along with the Fatou’s lemma, Markov’s inequality and (5.14) yields η(Xδ N) = lim inf k→∞ ZHk−1Zk 0 Pnω∈Ω : X(s, X0)∈ Xδ Nodsη(dX0) ⩾ZHlim inf k→∞ k−1Zk 0 Pnω∈Ω : X(s, X0)∈ Xδ Nodsη(dX0) ⩾1−ZHlim sup k→∞ k−1Zk 0 Pnω∈Ω : X(s, X0)/∈ Xδ Nodsη(dX0) ⩾1− N X l=1 ZHlim sup k→∞ k−1Zk 0 Pnω∈Ω : X(s, X0)/∈ Zδ lodsη(dX0) ⩾1− N X l=1 ZHlim sup k→∞ k−1Zk 0 Pnω∈Ω : X(s, X0)/∈ Yδ lodsη(dX0) ⩾1− N X l=1 ZHlim sup k→∞ k−1Zk 0 Pnω∈Ω : kX(s, X0)kVi0>(δ−1C022l) 1 αi0odsη(dX0) ⩾1− N X l=1 δ C022lZHlim sup k→∞ k−1Zk 0 E[kX(s, X0)kαi0 Vi0]dsη(dX0)⩾1− N X l=1 δ 22l⩾1−δ 3. Then we know η(Zδ)⩾η(Xδ N)−δ 3>1−δ. By (i) and [30, Theorem 6.1] we complete the proof.  Remark 5.2. Let η k=k−1Rk 0Pε 0,t(X0,·)dt, where P 0,t(X0,·)denotes the law of X(t, 0, X0). Then, as in Theorem 1.4, we know η kconverges (up to a subsequence) weakly to ηas k→ ∞, which is an invariant MEAN ATTRACTORS AND INVARIANT MEASURES OF SPDES 23 probability measure of (P 0,t)t⩾0. Set ¨ P(H) = {η:ηis a weak limt point of η kas k→ ∞}. Since we can prove that the estimates of Lemma 4.3 for X(t, 0, X0)are uniform for ∈[0,1/√2], following the proof of Theorem 1.4, we can prove that for every δ > 0,∈[0,1/√2] and η∈¨ P(H), there exists a compact set Zδ⊆H(independent of ) such that η k(Zδ)>1−δ, and hence η(Zδ)⩾lim sup k→∞ η k(Zδ)⩾ 1−δ. Then Sε∈[0,1/√2] ¨ P(H)is tight on H. 6. Evolution systems of measures of (1.1): nonautonomous case In this subsection we investigate the existence, uniqueness, forward strongly mixing, backward strongly mixing and global exponential mixing for evolution systems of probability measures of nonautonomous (1.1). 6.1. Proof of Theorem 1.8. Lemma 6.1. Let assumptions in Theorem 1.8 hold. Then, for each t∈Rand X0∈H, there exists ζt∈L2(Ω,Ft,P;H), independent of X0, such that lim τ→−∞X(t, τ, X0) = ζtin L2(Ω,Ft,P;H)and E[kX(t, τ, X0)−ζtk2 H]⩽4e[−λi1ϑi1+φ1](t−τ)kX0k2 H+Zτ −∞ e[λi1θi1−φ2](s−τ)φ3(s)ds, τ ⩽t. Proof. Given h > 0, t∈Rand X0∈H, let Z(t) := X(t, τ, X0)−X(t, τ −h, X0) for τ⩽t. Then, by Remark 1.3 with αi1= 2, we obtain E(kZ(t)k2 H)⩽e[−λi1ϑi1+φ1](t−τ)E(kZ(τ)k2 H). Thus, by Lemma 4.3, we see E(kX(t, τ, X0)−X(t, τ −h, X0)k2 H) ⩽2e[−λi1ϑi1+φ1](t−τ)kX0k2 H+e[−λi1θi1+φ2]hkX0k2 H+Zτ τ−h e[−λi1θi1+φ2](τ−s)φ3(s)ds = 2e[−λi1ϑi1+φ1](t−τ)kX0k2 H+e[−λi1θi1+φ2]hkX0k2 H + 2e[−λi1ϑi1+φ1]t+[λi1(ϑi1−θi1)+φ2−φ1]τZτ τ−h e[λi1θi1−φ2]sφ3(s)ds. (6.1) By the conditions on ϑi1,θi1and φ3we derive E[kX(t, τ, X0)−X(t, τ −h, X0)k2 H]→0 as τ→ −∞ and h→+∞. Then, by the completeness of L2(Ω,F,P;H) and a contradiction argument, there exists ζt(X0)∈L2(Ω,F,P;H) such that ζt(X0) = lim τ→−∞X(t, τ, X0) in L2(Ω,F,P;H). Note that, for any τ⩽t∈Rand X0, Y0∈H,E[kX(t, τ, X0)−X(t, τ, Y0)k2 H]⩽e[−λi1ϑi1+φ1](t−τ)kX0−Y0k2 H→0 as τ→ −∞, and hence ζt(X0) = ζt(Y0). Then ζt(X0) is independent of X0. Letting h→ ∞ in (6.1), we complete the proof.  Proof. Proof of Theorem 1.8. By Lemma 6.1 we deduce that, for all t∈Rand X0∈H, there exists ζt∈L2(Ω,F,P;H), independent of X0, such that lim τ→−∞X(t, τ, X0) = ζtin L2(Ω,Ft,P;H). Let ηtbe the law of ζt. Then, for all ϕ∈Lipb(H), we have |(Pτ,tϕ)(X0)−hϕ, ηti|2=|E[ϕ(X(t, τ, X0)) −ϕ(ζt)]|2 ⩽4kϕk2 Lipb e[−λi1ϑi1+φ1](t−τ)kX0k2 H + 4kϕk2 Lipb e[−λi1ϑi1+φ1]te(λi1(ϑi1−θi1)+φ2−φ1)τZτ −∞ e[λi1θi1−φ2]sφ3(s)ds. (6.2) This along with Proposition 4.1 implies (1.13) and lim τ→−∞(Pτ,tϕ)(X0) = RHϕ(y)ηt(dy) for any ϕ∈Cb(H). Letting τ→ −∞ in (Pτ,σϕ)(X0)=(Pτ,t(Pt,σϕ))(X0) for any σ⩾t⩾τ, and by the Feller property of (Pτ,t)t⩾τin Lemma 4.2, we find RHϕ(y)ησ(dy) = RHPt,σϕ(y)ηt(dy). So, {ηt}t∈Ris an evolution system of probability measures of (Pτ,t)t⩾τon H. 24 RENHAI WANG, TOM ´ AS CARABALLO, AND NGUYEN HUY TUAN Let ϕn(x) := kxk2 H∧nfor n∈Nand x∈H. Then ϕn∈Cb(H). By Lemma 4.3, we see hϕn, ηti⩽|(Pτ,tϕn)(0) −hϕn, ηti|+Ekφ(t, τ, 0)k2 H ⩽|(Pτ,tϕn)(0) −hϕn, ηti|+Zt −∞ e[λi1θi1−φ2](s−t)φ3(s)ds. Letting τ→ −∞ in the above inequality, we find ZHkxk2 H∧nηt(dx)⩽Zt −∞ e[λi1θi1−φ2](s−t)φ3(s)ds. (6.3) Letting n→ ∞ in (6.3), by Fatou’s lemma we find (1.12). Note that for any {µt}t∈R⊆ P(H) satisfying RHkxk2 Hµt(dx)⩽Rt −∞ e[λi1θi1−φ2]sφ3(s)ds, we have dP W(Qτ,tµτ, ηt) = sup ϕ∈Lipb(H),kϕkLipb ⩽1ZHZH E[ϕ(X(t, τ, X0)) −ϕ(X(t, τ, Y0))]µτ(dX0)ητ(dY0) ⩽ZHZH E[kX(t, τ, X0)−X(t, τ, Y0)k2 H]µτ(dX0)ητ(dY0)1/2 ⩽√2e1 2[−λi1ϑi1+φ1](t−τ)ZHkX0k2 Hµτ(dX0) + ZHkY0k2 Hητ(dY0)1/2 ⩽2e1 2[−λi1ϑi1+φ1](t−τ)Zτ −∞ e[λi1θi1−φ2](s−τ)φ3(s)ds1/2 . This proves (1.14). The uniqueness of {ηt}t∈Rcan be proved similarly.  7. Applications to models In this section we will present two typical examples (do not fall within the previous frameworks in the literature) for our abstract results. From now on, we let Obe a bounded open subset of RNwith smooth boundary ∂O. Lemma 7.1. For p∈[1,∞)and a, b ∈R, there exists a constant γ > 0, depending only on p, such that (|a|p−1a−|b|p−1b)(a−b)⩾γ|a−b|p+1,(7.1a) (a|a|p−1−b|b|p−1)(a−b)⩾1 2(|a|p−1+|b|p−1)(a−b)2.(7.1b) Proof. We only focus on (7.1b). Rearranging (a|a|p−1−b|b|p−1)(a−b), as in [36], to find (a|a|p−1−b|b|p−1)(a−b)=(a|a|p−1−b|a|p−1+b|a|p−1−a|b|p−1+a|b|p−1−b|b|p−1)(a−b) = (|a|p−1+|b|p−1)(a−b)2+ (b|a|p−1−a|b|p−1)(a−b) = (|a|p−1+|b|p−1)(a−b)2+ (|a|p−1+|b|p−1)ab −a2|b|p−1−b2|a|p−1 =1 2(|a|p−1+|b|p−1)(a−b)2+1 2|a|p+1 +|b|p+1 −|a|p−1b2−|b|p−1a2 ⩾1 2(|a|p−1+|b|p−1)(a−b)2+1 2|a|p+1 +|b|p+1 −p−1 p+ 1|a|p+1 −2 p+ 1bp+1 −p−1 p+ 1|b|p+1 −2 p+ 1ap+1 ⩾1 2(|a|p−1+|b|p−1)(a−b)2.  MEAN ATTRACTORS AND INVARIANT MEASURES OF SPDES 25 7.1. Fractional (s, p)-Laplacian equations. Fractional calculus has many applications in modeling complex phenomena arriving from a wide range of fields within finance, engineering, physics, chemistry, biology, and others, see e.g., [2]. In particular, the solutions and their associated dynamical behavior of SPDEs with fractional Laplacian driven by nonlinear white noise have been studied in [51]. In contrast, both well-posedness and dynamics of SPDEs with fractional (s, p)-Laplacian driven by nonlinear white noise have not been studied in the literature. However, our abstract frameworks can be used to study the well-posedness and dynamics of a class of fractional (s, p)-Laplacian equations driven by nonlinear white noise for any s∈(0,1) and p⩾2. For s∈(0,1) and p⩾2, we define the fractional (s, p)-Laplacian operator by, for x∈RN, −(−∆)s pu(x) = −C(N, p, s)ZRN |u(x)−u(y)|p−2u(x)−u(y) x−yN+ps dy =C(N, p, s) 2ZRN |u(x)−u(x+y)|p−2u(x+y)−u(x)+|u(x)−u(x−y)|p−2u(x−y)−u(x) |y|N+ps dy, where the normalized constant C(N, p, s) = s4sΓps+p+N−2 2 πN 2Γ(1−s)is defined by the Gamma function. If p= 2, then the fractional (s, p)-Laplacian operator reduced to the standard fractional Laplacian operator. The fractional Sobolev space Ws,p(RN) is defined by Ws,p(RN) = nu∈Lp(RN) : RRNRRN|u(x)−u(y)|p |x−y|N+ps dxdy < ∞o. The norm of Ws,p(RN) is defined by kukWs,p(RN)=ku(x)kp Lp(RN)+RRNRRN|u(x)−u(y)|p |x−y|N+ps dxdy1/p .The so-called Gagliardo semi-norm of Ws,p(RN) is defined by kuk˙ Ws,p(RN)=RRNRRN|u(x)−u(y)|p |x−y|N+ps dxdy1/p Then kukp Ws,p(RN)=ku(x)kp Lp(RN)+kukp ˙ Ws,p(RN). Due to the non-local nature of the fractional (s, p)- Laplacian, we introduce the spaces Lr(O) := u∈Lr(RN) : u= 0 a.e. on RN\Oand Ws,p(O) := u∈Ws,p(RN) : u= 0 a.e. on RN\Ofor s∈(0,1), p⩾2 and r⩾1. Recall the fractional Poincar´e inequality [16, Theorem 6.5]: kukp ˙ Ws,p(O)=ZOZO |u(x)−u(y)|p |x−y|N+ps dxdy ⩾cZO|u(x)|pdx, ∀u∈Ws,p(O),(7.2) where cis a positive constant depending only on p,s,Nand O. Then k·k ˙ Ws,p(O)is an equivalent norm of k·kWs,p(O). For convenience we agree k·kWs,p(O)=k·k˙ Ws,p(O). Given s∈(0,1), p⩾2, ν > 0 and τ∈R, we consider the fractional (s, p)-Laplacian equation on O:            ∂ ∂tu(t) + ν(−∆)s pu(t) = m X i=2 Fi(t, x, u(t)), t > τ, u(t, x) = 0, x ∈RN\O, t > τ, u(τ, x) = u0(x), x ∈ O. (7.3) Here Fi:R×O×Ω×R→Ris a continuous nonlinear function satisfying Fi(t, x, ω, s)s⩽−3 2θi|s|αi+ψi(t, x, ω),(7.4) |Fi(t, x, ω, s)|⩽˙ ψi(t, x, ω)|s|αi−1+¨ ψi(t, x, ω),(7.5) (Fi(t, x, ω, s1)−Fi(t, x, ω, s2))(s1−s2)⩽−ϑi|s1−s2|αi+... ψi(t, x, ω)|s1−s2|2,(7.6) where θi, ϑi>0, αi⩾2, ψi∈L1([τ, τ +T]×Ω, dt ×P;L1(O)), ˙ ψi∈L αi αi−1([τ, τ +T]×Ω, dt ×P;L∞(O)), ¨ ψi∈L αi αi−1([τ, τ +T]×Ω, dt ×P;L αi αi−1(O)) and ... ψi∈L∞([τ, τ +T]×Ω, dt ×P;L∞(O)) are Ft-adapted nonnegative processes. Let H:= L2(O), V1:= Ws,p(O), α1:= p,Vi:= Lαi(O), i= 2, . . . , m ∈N, and V:= Ti=1,2,...,m Vi. Then we get V⊆H≡H∗⊆V∗and Vi⊆H≡H∗⊆V∗ i. Define A1=−(−∆)s α1:V1→V∗ 1by V∗ 1hA1(v), ξiV1=−νC(N, α1, s) 2ZRNZRN |v(x)−v(y)|α1−2v(x)−v(y)ξ(x)−ξ(y) |x−y|N+α1sdxdy, v, ξ ∈V1. 32 RENHAI WANG, TOM ´ AS CARABALLO, AND NGUYEN HUY TUAN Example of Bin (7.14) Define a mapping B0:V→Hby B0(v) = v+ sin(v) + εB(v,h) with h∈D(A). Then for any v1,v2,v∈V, we have kB0(v)k2 H⩽c+ckvk2 H+εckA(h)k2 Hkvk2 Vand kB0(v1)−B0(v2)k2 H⩽ckv1−v2k2 H+εckA(h)k2 Hkv1−v2k2 V. Let e0∈Hwith ke0kH= 1, and U:= span{e0}. Given v∈V, define B:V→ L2(U, H) by B(v)u=hu,e0iUB0(v), u∈U. Then kB(v)kL2(U,H)=kB0(v)kHand kB(v1)−B(v2)kL2(U,H)= kB0(v1)−B0(v2)kH. By taking εsmall enough, we find that Bsatisfies the conditions in Example (7.9). Acknowledgements Renhai Wang supported by the National Natural Science Foundation of China (12301299), the research fund of Qianshixinmiao[2022]B16, and the Natural Science Research Project of Guizhou Provincial Department of Education (No. QJJ[2023]011). 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Caraballo: [email protected] (Nguyen Huy Tuan) Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam Faculty of Technology, Van Lang University, Ho Chi Minh City, Vietnam Email address, N.H. Tuan: [email protected]