# Ergodicity of Stochastic Burgers’ System with Dissipative Term

Authors: Guoli Zhou; Boling Guo; Daiwen Huang; Yongqian Han
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A 2-dimensional stochastic Burgers equation with dissipative term
perturbed by Wiener noise is considered. The aim is to prove the well-posedness,
existence, and uniqueness of invariant measure as well as strong
law of large numbers and convergence to equilibrium.


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## Body

## 1. Introduction

The paper is concerned with the 2-dimensional Burgers equation in a bounded domain with Wiener noise as the body forces like this(1)du=(νΔu+(u·∇)u)dt+dW,on[0,T]×D,u(t,x)=0,t∈[0,T],x∈∂D,u(0,x)=u0(x),x∈D,
where u(t,x)=(u1(t,x),u2(t,x)) is the velocity field, ν>0 is viscid coefficient, Δ denotes the Laplace operator, ∇ represents the gradient operator, W stands for the Q-Wiener process, and D is a regular bounded open domain of ℝ2. Burgers equation has received an extensive amount of attention since the studies by Burgers in the 1940s (and it has been considered even earlier by Beteman [1] and Forsyth [2]). But it is well known that the Burgers’ equation is not a good model for turbulence since it does not perform any chaos. Even if a force is added to equation, all solutions will converge to a unique stationary solution as time goes to infinity. However, if the force is a random one, the result is completely different. So, several authors have indeed suggested to use the stochastic Burgers’ equation to model turbulence, see [3–6]. The stochastic equation has also been proposed in [7] to study the dynamics of interfaces.So far, most of the monographs concerning the equation focus on one-dimensional case, for example, Bertini et al. [8] solved the equation with additive space-time white noise by an adaptation of the Hopf-cole transformation. Da Prato et al. [9] studied the equation via a different approach based on semigroup property for the heat equation on a bounded interval. The more general equation with multiplicative noise was considered by Da Prato and Debussche [10]. With a similar method, Gyöngy and Nualart [11] extended the Burgers equation from bounded interval to real line. A large deviation principle for the solution was obtained by Gourcy [12]. Concerning the ergodicity, an important paper by Weinan et al. [13] proved that there exists a unique stationary distribution for the solutions of the random inviscid Burgers equation, and typical solutions are piecewise smooth with a finite number of jump discontinuities corresponding to shocks. For model with jumps, Dong and Xu [14] proved that the global existence and uniqueness of the strong, weak, and mild solutions for a one-dimensional Burgers equation perturbed by Lévy noise. When the noise is fractal, Wang et al. [15] get the well-posedness.The main aim in our paper is to study the large time behavior of stochastic system. There are lots of the literature about the topic (see [16–20]).Burgers system is a well-known model for mechanics problems. But as far as we know, there are no results about the long-term behavior of stochastic Burgers’ system. We think that the difficulty lies in the fact that the dissipative termΔu cannot dominate the nonlinear term (u·∇)u. However, in many practical cases, we cannot ignore the energy dissipation and external forces, especially considering the long-term behavior. Therefore, we introduce dissipative term f(u) and study the ergodicity of the following equation:
(2)du=[Δu+(u·∇)u-f(u)]dt+dW,on[0,T]×D,u(t,x)=0,t∈[0,T],x∈∂D,u(0,x)=u0(x),x∈D,
where f(u)=ϑ|u(t,x)|2u(t,x),  ϑ>0,|·| denote the absolute value or norm for the real number or two-dimensional vector, respectively.We believe that our work is new and is worth researching. The methods and results in this paper can be applied to stochastic reaction diffusion equations and stochastic real valued Ginzburg Landau equation in high dimensions. But we cannot extend our result to dynamical systems with state-delays. Since in order to show the existence of an invariant measure, we should consider the segments of a solution. In contrast to the scalar solution process, the process of segments is a Markov process. We show that the process of segments is also Feller and that there exists a solution of which the segments are tight. Then, we apply the Krylov-Bogoliubov method. Since the segment process has values in the infinite-dimensional spaceC([-r,0],H), boundedness in probability does not automatically imply tightness. For solution processes of infinite-dimensional equations, one often uses compactness of the orbits of the underlying deterministic equation to obtain tightness. For an infinite-dimensional formulation of the functional differential equation, however, such a compactness property does not hold. For ergodicity of stochastic delay equations, we can see [21]. We believe that stochastic Burgers’ system with state-delays is a very interesting problem.In order to study ergodicity of problem (2), we use a remarkable dissipativity property of the stochastic dynamic to obtain the existence of the invariant measure. For uniqueness, we try to use the method from [22] to prove that the distributions P(t,x,·) induced by the solution are equivalent. It is well known that the equivalence of the distributions implies uniqueness, a strong law of large numbers, and the convergence to equilibrium.The remaining of this paper is organized as follows. Some preliminaries are presented in Section2, the local existence and global existence are presented, respectively, in Sections 3 and  4. In Section 5, we obtain the existence and uniqueness of the invariant measure as well as strong law of large numbers, and convergence to equilibrium. As usual, constants C may change from one line to the next; we denote by Ca a constant which depends on some parameter a.


## 2. Preliminaries on the Burgers Equation

Letu(t,x)=(u1(t,x),u2(t,x)) be a row vector valued function on [0,∞)×ℝ2. And it denotes the following:
(3)|u|2≔∑i=12|ui|2,∂iuj≔∂uj∂xi,i,j=1,2.
Let [C∞(D)]2 be infinitely differentiable 2-dimensional vector field on D, and let [C0∞(D)]2 be infinitely differentiable 2-dimensional vector field with compact support strictly contained in D. We denote by Hα the closure of [C∞(D)]2 in [Hα(D)]2, whose norms are denoted by ∥·∥Hα, when α≠0. Let H01,H be the closure of [C0∞(D)]2 in [H1(D)]2 and [L2(D)]2 whose norms are denoted by ∥·∥H1 and ∥·∥H, respectively. Without confusion, set 〈·,·〉 as the inner product in H or L2(D). For p>0, let ∥·∥Lp be the norm of vector filed in Lebesgue spaces [Lp(D)]2. |·|Hα represents the norm in the usual sobolev spaces Hα(D) for real valued functions on D and α∈ℝ; |·|Lp stands for the norm in the usual Lebesgue spaces Lp(D) for real valued functions on D. Denote A≔-Δ; then A:D(A)⊂H→H and D(A)=[H2(D)]2∩H01. Since H01 coincides with D(A1/2), we can endow H01 with the norm ∥u∥H1=∥A1/2u∥H. The operator A is positive self-adjoint with compact resolvent; we denote by 0<α1≤α2≤⋯ the eigenvalues of A, and by e1,e2,… the eigenvectors which is a corresponding complete orthonormal system in H satisfying
(4)(i)ei∈[C0∞(D)]2,(ii)|ei(x)|≤C,|∇ei(x)|≤Cαi,(ii)|ei(x)|≤C,x∈D,i=1,2,….,
for some positive constant C. We remark that ∥u∥H12≥α1∥u∥H2. We define the bilinear operator B(u,v):H1×H1→H-1 as
(5)〈B(u,v),z〉=∫Dz(x)·(u(x)·∇)v(x)dx,
for all z∈H1. Then, (2) is equivalent to the following abstract equation:
(6)du(t)+[Au(t)+B(u(t),u(t))+f(u(t))]dt=dW(t).W is the Q Wiener process having the following representative:
(7)W(t)=∑n=1∞λnenβn(t),t∈[0,T],
in which ∑n=1∞‍λn<∞ and βk are a sequence of mutually independent 1-dimensional Brownian motions in a fixed probability space (Ω,ℱ,P) adapted to a filtration {ℱt}t≥0.It can be derived from [23] that the solution to the linear problem corresponding to (2) with the following initial condition:
(8)du=Δudt+dW,u(t,x)=0,t∈[0,T],x∈∂D,u(0,x)=u0(x),x∈D,
is unique, and when u0=0, it has the form of
(9)WA(t)=∫0te(t-s)AdW(s).
Let
(10)v(t)=u(t)-WA(t),t≥0,
then u is a solution to (2) if and only if it solves the following evolution equation:
(11)∂vdt+Av+B(v+WA,v+WA)+f(v+WA)=0,v(t,x)=0,t∈[0,T],x∈∂D,v(0)=u0.
So, we see that when w∈Ω is fixed, this equation is in fact a deterministic equation. From now on, we will study the equation of the form (11) to get the existence and uniqueness of the solution a.s. w∈Ω.


## 3. Local Existence in Time

Definition 1 (see Definition 5.1.1 in [24]).
We say a(ℱ(t))t≥0 adapted process v(t) is a mild solution to (11), if v(t)∈C([0,T];H01) and it satisfies
(12)v(t)=etAv0+∫0te(t-s)AB(v+WA,v+WA)ds-∫0te(t-s)Af(v+WA)ds,t∈[0,T].Lemma 2.
For anyθ∈(0,1), if ∑i=1∞λi(αi)θ<∞, then A1/2WA has a version which is α-Hölder continuous with respect to t∈[0,T],  x∈D with any α∈]0,θ/2[.Proof.
LetT>0 and s,t∈[0,T]; then
(13)E|A1/2WA(t,x)-A1/2WA(s,x)|2=∑i=1∞λi∫st|A1/2S(t-τ)ei(x)|2ds+∑i=1∞λi∫0s|A1/2[S(t-τ)-S(s-τ)]ei(x)|2dτ=:I1(t,s,x)+I2(t,s,x).
Then, we have
(14)I1(t,s,x)≤C∑i=1∞λiαi∫ste-2(t-τ)αidτ=C∑i=1∞λiαi(1-e-2(t-s)αi2αi)≤C∑i=1∞λi(αi)θ|t-s|θ,I2(t,s,x)≤12C∑i=1∞λiαi∫0s|[e-(t-τ)αi-e-(s-τ)αi]|2dτ=C∑i=1∞λiαi12αi[(e-(t-s)αi-1)2-(e-tαi-e-sαi)2]≤C∑i=1∞λi(αi)θ|t-s|θ.
So, by the estimate of I1 and I2, we arrive at
(15)E|A1/2WA(t,x)-A1/2WA(s,x)|2≤C∑i=1∞λi(αi)θ|t-s|θ.
For t∈[0,T],  x,y∈D, we get
(16)E|A1/2WA(t,x)-A1/2WA(t,y)|2=∑i=1∞λiαi∫0te-2αi(t-s)|ei(x)-ei(y)|2ds≤∑i=1∞λi|ei(x)-ei(y)|2≤∑i=1∞λi(αi)θ|x-y|θ.
Therefore,
(17)E|A1/2WA(t,x)-A1/2WA(s,y)|2≤C(|t-s|θ+|x-y|θ).
As A1/2WA(t,x)-A1/2WA(s,y) is a Gaussian random variable, we obtain
(18)E|A1/2WA(t,x)-A1/2WA(s,y)|2m≤C(|t-s|mθ+|x-y|mθ),
for m=1,2,… By Kolmogorov’ test theorem, we get the conclusion.Remark 3.
An example of the noise satisfying condition of Lemma2 is
(19)dW(t)=∑n=1∞λnendβn(t),
where {βn} is a sequence of independent 1-dimensional Brownian motion, and {λn} satisfies
(20)λn=n-(1+2θ),αn=n∀n∈ℕ.
It is so because the eigenvalues αn of the operator A, in 2-dimensional space, behave like n.Remark 4.
Another example of stochastic noise satisfying Lemma2 is
(21)A-γLdW(t),
where W(t)=∑n=1∞‍enβn(t),  L is an isomorphism in H, and
(22)γ≥12+θ.To prove the local existence of the solution of (1) in sense of Definition 1, we introduce the space ℬm defined by
(23)ℬm={v:v∈C([0,T*];H01),∥v∥H1≤m,∀t∈[0,T*]},
where T*≥0 which in fact is a stopping time and m>0, p>0.Lemma 5.
Foru0=(u1(0),u2(0)),∥u0∥H1<m, and ui(0) is adapted to ℱ0, i=1,2; then there exists a unique mild solution v in sense of Definition 1 to (11) in ℬm.Proof.
Choose av in ℬm, and set
(24)ℒ(v)≔e-tAu0+∫0te-(t-s)A[(v+WA)·∇](v+WA)ds-∫0te-(t-s)Af(v+WA)ds.
Then,
(25)∥ℒ(v)∥H1≤∥e-tAu0∥H1+∥∫0t‍e-(t-s)A[(v+WA)·∇](v+WA)ds∥H1+∥∫0te-(t-s)Af(v+WA)ds∥H1.
For the second term on the right hand side of (25),
(26)∥e-(t-s)A[(v+WA)·∇](v+WA)∥H1=∥e-(t-s)A[u·∇]u∥H1≤12|e-(t-s)A∂1(u1)2|H1+12|e-(t-s)A∂2(u2)2|H1+|e-(t-s)Au2∂2u1|H1+|e-(t-s)Au1∂1u2|H1≔I1+I2+I3+I4.
In the following, we will estimate Ii, respectively, i=1,2,3,4. Since {e-tA}t≥0 is contraction on Lp(D),p≥1, it is known that
(27)|e-tAz|Ws2,r≤C1t(s1-s2)/2|z|Ws1,r,
for all z∈Ws1,r(D),  s1,s2∈ℝ,  s1≤s2,  r≥1, and C1 only depends on s1,  s2, and r. Before calculating each Ii, we outline the Sobolev embedding principle in fractional Sobolev spaces as follows:
(28)Wη1,p1(D)⊂Wη2,q1(D),
when
(29)1p1-1n(η1-η2)≤1q1≤1p1,
where n is the dimension of the spatial. Let η1=3/4,  p1=2,  η2=1/4, q1=4 satisfying (29) such that
(30)W3/4,2(D)⊂W1/4,4(D).
ForI1, by (27) and Theorem A.8 in [25], we get
(31)I1≤C1|t-s|-7/8|∂1(u1)2|H-3/4=C1|t-s|-7/8|A1/8(u1)2|H=C1|t-s|-7/8|2u1A1/8u1+R|H,
where
(32)R=A1/8(u1)2-2A1/8u1,
satisfying
(33)|R|H≤|A1/16u1|L42≤|u1|H12.
The last inequality follows by (30). For the other term added to R, we have
(34)|2u1A1/8u1|H≤|u1|L42+|A1/8u1|L42≤2|u1|H12.
So, by (31)–(34), we have
(35)I1≤3C1|t-s|-7/8|u1|H12.
Similarly, we get for I2 that
(36)I2≤3C1|t-s|-7/8|u2|H12.
For I3, by Theorem A.8 in [25], we get
(37)I3≤|e-(t-s)Au2A1/2u1|H1=|e-(t-s)A[A1/4(u2A1/4u1)-(A1/4u1)(A1/4u2)-R1]|H1,
where
(38)R1=A1/4(u2A1/4u1)-[A1/4u2][A1/4u1]-u2A1/2u1.
ForR1, we have
(39)|e-(t-s)AR1|H1≤C1|t-s|-1/2|R1|H≤C1|t-s|-1/2|A1/4u1|L4·|A1/4u2|L4≤C1|t-s|-1/2(|u1|H12+|u2|H12).
For the first term on the right hand side of (37), by (27), we have
(40)|e-(t-s)AA1/4(u2A1/4u1)|H1=|e-(t-s)AA3/4(u2A1/4u1)|H≤C1|t-s|-3/4|u2A1/4u1|H≤C1|t-s|-3/4(|u2|L42+|A1/4u1|L42)≤C1|t-s|-3/4(|u2|H12+|u1|H12).
For the second term on the right hand side of (37), by (27), we obtain
(41)|e-(t-s)A[A1/4u2·A1/4u1]|H1≤C1|t-s|-1/2|A1/4u2·A1/4u1|H≤C1|t-s|-1/2(|u1|H12+|u2|H12).
From (37) to (41), we get for I3 that
(42)I3≤C(|t-s|-1/2+|t-s|-3/4)(|u1|H12+|u2|H12).
Analogously, for I4, we get
(43)I4≤C(|t-s|-1/2+|t-s|-3/4)(|u1|H12+|u2|H12).
By (26), (35), (36), (42), and (43), we have
(44)∥e-(t-s)A[(v+WA)·∇](v+WA)∥H1≤C(|t-s|-1/2+|t-s|-3/4+|t-s|-7/8)×(|u1|H12+|u2|H12).
As u=v+WA, by (44), for t≤T*, we have
(45)∫0tds∥e-(t-s)A[(v+WA)·∇](v+WA)∥H1≤C(t1/8+t1/4+t1/2)(supt∈[0,T*]∥v∥H12+supt∈[0,T]∥WA∥H12).
Since by Lemma 2,
(46)supt∈[0,T]∥WA∥H12<∞.
For the last term on the right hand side of (25), we have
(47)∥e-(t-s)Af(v+WA)∥H1≤C|t-s|-1/2(∥v+WA∥L63)≤C|t-s|-1/2(∥WA∥H13+∥v∥H13).
Therefore,
(48)∥∫0t‍e-(t-s)Af(v+WA)ds∥H1≤C(1+m3)∫0t|t-s|-1/2ds≤C(1+m3)T*1/2.
So by (25), (45), and (48), when T* is small enough,
(49)∥ℒ(v)∥H1≤m.
For each v1,v2∈ℬm, set ui=vi+WA, i=1,2. To simplify the notation in the following calculation, we denote ui=(ui1,ui2), i=1,2. Then,
(50)ℒ(v1)-ℒ(v2)=∫0te-(t-s)A[(u1·∇)u1-(u2·∇)u2]ds+∫0te-(t-s)A[f(u1)-f(u2)]ds.
So,
(51)∥ℒ(v1)-ℒ(v2)∥H1≤∫0t∥e-(t-s)A[(u1·∇)u1-(u2·∇)u2]∥H1ds+∫0t∥e-(t-s)A[f(u1)-f(u2)]∥H1ds.
In order to simplify the notation, we set
(52)(u1·∇)u1-(u2·∇)u2=(f1+f2,f3+f4),
where
(53)f1=12∂1[(u11)2-(u21)2],f2=u12∂2u11-u22∂2u21,f3=12∂2[(u12)2-(u22)2],f4=u11∂1u12-u21∂1u22.
Then, we estimate fi, i=1,2,3,4, respectively. For f1, we have
(54)|e-(t-s)Af1|H1=|e-(t-s)AA1/2[(u11-u21)(u11+u21)]|H1≤C|t-s|-7/8|A1/8[(u11)2-(u21)2]|H=C|t-s|-7/8|A1/8[(u11-u12)(u11+u12)]|H=C|t-s|-7/8|[A1/8(u11-u12)](u11+u12)|H+C|t-s|-7/8|[A1/8(u11+u12)](u11-u12)+R2|H.
We first consider
(55)|R2|H≤C|A1/16(u11-u21)|L4·|A1/16(u11+u21)|L4≤C|u11-u21|H1·|u11+u21|H1.
For the other term added to R2,
(56)|[A1/8(u11+u21)](u11-u21)|H≤|u11+u21|H1·|u11-u21|H1.
By (54)–(56),
(57)|e-(t-s)Af1|H1≤C|t-s|-7/8|u11-u21|H1·|u11+u21|H1.
Analogously, for f3,
(58)|e-(t-s)Af3|H1≤C|t-s|-7/8|u11-u21|H1·|u11+u21|H1.
For f2, by (53), we have
(59)|e-(t-s)Af2|H1=|e-(t-s)A(u12∂2u11-u22∂2u21)|H1≤|e-(t-s)A(u12(∂2u11-∂2u21))|H1+|e-(t-s)A((u12-u22)∂2u21)|H1.
For the first term on the right hand side of (59), we have
(60)|e-(t-s)A(u12(∂2u11-∂2u21))|H1≤|e-(t-s)A(u12A1/2(u11-u21))|H1=|e-(t-s)A{A1/4[u12A1/4(u11-u21)]-[A1/4u12,A1/4(u11-u21)]-R3}|H1.
For R3,
(61)|e-(t-s)AR3|H1≤|t-s|-1/2|R3|H≤|t-s|-1/2|A1/4u12|L4·|A1/4(u11-u21)|L4=|t-s|-1/2|u12|H1·|u11-u21|H1.
For the first term on the right hand side of (60), we arrive at
(62)|e-(t-s)AA1/4[u12A1/4(u11-u21)]|H1=|e-(t-s)AA3/4[u12A1/4(u11-u21)]|H≤|t-s|-3/4|u12A1/4(u11-u21)|H≤|t-s|-3/4|u12|L4·|A1/4(u11-u21)|L4≤|t-s|-3/4|u12|H1·|u11-u21|H1.
For the second term on the right hand side of (60), we obtain
(63)|e-(t-s)A[(A1/4u12)(A1/4(u11-u21))]|H1≤|t-s|-1/2|[A1/4u12]·[A1/4(u11-u21)]|H≤|t-s|-1/2|A1/4u12|L4·|A1/4(u11-u21)|L4≤|t-s|-1/2|u12|H1·|u11-u21|H1.
By (59)–(63), we get for f2 that
(64)|e(t-s)Af2|H1≤C(|t-s|-1/2+|t-s|-3/4)×(∥u1∥H1+∥u2∥H1)∥u1-u2∥H1.
Similarly, we get for f4 that
(65)|e(t-s)Af4|H1≤C(|t-s|-1/2+|t-s|-3/4)×(∥u1∥H1+∥u2∥H1)∥u1-u2∥H1.
By (52), (53), (57), (58), (64), and (65), we have
(66)∥e-(t-s)A[(u1·∇)u1-(u2·∇)u2]∥H1≤∑i=14|e-(t-s)Afi|H1≤C(|t-s|-1/2+|t-s|-3/4+|t-s|-7/8)×(∥u1∥H1+∥u2∥H1)∥u1-u2∥H1≤C(2m+1)(|t-s|-1/2+|t-s|-3/4+|t-s|-7/8)×∥v1-v2∥H1.
For the second term on the right hand side of (51), we have
(67)f(u1)-f(u2)=(h1,h2),
where
(68)h1=|u1|2u11-|u2|2u21,h2=|u1|2u12-|u2|2u22.
Then,
(69)|e(t-s)Ah1|H1≤C|t-s|-1/2||u1|2u11-|u2|2u21|H≤C|t-s|-1/2||u1|2u11-|u2|2u11|H+C|t-s|-1/2||u2|2u11-|u2|2u21|H≤C|t-s|-1/2||v1-v2|·(|v1|+|v2|+2|WA|)·|u11||H+C|t-s|-1/2||u2|2·|v11-v21||H≤C|t-s|-1/2∥v1-v2∥L4·|(|v1|+|v2|+2|WA|)·|u11||L4+C|t-s|-1/2∥v1-v2∥L4·∥u2∥L42≤C|t-s|-1/2∥v1-v2∥L4(∥v1∥L82+∥v2∥L82+1)≤C|t-s|-1/2∥v1-v2∥H1(∥v1∥H12+∥v2∥H12+1).
Similarly, we can get the same estimate for h2. So, we have
(70)∫0t∥e(t-s)A[f(u1)-f(u2)]∥H1ds≤C(1+m2)T*1/2supt∈[0,T*]∥v1(t)-v2(t)∥H1.
By (51), (66), and (70), we have
(71)∥ℒ(v1)-ℒ(v2)∥H1≤C[T*1/2+T*1/4+T*1/8]·(supt∈[0,T*]∥v1-v2∥H1).
By (49), (71), and fixed point principle, we get the conclusion.Remark 6.
By making some minor modifications in the proof of Lemma5, we can see that the conclusion in Lemma 5 is also true for (1). Our original aim is to get the global well-posedness of (1), but we find that the dissipative term Δu cannot dominate the nonlinear term (u·∇)u. So, we introduce the dissipative term |u|2u which will also play an important role in obtaining the ergodicity.


## 4. Global Existence

Theorem 7.
With conditions in Lemma2, for v∈C([0,T];H01) satisfying (12), when ϑ>1/16, one has
(72)∥v∥H1≤(CT+∥v0∥H12)eCT.
Subsequently, one gets the existence of the global solution belonging to C([0,T];H01).Proof.
Let{un0}n≥1 be a sequence of vectors which satisfies un0=(un0,1,un0,2) and un0,i∈C0∞(D),  i=1,2,n≥1, such that
(73)un0⟶u0,asn⟶∞,
in sense of ∥·∥H1. Let {Wn}n≥1 be a sequence of regular process, such that
(74)Aa/2WAn≔Aa/2∫0te(t-s)AdWn(s)⟶Aa/2WA(t),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhasn→∞,
in C(T×D) when a=0 or a=1. For h=(h1,h2), hi∈C([0,T]×D;ℝ),  ∥h∥C(T×D)≔∑i=12|hi|C(T×D), where |hi|C(T×D)=sup(t,x)∈[0,T]×D|hi|. Then, by (74), we have
(75)sup{n≥1}∥WAn∥C(T×D)<∞,(76)sup{n≥1}supt∈[0,T]|A1/2WAn|<∞.
If vn satisfies
(77)vn=etAun0+∫0te(t-s)A[(vn+WA)·∇](vn+WA)ds-∫0te(t-s)Af(vn+WA),
then, vn is regular, such that
(78)∂vn∂t+Avn+B(vn+WAn,vn+WAn)+f(vn+WAn)=0.
Taking inner product with respect to vn in (78), we have
(79)〈∂vn∂t,vn〉+〈Avn,vn〉+〈B(vn+WAn,vn+WAn),vn〉+〈f(vn+WAn),vn〉=0.
For simplicity, we calculate the third term on the left hand side of (79) first as follows:
(80)〈B(vn+WAn,vn+WAn),vn〉=〈(vn1+WA,1n)∂1(vn1+WA,1n),vn1〉+〈(vn2+WA,2n)∂2(vn1+WA,1n),vn1〉+〈(vn1+WA,1n)∂1(vn2+WA,2n),v2n〉+〈(vn2+WA,2n)∂2(vn2+WA,2n),vn2〉=I1+I2+I3+I4,
where WAn=(WA,1n,WA,2n). For I1, we have
(81)I1=〈(vn1+WA,1n)∂1(vn1+WA,1n),vn1〉=〈vn1∂1vn1,vn1〉+〈WA,1n∂1vn1,vn1〉+〈vn1∂1WA,1n,vn1〉+〈WA,1n∂1WA,1n,vn1〉.
In the following, we estimate the four terms for I1, respectively. For the first term,
(82)〈vn1∂1vn1,vn1〉=∫D(vn1)2∂1vn1dx=∫D∂1[(vn1)33]dx=0.
For the second term, by (75), we have
(83)〈WA,1n∂1vn1,vn1〉≤C|vn1|H2+ɛ∫D(∂1vn1)2dx≤C|vn1|H2+ɛ|vn1|H12.
similarly, for the third term,
(84)|〈vn1∂1WA,1n,vn1〉|=|∫D(vn1)2∂1WA,1ndx|=|∫DWA,1n∂1(vn1)2dx|≤C|∫Dvn1∂1vn1dx|≤C|vn1|H2+ɛ|vn1|H12.
For the last term, by (75) and (76),
(85)|〈WA,1n∂1WA,1n,vn1〉|≤C|∫D∂1vn1dx|≤C+ɛ|vn1|H12.
By (81)–(85), it follows that
(86)I1≤C(1+∥vn∥H2)+4ɛ∥vn∥H12.
Similarly,
(87)I4≤C(1+∥vn∥H2)+4ɛ∥vn∥H12.
For I3,
(88)I3=〈vn1∂1vn2,vn2〉+〈vn1∂1WA,2n,vn2〉+〈WA,1n∂1vn2,vn2〉+〈WA,1n∂1WA,2n,vn2〉.
For the first term on the right hand side of (88), we deduce that
(89)|〈vn1∂1vn2,vn2〉|=12|∫Dvn1∂1(vn2)2dx|=12|∫D∂1vn1·(vn2)2dx|≤12|vn2|L42·|vn1|H1≤14ϵ|vn2|L44+14ϵ|vn1|H12,
where ϵ>0. For the second term on the right hand side of (88), we have
(90)|〈vn1∂1WA,2n,vn2〉|≤ɛ∥vn∥H12+C∥vn∥H2.
Analogously, for the third term on the right hand side of (88), we see that
(91)|〈WA,1n∂1vn2,vn2〉|≤C∥vn∥H2+ɛ∥vn∥H12.
For the last term, by (75) and (76), we have
(92)|〈WA,1n∂xWA,2n,vn2〉|≤C+ɛ∥vn∥H12.
By (88)–(92), we get
(93)I3≤14ϵ∥vn2∥L44+ϵ4∥vn1∥H12+3ɛ∥vn∥H12+C∥vn∥H2+C.
Analogously, for I2, it follows that
(94)I2≤14ϵ∥vn1∥L44+ϵ4∥vn2∥H12+3ɛ∥vn∥H12+C∥vn∥H2+C.
By (80) and the estimates of I1,I2,I3, and I4, see (86), (87), (93), and (94), we have
(95)〈B(vn+WAn,vn+WAn),vn〉≤C(1+∥vn∥H2)+(ϵ4+14ɛ)∥vn∥H12+14ϵ∥vn∥L44.
For the last term on the left hand side of (79), we have
(96)〈f(vv+WAn),vn〉=ϑ∥vn∥L44+3ϑ∫D|vn|2(vn1WA,1n+vn2WA,2n)dx+ϑ∫D|vn|2|WAn|2dx+ϑ∫D(vn1WA,1n+vn2WA,2n)|WAn|2dx+2ϑ∫D(|WA,1n|2|vn1|2+|WA,2n|2|vn2|2)dx+4ϑ∫DWA,1nWA,2nvn1vn2dx≤(ϑ+ɛ)∥vn∥L44+C(1+∥vn∥H2).
By (79), (95), and (96), we get
(97)12∂∂t∥vn∥H2+∥vn∥H12+ϑ∥vn∥L44≤C(1+∥vn∥H2)+(ϵ4+14ɛ)∥vn∥H12+(14ϵ+ɛ)∥vn∥L44.
Rearranging the above inequality, we deduce that
(98)12∂∂t∥vn∥H2+(1-ϵ4-14ɛ)∥vn∥H12+(ϑ-14ϵ-ɛ)∥vn∥L44≤C(1+∥vn∥H2).
Let ϵ∈(1/4ϑ,4), and ɛ be small enough, such that
(99)1-ϵ4-14ɛ>0,ϑ-14ϵ-ɛ>0.
So, we integrate with respect to t on both sides of (98) to obtain
(100)∥vn(t)∥H2+Cϵ∫0t∥vn(s)∥H12ds≤∥vn(0)∥H2+Ct+C∫0t∥vn(s)∥H2ds,
where Cϵ=2(1-ϵ/4-14ɛ), by Gronwall’s inequality, we arrive at
(101)∥vn(t)∥H2≤(∥vn(0)∥H2+Ct)eCt≤CT.
By (100) and (101), we have
(102)∫0t∥vn(s)∥H12ds≤CT.
Multiplying Avn on both sides of (78), and integrating with respect to x∈D, we have
(103)〈∂vn∂t,Avn〉+〈Avn,Avn〉+〈f(vn+WAn),Avn〉=〈B(vn+WAn,vn+WAn),Avn〉,
which is equivalent to
(104)12∂∂t∥vn∥H12+∥vn∥H22=-〈f(vn+WAn),Avn〉+〈B(vn+WAn,vn+WAn),Avn〉.
We first estimate the second term on the right hand side of (104) as follows:
(105)〈B(vn+WAn,vn+WAn),Avn〉=〈vn1+WA,1n∂1(vn1+WA,1n),Avn1〉+〈vn2+WA,2n∂2(vn1+WA,1n),Avn1〉+〈vn1+WA,1n∂1(vn2+WA,2n),Avn2〉+〈vn2+WA,2n∂2(vn2+WA,2n),Avn2〉=J1+J2+J3+J4.
For J1, we have
(106)J1=〈vn1∂1vn1,Avn1〉+〈vn1∂1WA,1n,Avn1〉+〈WA,1n∂1vn1,Avn1〉+〈WA,1n∂1WA,1n,Avn1〉=k1+k2+k3+k4.
For k1, we have
(107)k1≤ɛ|vn1|H22+C|vn1|L42·|vn1|W1,42.
By interpolation inequality, there exists some C>0, such that
(108)|vn1|L4≤C|vn1|H1/2|vn1|H11/2,|vn1|W1,4≤C|vn1|H1/4|vn1|H23/4.
Then,
(109)k1≤ɛ|vn1|H22+C|vn1|H3/2·|vn1|H1·|vn1|H23/2≤ɛ|vn1|H22+ɛ|vn1|H22+C|vn1|H6|vn1|H14≤2ɛ|vn1|H22+CT|vn1|H14,
where the last inequality follows from (101). For k2, we deduce that
(110)k2≤ɛ|vn1|H22+C∫D(vn1)2(∂1WA,1n)2dx≤ɛ|vn1|H22+C|vn1|H2≤ɛ|vn1|H22+CT.
For k3, we arrive at
(111)k3≤C∫D|∂1vn1·Avn1|dx≤ɛ|vn1|H22+C|vn1|H12.
For k4, we obtain
(112)k4≤C+ɛ|vn1|H22.
By (106) and (109)–(112),
(113)J1≤5ɛ|vn1|H22+CT|vn1|H14+C|vn1|H12+CT.
Similarly, for J4, we infer that
(114)J4≤5ɛ|vn2|H22+CT|vn2|H14+C|vn2|H12+CT.
For J2, we have
(115)J2=〈vn2∂2vn1,Avn1〉+〈WA,2n∂2vn1,Avn1〉+〈vn2∂2WA,1n,Avn1〉+〈WA,2n∂2WA,1n,Avn1〉=l1+l2+l3+l4.
By interpolation inequality and (101), we deduce that
(116)l1≤ɛ|vn1|H22+C|vn2|L42·|vn1|W1,42≤ɛ|vn1|H22+C|vn2|H·|vn2|H1·|vn1|H1/2·|vn1|H23/2≤2ɛ|vn1|H22+CT|vn2|H14.
For l2, we have
(117)l2≤C∫D|∂2vn1|·|Avn1|dx≤ɛ|vn1|H22+C|vn1|H12.
Similarly, for l3,
(118)l3≤C∫D|vn2|·|Avn1|dx≤ɛ|vn1|H22+CT.
As for l4, we get
(119)l4≤ɛ|vn1|H22+CT.
By (115)-(119), we arrive at
(120)J2≤5ɛ∥vn1∥H22+CT∥vn2∥H14+C∥vn1∥H12+CT.
Analogously to J2, we have
(121)J3≤5ɛ∥vn2∥H22+CT∥vn1∥H14+C∥vn2∥H12+CT.
By (105) and the estimates of J1-J4, see (113), (114), (120), and (121), we get that
(122)〈B(vn+WAn,vn+WAn),Avn〉≤10ɛ∥vn∥H22+CT∥vn∥H14+C∥vn∥H12+CT.
For the first term on the right hand side of (104), we have
(123)|〈f(vn+WAn),Avn〉|≤ɛ∥vn∥H22+C∥vn+WAn∥L66≤ɛ∥vn∥H22+C∥vn∥L66+CT≤ɛ∥vn∥H22+CT∥vn∥H12∥vn∥H4+CT≤ɛ∥vn∥H22+CT(1+∥vn∥H12).
By (104), (122), and (123),
(124)12∂∂t∥vn∥H12+∥vn∥H22≤11ɛ∥vn∥H22+CT(1+∥vn∥H12)∥vn∥H12+CT.
By the Gronwall inequality, we get
(125)∥vn(t)∥H12≤(∥vn(0)∥H12+CT)eCT∫0t‍(1+∥vn(s)∥H12ds)≤(∥vn(0)∥H12+CT)eCT.
Let n→∞, by Fatou Lemma,
(126)∥v(t)∥H12≤(∥v(0)∥H12+CT)eCT.


## 5. Invariant Measures

### 5.1. Existence

In this section, we will establish the existence of invariant measure for (2). Analogously to [24], we extend the Wiener process W(t) to ℝ by setting
(127)W(t)≔W1(t),t≤0,
where W1(t) is another H-valued Wiener process satisfying conditions in Lemma 2 and being independent of W(t). For any τ≥0, we consider the following equation:
(128)duτ+[Auτ+B(uτ,uτ)+f(uτ)]dt=dW,hhhhhhhhhhhhon[0,T]×D,uτ(-τ)=0.
By Theorem 7, we know that there exists unique solution. In order to obtain the invariant measure, we should show that the family of laws {ℒ(uτ(0))}τ≥0 is tight. Since H1+δ⊂H1 is compact, for any δ>0, we only need to show that  {ℒ(uτ(0))}τ≥0 is bounded in probability in H1+δ. As we know,
(129)WA(t)=∫-∞te-(t-s)AdW(s),t∈ℝ
is the mild solution of (8) with the following initial condition:
(130)WA(0)=∫-∞0esAdW(s).
Making the classical change of variable vτ(t)=uτ(t)-WA(t), (128) is equivalent to
(131)dvτ(t)dt=Avτ(t)+B(vτ(t)WA(t),vτ(t)++WA(t),vτ(t)+WA(t))+f(vτ(t)+WA(t)),
with initial condition
(132)vτ(-τ)=-WA(-τ).
In order to get the invariant measure of (131), it is enough to show that  vτ(0) is bounded in probability in H1+δ, for some δ>0. That is what we have to do in Theorem 8 below.Theorem 8.
With conditions in Lemma2, when ϑ>1/4, there exists an invariant measure for (2).Proof.
Multiplying (131) by vτ and integrating on D, we get
(133)12ddt∥vτ(t)∥H2+∥vτ(t)∥H12+〈f(vτ(t)+WA(t)),vτ(t)〉=〈B(vτ(t)+WA(t),vτ(t)+WA(t)),vτ(t)〉.
For the third term on the left hand side of (133), we deduce that
(134)〈f(vτ(t)+WA(t)),vτ(t)〉=ϑ〈|vτ(t)+WA(t)|2(vτ(t)+WA(t)),vτ(t)+WA(t)〉-ϑ〈|vτ(t)+WA(t)|2(vτ(t)+WA(t)),WA(t)〉=ϑ∥vτ(t)+WA(t)∥L44-ϑ〈∥vτ(t)+WA(t)∥2(vτ(t)+WA(t)),WA(t)〉≥ϑ[∥vτ(t)∥L4-∥WA(t)∥L4]4-ϑ〈∥vτ(t)+WA(t)∥2(vτ(t)+WA(t)),WA(t)〉≥ϑ∥vτ(t)∥L44-4ϑ∥vτ(t)∥L43∥WA(t)∥L4-4ϑ∥vτ(t)∥L41∥WA(t)∥L43-ϑ〈∥vτ(t)+WA(t)∥2(vτ(t)+WA(t)),WA(t)〉.
Substituting (134) into (133), we have
(135)12ddt∥vτ(t)∥H2+∥vτ(t)∥H12+ϑ∥vτ(t)∥L44≤4ϑ∥vτ(t)∥L43∥WA(t)∥L4+4ϑ∥vτ(t)∥L4∥WA(t)∥L43+ϑ〈∥vτ(t)+WA(t)∥2(vτ(t)+WA(t)),WA(t)〉+〈B(vτ(t)+WA(t),vτ(t)+WA(t)),vτ(t)〉.
For the third term on the right hand side of (135), we get by the Young inequality that
(136)ϑ〈∥vτ(t)+WA(t)∥2(vτ(t)+WA(t)),WA(t)〉≤ɛ∥vτ(t)∥L44+C∥WA(t)∥L44.
For the last term on the right hand side of (135),
(137)〈B(vτ(t)+WA(t),vτ(t)+WA(t)),vτ(t)〉=〈(vτ(t)·∇)vτ(t),vτ(t)〉+〈(WA(t)·∇)vτ(t),vτ(t)〉+〈(vτ(t)·∇)WA(t),vτ(t)〉+〈(WA(t)·∇)WA(t),vτ(t)〉=r1+r2+r3+r4.
Sincevτ(t) is vector field, we denote it by vτ(t)=(vτ1(t),vτ2(t)), where vτi(t) is real valued function, i=1,2. For r1, we have
(138)r1=〈vτ1(t)∂1vτ1(t)+vτ2(t)∂2vτ1(t),vτ1(t)〉+〈vτ1(t)∂1vτ2(t)+vτ2(t)∂2vτ2(t),vτ2(t)〉=〈vτ2(t)∂2vτ1(t),vτ1(t)〉+〈vτ1(t)∂1vτ2(t),vτ2(t)〉≤-12〈∂2vτ2(t),(vτ1(t))2〉-12〈∂1vτ1(t),(vτ2(t))2〉≤14|∂1vτ1(t)|H2+14|vτ2(t)|L44+14|∂2vτ2(t)|H2+14|vτ1(t)|L44≤14∥vτ(t)∥H12+14∥vτ(t)∥L44.
Similarly for r2,
(139)r2=〈WA,1(t)∂1vτ1(t)+WA,2(t)∂2vτ1(t),vτ1(t)〉+〈WA,1(t)∂1vτ2(t)+WA,2(t)∂2vτ2(t),vτ2(t)〉=-〈∂1WA,1(t),(vτ1(t))2〉-〈∂2WA,2(t),(vτ1(t))2〉-〈∂1WA,1(t),(vτ2(t))2〉-〈∂2WA,2(t),(vτ2(t))2〉≤ɛ∥vτ(t)∥L44+C∥WA(t)∥H12.
Analogously to r1, we deduce that
(140)r3=〈vτ1(t)∂1WA,1(t)+vτ2(t)∂2WA,1(t),vτ1(t)〉+〈vτ1(t)∂1WA,2(t)+vτ2(t)∂2WA,2(t),vτ2(t)〉≤ɛ∥vτ(t)∥L44+C∥WA(t)∥H12.
For r4, we have
(141)r4=〈WA,1(t)∂1WA,1(t)+WA,2(t)∂2WA,1(t),vτ1(t)〉+〈WA,1(t)∂1WA,2(t)+WA,2(t)∂2WA,2(t),vτ2(t)〉≤ɛ|vτ1(t)|H2+C|WA,1(t)∂1WA,1(t)|H2+ɛ|vτ1(t)|H2+C|WA,2(t)∂2WA,1(t)|H2+ɛ|vτ2(t)|H2+C|WA,1(t)∂1WA,2(t)|H2+ɛ|vτ2(t)|H2+C|WA,2(t)∂2WA,2(t)|H2≤ɛ|vτ1(t)|H2+|WA,1(t)|L42·|WA,1(t)|W1,42+C|WA,2(t)|L42·|WA,1(t)|W1,42+ɛ|vτ2(t)|H2+C|WA,1(t)|L42·|WA,2(t)|W1,42+C|WA,2(t)|L42·|WA,2(t)|W1,42≤ɛ∥vτ(t)∥H2+C∥WA(t)∥L42·∥WA(t)∥W1,42.
Since {A1/2WA(t)}t∈ℝ is a Gaussian process, we infer that
(142)E(|A1/2WA(t)|4)≤C[E(|A1/2WA(t)|2)]2.
Then, with the proof of Lemma 2, we know that ∥WA(t)∥W1,42 is continuous with respect to t. By (137)–(141), we have
(143)〈B(vτ(t)+WA(t),vτ(t)+WA(t)),vτ(t)〉≤(14+ɛ)∥vτ(t)∥H12+(14+2ɛ)∥vτ(t)∥L44+C(∥WA(t)∥H12+∥WA(t)∥L42∥WA(t)∥W1,42).
By (135), (136), and (143), we arrive at
(144)12ddt∥vτ(t)∥H2+∥vτ(t)∥H12+ϑ∥vτ(t)∥L44≤(14+3ɛ)∥vτ(t)∥L44+(14+ɛ)∥vτ(t)∥H12+C∥WA(t)∥W1,44+C.
It is equivalent to
(145)12ddt∥vτ(t)∥H2+(34-ɛ)∥vτ(t)∥H12+(ϑ-14-3ɛ)∥vτ(t)∥L44≤C(1+∥WA(t)∥W1,44).
Since ϑ>1/4, let ɛ be small enough, such that
(146)34-ɛ>0;ϑ-14-3ɛ>0.
Then, the above estimates can be changed into
(147)ddt∥vτ(t)∥H2+α1∥vτ(t)∥H12+Cν∥vτ(t)∥L44≤C(1+∥WA(t)∥W1,44).
By the Gronwall inequality, we get
(148)∥vτ(t)∥H2≤∥WA(-τ)∥H2e-α1(τ+t)+C∫-τt(1+∥WA(s)∥W1,44)eα1(s-t)ds≤∥WA(-τ)∥H2e-α1(τ+t)+C∫-∞0(1+∥WA(s)∥W1,44)eα1(s-t)ds.
Similarly to the argument of [26], we will prove that ∥WA(t)∥W1,4 has at most polynomial growth, when t→-∞ a.s. So, we conclude that
(149)sup0≤τ;t≤T∥vτ(t)∥H2<∞.a.s.
Multiplying eδt on both sides of (147) and integrating with respect to t, we have
(150)∫-τteδs∥vτ(s)∥H12ds≤e-δτ∥WA(-τ)∥H2+α1∫-τteδs∥vτ(s)∥H2ds+C∫-τt(1+∥WA(t)∥W1,44)eδsds≤e-δτ∥WA(-τ)∥H2+α1∫-∞0eδs∥vτ(s)∥H2ds+C∫-∞0(1+∥WA(t)∥W1,44)eδsds.
As
(151)∫-∞0(1+∥WA(t)∥W1,44)eδsds<∞,
by (149), we have
(152)sup0≤τ;t≤T∫-τteδs∥vτ(s)∥H12ds<∞.a.s.
By Theorem 7, we know that for problem (131) there exists unique mild solution, which has the following:
(153)vτ(0)=eτAWA(-τ)+∫-τ0‍etAB((vτ(t)+WA(t),vτ(t)+WA(t)))dt+∫-τ0etAf(vτ(t)+WA(t))dt.
Then, for any ζ∈(0,θ)∩(0,1/4), where the θ is the parameter in Lemma 2,
(154)∥A(1+ζ)/2vτ(0)∥H≤∥eτAA(1+ζ)/2WA(-λ)∥H+∫-τ0∥A(1+ζ)/2etAB((vτ(t)+WA(t),vτ(t)+WA(t))∥Hdt+∫-τ0∥A(1+ζ)/2etAf(vτ(t)+WA(t))∥Hdt.
Since
(155)B((vτ(t)+WA(t),vτ(t)+WA(t)))=(vτ(t)·∇)vτ(t)+(vτ(t)·∇)WA(t)+(WA(t)·∇)vτ(t)+(WA(t)·∇)WA(t),
then,
(156)∥A(1+ζ)/2etAB((vτ(t)+WA(t),vτ(t)+WA(t)))∥H≤∥A(1+ζ)/2etA[vτ(t)·∇]vτ(t)∥H+∥A(1+ζ)/2etA[vτ(t)·∇]WA(t)∥H+∥A(1+ζ)/2etA[WA(t)·∇]vτ(t)∥H+∥A(1+ζ)/2etA[WA(t)·∇]WA(t)∥H=z1+z2+z3+z4.
For z1, we have
(157)z1≤|A(1+ζ)/2etA[vτ1(t)∂1vτ1(t)]|H+|A(1+ζ)/2etA[vτ2(t)∂2vτ1(t)]|H+|A(1+ζ)/2etA[vτ1(t)∂1vτ2(t)]|H+|A(1+ζ)/2etA[vτ2(t)∂2vτ2(t)]|H=z1,1+z1,2+z1,3+z1,4.
In the following, we use Theorem 6.13 in chapter two of [27] to estimate them respectively as follows:
(158)z1,1=12|etAA(1+ζ)/2∂1(vτ1)2|H≤12|etAA(3+2ζ)/4(vτ1)2|H1/2≤C|t|-(3+2ζ)/4eδt|(vτ1)2|H1/2≤C|t|-(3+2ζ)/4eδt|2vτ1(t)A1/4vτ1(t)+R4|H,
the last inequality follows by Theorem A.8 in [25], where δ>0,  R4=A1/4(vτ1)2-2vτ1A1/4vτ1, and |R4|H≤C|A1/8vτ1(t)|L42≤C|vτ1(t)|H12. So, by Hölder inequality and interpolation inequality, we have
(159)z1,1≤C|t|-(3+2ζ)/4eδt∥vτ(t)∥H12.
For z1,2, we have
(160)z1,2=|A(1+ζ)/2etA[vτ2(t)∂2vτ1(t)]|H≤|A(1+ζ)/2etA[A1/4(vτ2(t)A1/4vτ1(t))]|H+|A(1+ζ)/2etA[A1/4vτ1(t)A1/4vτ2(t)]|H+|A(1+ζ)/2etAR5|H,
where
(161)R5=A1/4[vτ2(t)A1/4vτ1(t)]-[A1/4vτ1(t),A1/4vτ2(t)]-vτ2(t)A1/2vτ1(t).
Analogously to estimating z1,1, we have
(162)z1,2≤C|t|-(3+2ζ)/4eδt|vτ2(t)|L4|vτ1(t)|W1/2,4+C|t|-(1+ζ)/2eδt|vτ1(t)|W1/2,4|vτ2(t)|W1/2,4≤C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)eδt∥vτ(t)∥W1/2,42≤C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)eδt∥vτ(t)∥H12.
Similarly, we can get the same estimates for z1,3 and z1,4. Therefore,
(163)z1≤C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)eδt∥vτ(t)∥H12.
Analogously to estimating z1, we can get for z2,z3, and z4 that
(164)z2≤C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)×eδt(∥WA(t)∥H12+∥vτ(t)∥H12),z3≤C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)×eδt(∥WA(t)∥H12+∥vτ(t)∥H12),z4≤C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)×eδt∥WA(t)∥H12.
So, by (163)–(164) and (156), we get
(165)∥A(1+ζ)/2etAB((vτ(t)+WA(t),vτ(t)+WA(t)))∥H≤C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)×eδt(∥WA(t)∥H12+∥vτ(t)∥H12).
For the third term on the right hand side of (154), we obtain
(166)∥A(1+ζ)/2etAf(vτ(t)+WA(t))∥H≤C|t|-(1+ζ)/2eδt(∥vτ(t)∥L63+∥WA(t)∥L63)≤C|t|-(1+ζ)/2×eδt(∥vτ(t)∥H·∥vτ(t)∥H12+∥WA(t)∥H13)≤C|t|-(1+ζ)/2eδt×eδt(∥vτ(t)∥H12+∥WA(t)∥H12),
since ∥vτ(t)∥H and eδt∥WA(t)∥H12 are bounded for t,τ∈(-∞,T], the last inequality follows. For the first term on the right hand side of (154), we have
(167)∥eτAA(1+ζ)/2WA(-τ)∥H≤e-δτ∥A(1+ζ)/2WA(-τ)∥H.
Similar to [26], we can prove that ∥A(1+ζ)/2WA(-τ)∥H has at most polynomial growth when τ→∞. For the reader convenience, we sketch a proof. By Lemma 2, we know that W(t)-W(s) is a D(Aθ/2) valued Brownian motion, for s≤t≤0. So, by the law of iterated logarithm, we have
(168)wn≔supn≤s≤t≤n+1∥W(t)-W(s)∥Hθ|t-s|1/2+(ζ-θ)/4<∞,a.s.n∈ℤ.
Obviously, wn is a i.i.d sequence. By the law of large numbers, there exists an integer-valued random variable n0(w)>0, when n≥n0(w), we have
(169)w-nn≤w-n+⋯+w-1n≤Ew0+1<∞.
This implies that
(170)w-n≤C0(w)n,
for all n>0. In other words,
(171)∥W(t)-W(s)∥Hθ≤C0(w)|[s]|·|t-s|1/2+(ζ-θ)/4,
when s≤t≤[s]+1. By the law of iterated logarithm, we have
(172)∥W(t)∥Hθ≤C1(w)|t|,t∈(-∞,0],
for some positive random variable. By Theorem 5.14 in [23], we know that
(173)WA(t)=∫-∞tAe-(t-s)A(W(t)-W(s))ds.
So, we have that
(174)∥A(1+ζ)/2WA(t)∥H≤∫-∞t∥A1+1/2+ζ/2e-(t-s)A(W(t)-W(s))∥Hds=∫-∞t∥A1+1/2+(ζ-θ)/2e-(t-s)A[Aθ/2(W(t)-W(s))]∥Hds≤∫-∞te-δ(t-s)|t-s|1+1/2+(ζ-θ)/2∥W(t)-W(s)∥Hθds≤∫[t]-1te-δ(t-s)|t-s|1+(ζ-θ)/4·∥W(t)-W(s)∥Hθ|t-s|1/2+(ζ-θ)/4+∫-∞[t]-1e-δ(t-s)|t-s|1+(ζ-θ)/4·C1(w)(|t|+|s|)|t-s|1/2+(ζ-θ)/4≤∫[t]-1te-δ(t-s)|t-s|1+(ζ-θ)/4·C0(w)|[s]|ds+∫-∞[t]-1e-δ(t-s)C1(w)(|t|+|s|)≤(C0(w)+C1(w))(|t|+1),
since s≤[t]-1, the fourth inequality follows. By (167) and (174), we know that
(175)supτ≥0∥eτAA(1+ζ)/2WA(-τ)∥H<∞,a.s.
If we let ζ=1/2<θ, repeating the argument of (174), we can see that ∥WA(t)∥W1,4 also has at most polynomial growth, when t→-∞ a.s., since we have the Sobolev embedding H3/2⊂W1,4. Consider the second term on the right hand side of (154), by (165),
(176)∫-τ0∥A(1+ζ)/2etAB((vτ(t)+WA(t),vτ(t)+WA(t)))∥Hdt≤∫-τ0C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)×eδt(∥WA(t)∥H12+∥vτ(t)∥H12)dt≤∫-10C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)×eδt(∥WA(t)∥H12+∥vτ(t)∥H12)dt+∫-∞-1Ceδt(∥WA(t)∥H12+∥vτ(t)∥H12)dt<∞,
where the last inequality follows by (152). Analogously, we can prove that
(177)∫-τ0∥A(1+ζ)/2etAf(vτ(t)+WA(t))∥Hdt≤∫-τ0C(|t|-(1+ζ)/2)×eδt(∥vτ(t)∥H∥vτ(t)∥H12+∥WA(t)∥H13)<∞,
where we used (149) and (152) for the last inequality. By (154) and (175)–(177), we get
(178)∥A(1+ζ)/2vτ(0)∥H≤ξ(w),a.s.,
for some positive random variable ξ(w). As H1+δ⊂H1 is compact, by Prohorov Theorem, we know that the family of laws for (vτ(0))τ≥0 taking values in H1 is tight. Since vτ(0)=uτ(0)-WA(0), then so does the law of (uτ(0))τ≥0 taking values in the same space. For t≥0, set
(179)(Ptf)(x)=Ef(u(t,.;0,x)),
where f∈Cb(H01). Following the arguments in [24], for all t0<s<t and all ut0∈H01, by proving
(180)E(f(u(t;t0,ut0))∣ℱs)=Pt-s(u(s;t0,ut0)),
we can show that u is a Markov process. Here, ℱs is the σ-algebra generated by W(r) for r≤s. So, (Pt)t≥0 is the Markov semigroup. Define a dual semigroup Pt* in the space P(H01) of probability measures on H01 as follows:
(181)∫H01fd(Pt*μ)=∫H01Ptfdμ.
Let μτ be the law of uτ(0), which is the solution of (2) with initial condition u(-τ)=0. Then, we have
(182)μτ(f)=Ef(u-τ(0))=Ef(u(τ,·;0,0))=(Pτf)(0)=∫H01Pτfdδ0=∫H01fd(Pτ*δ0),
where we use the fact that u(τ,·;0,0) and uτ(0) have the same law, the second equality follows. Therefore,
(183)Pτ1*μτ=μτ+τ1.
Since (μτ)τ≥0 is tight, then by Prokhorov theorem, we know that (μτ)τ≥0 is relatively compact. We can choose a subsequence of (μτ)τ≥0 denoted by (μτn)n∈ℕ such that for μ∈P(Hσ),
(184)∫H01(Ptf)(x)μ(dx)=limn→∞∫H01(Ptf)(x)μτn(dx)=limn→∞∫H01f(x)Pt*μτn(dx)=limn→∞∫H01f(x)μτn+t(dx)=∫H01f(x)μ(dx).


### 5.2. Uniqueness

The main result of this part is as follows.Theorem 9.
Assumeθ>1/2 in Lemma 2 and ϑ>1/4; then,(i)
the stochastic Burgers equation (2) has a unique invariant measure μ;(ii)
for allu0∈H01 and all Borel measurable functions φ, H01→ℝ, such that ∫H01‍|φ|dμ<∞,
(185)limT→∞1T∫0Tφ(u(t;u0))dt=∫H01φdμa.s.;(iii)
for every Borel measureμ* on H01, one has that
(186)∥Pt*μ*-μ∥TV⟶0ast⟶∞,where ∥·∥TV stands for the total variation of a measure. In particularly, one has that
(187)Pt*μ*(B)⟶μ(B),ast⟶∞,
for every Borel set B∈ℬ(H01)(the Borel σ-algebra of H01).In order to prove Theorem9, we only need Theorem 10 below, see [28, Theorem 4.2.1]. We define P(t,x,·),t>0, x∈H01, to be the transition probability measure that is,
(188)P(t,x,B)=Pt*δx(B)=P(u(t;x)∈B)
for B∈ℬ(H01).Theorem 10.
Assume that the probability measuresP(t,x,·),t>0,x∈H01, are all equivalent, in the sense that they are mutually absolutely continuous. Then, Theorem 9 holds true.In the following, we will prove the irreducibility and the strong Feller property inH01 to get the equivalence of the measure P(t,x,·). For the two notations, we outline them below. For y∈H01,ɛ>0, let
(189)B(y,ɛ)={x∈H01;∥x-y∥H1<ɛ}.(I)
For anyx,y∈H01, such that for all ɛ>0,
(190)P(t,x,B(y,ɛ))>0for eacht>0.(S)
For allO∈ℬ(H01), every t>0, and all xn,x∈H01 such that xn→x in H01, it holds that
(191)P(t,xn,O)⟶P(t,x,O).Before checking the condition (I), we need Lemma11 below. For x∈H01 and ϕ:[0,T]→H01, set
(192)u(t,x,ϕ)=v(t,x,ϕ)+ϕ(t),
where v(t,x,ϕ) is solution of the following equation:
(193)dvdt+Av+B(v+ϕ,v+ϕ)+f(v+ϕ)=0,
for t∈[0,T], with initial condition v(0)=x. As it is proved in previously this equation has a unique solution as follows:
(194)v∈C([0,T];H01),
when x∈H01 and ϕ∈C([0,T];H01).Lemma 11.
DefineΨ(ϕ)=u(·,x,ϕ); then, (i)
the mapping(195)Ψ:C0([0,T];H3/2)⟶C([0,T];H01)is continuous, whereC0([0,T];B)≔{h∈C([0,T];B);h(0)=0} for Banach space B;(ii)
for everyx,y∈H3/2 and T>0 there exists z¯∈C0([0,T];H3/2) such that u(T,x,z¯)=y.Proof.
(i) is proved by(A.30) in the Appendix. To prove (ii), let x,y,∈H3/2 and T>0, define u¯ as
(196)u¯(t)=e-tAx,t∈[0,t0],u¯(t)=e-(T-t)Ay,t∈[t1,T],u¯(t)=u¯(t0)+t-t0t1-t0(u¯(t1)-u¯(t0)),hhhhhhhhhhhhhhhhhhhhhht∈(t0,t1).
Obviously, u¯(t)∈C([0,T];H3/2). Define v¯ as the solution of the following equation:
(197)ddtv¯+Av¯+B(u¯,u¯)+f(u¯)=0,
with initial condition v¯(0)=x; then v¯∈C([0,T];H3/2). Set z¯=u¯-v¯; then it satisfies all the requirements of the lemma.Proposition 12.
With conditions in Theorem9, the irreducibility property (I) is satisfied.Proof.
Letx∈H3/2 and z¯ be the same as (ii) in Lemma 11. By the above lemma, we have that for ɛ>0, we can find δ>0, such that
(198)∥z-z¯∥C0([0,T];H3/2)<δ
implies that
(199)∥u(·,x,z)-u(·,x,z¯)∥C([0,T];H1)<ɛ.
If θ>1/2 in Lemma 2, and denote z and z¯ the corresponding Ornstein-Uhlenbeck process satisfying conditions in the lemma, then z,z¯∈C([0,T];H3/2). Choose δ1>0 such that δ1<δ and
(200)z∈Uδ1=:{z∈C0([0,T];H3/2);∥z-z¯∥C([0,T];H3/2)<δ1}.
Then, for z∈Uδ1, we have that
(201)∥u(T,,x,z)-y∥H1<ɛ.
Recall now that the solution u of the stochastic Burgers equation is equal to Ψ(z), z being the Ornstein-Uhlenbeck process. Then, it remains to show that
(202)P{z(·,w)∈Uδ1}>0.
But this is obviously true. So far, we have proved that for for all t>0, forallx,y∈H3/2, for all ɛ>0,
(203)P(t,x,B(y,ɛ))>0.
Next, we will prove for all x0∈H01,y0∈H3/2, the above inequality also holds. Indeed, for 0<h<t, by Chapman-Kolmogorov equation, we have
(204)P(t,x0,B(y0,ɛ))=∫H01P(t-h,x0,dy)P(h,y,B(y0,ɛ))=∫H3/2P(t-h,x0,dy)P(h,y,B(y0,ɛ))>0.
Since P(t-h,x0,H3/2)=1, we will extend (204) to the case for all x0∈H01,y0∈H01. If this is not true, there exists t0>0,  x0,y0∈H01, ɛ>0 such that
(205)P(t0,x0,B(y0,ɛ))=0.
Then, we can choose y1∈H3/2,  ɛ1>0 such that B(y1,ɛ1)⊂B(y0,ɛ). By (204), we have
(206)P(t0,x0,B(y1,ɛ1))>0,
which is contrary to (205).In this part, it is time to check the condition (S).We will first obtain the strong Feller property inH01 for modified Burgers equation (208) below, then let R→∞ to check the condition (S).FixR>0, let KR:[0,∞[→[0,∞[ satisfy KR∈C1(ℝ+) such that |KR|≤1,  |KR′|≤2 and
(207)KR=1,ifx<R,KR=0,ifx≥R+1.
Consider the following equation:
(208)duR(t)+AuR(t)dt+KR(∥uR(t)∥H12)B(uR(t),uR(t))dt+KR(∥uR(t)∥H12)f(uR)(t)=dW(t).Proposition 13.
There exists a unique mild solutionuR(·,w)∈C([0,T];H01) for (208) which is Markov process with the Feller property in H01, that is for every R>0, t>0, there exists a constant L=L(t,R)>0 such that
(209)|Pt(R)ϕ(x)-Pt(R)ϕ(y)|≤L∥x-y∥H1
holds for all x,y∈H01, and all ϕ∈Cb(H01)≤1, where Pt(R)ϕ(x)≔∫H01ϕ(y)PR(t,x,dy),PR(t,x,·) is the transition probabilities corresponding to (204).Proof.
The proof of existence and uniqueness is similar to Section2. Let ϕ1=ϕ2 in (A.28), by the Gronwall inequality, we know that uR is Lipschitz continuous with respect to initial value. Using the method in Proposition 4.3.3 in [24], we can prove that the solution is a Markov process. To prove the Fell property, we first consider the following Galerkin approximations of (208). Let Pn be the orthogonal projection in  H defined as Pnx=∑j=1n‍〈x,ej〉ej,x∈H. Clearly, Hn≔PnH for every n. Consider the equation in Hn as follows:
(210)dun(R)(t)+Aun(R)(t)dt+KR(∥un(R)(t)∥H12)PnB(un(R)(t),un(R)(t))+KR(∥un(R)(t)∥H12)f(un(R))(t)=dW(t),
with initial condition un(R)(0)=Pnu0. This is a finite-dimensional equation with globally Lipschitz nonlinear functions, so it has a unique progressively measurable solution with P-a.e. trajectory un(R)(·,w)∈C([0,T];Hn), which is also a Markov process in Hn with associated semigroup Pn,t(R) defined as
(211)Pn,t(R)ϕ(x)=Eϕ(un(R)(t;x)),
for all x∈Hn and ϕ∈Cb(Hn). For every R>0,t>0, we can prove that there exists a constant L=L(t,R)>0 such that
(212)|Pn,t(R)ϕ(x)-Pn,t(R)ϕ(y)|≤L∥x-y∥H1
hold for all n∈ℕ,x,y∈Hn, and all ϕ∈Cb(Hn) with ∥ϕ∥H1≤1. Indeed, the following remarkable formula holds true for the differential in x of Pn,t(R)ϕ [29]:
(213)DxPn,t(R)ϕ(x)·h=1tE(ϕ(un(R)(t;x))∫0t〈(PnQQ*Pn)-1/2Dxun(R)(s;x)·h,dβn(s)(PnQQ*Pn)-1/2Dxun(R)〉ϕ(un(R)(t;x))∫0t〈(PnQQ*Pn)-1/2Dxun(R)(s;x)),
for all h∈Hn, where βn is a n-dimensional standard Wiener process with incremental covariance PnQ and Q is the covariance operator of W(t). Obviously, Q is nonnegative, adjoint, Hilbert-Schmidt operator with inverse. Since the eigenvalues αn of the Stokes operator A, in 2-space dimension, behave like n, let θ=1/2+ɛ for some ɛ>0, in Lemma 2, we have D(A)⊂ℛ(Q)⊂D(A3/4), where ℛ(Q) is the image of Q. Therefore,
(214)|DxPn,t(R)ϕ(x)·h|≤1tE(∫0t∥(PnQQ*Pn)-1/2Dxun(R)(s;x)·h∥H2ds)1/2.
Since for y∈Hn(215)∥(PnQQ*Pn)-1/2y∥H2=〈(PnQQ*Pn)-1y,y〉=〈(APnQQ*PnA)-1Ay,Ay〉≤C∥y∥H22,
it follows that
(216)|DxPn,t(R)ϕ(x)·h|≤1tCE(∫0t∥Dxun(R)(s;x)·h∥H22ds)1/2≤1tC(R)∥h∥H1,
where the last inequality follows by the Estimate 4 of the Appendix (note that C(R) is independent of x∈Hn and n∈ℕ). Indeed, un(R)(t,x) is given by vn(t,x)+Pnz(t), where z is the Ornstein-Uhlenbeck process, and vn is the solution of (A.2). Therefore,
(217)|Pn,t(R)ϕ(x)-Pn,t(R)ϕ(y)|≤sup∥h∥H1≤1,k∈Hn|DxPn,t(R)ϕ(k)·h|·∥x-y∥H1≤1tC(R)∥x-y∥H1.
In the following step, we will let n→∞ to get the Fell property for (208). Let x∈H01 and ϕ∈Cb(H01) be given. From the Appendix, Remark A.1, we know that un(R)(t) converges to u(R)(t) strongly in L2(0,T;H01), p-a.s.. By the boundedness and continuous of ϕ as well as Lebesgue dominated convergence theorem, we have
(218)E∫0T|ϕ(un(R)(,;x))-ϕ(u(R)(,;x))|dt⟶0,
which implies that for some subsequence nk,
(219)Eϕ(unk(R)(,;x))⟶Eϕ(u(R)(,;x)),
for a.e.  t∈[0,T]. Take x,y∈H01, by the previous argument, we can find a subsequence nk such that the previous almost sure convergence in t∈[0,T] holds true both x and y.
Thus, from (212), we have
(220)|Pt(R)ϕ(x)-Pt(R)ϕ(y)|≤L∥x-y∥H1,
for a.e. t∈[0,T]. As u(R)(t;x) has continuous trajectories with values in H01, the above inequality holds for all t∈[0,T].Proposition 14.
Under conditions of Theorem9, (S) holds true.Proof.
Taket>0,xn,x∈H01 satisfying xn→x in H1. For every R>0, we have that
(221)∥PR(t,xn,·)-PR(t,x,·)∥TV=sup∥ϕ∥Cb(H1)≤1|Pt(R)ϕ(xn)-Pt(R)ϕ(x)|≤L∥xn-x∥H1⟶0,
as n→∞ by Proposition 13. Then,
(222)∥PR(t,xn,·)-P(t,xn,·)∥TV+∥PR(t,x,·)-P(t,x,·)∥TV=sup∥ϕ∥Cb(H1)≤1|Pt(R)ϕ(xn)-Ptϕ(xn)|+sup∥ϕ∥Cb(H1)≤1|Pt(R)ϕ(x)-Ptϕ(x)|=sup∥ϕ∥Cb(H1)≤1|Eϕ(uR(t;xn))-Eϕ(u(t;xn))|+sup∥ϕ∥Cb(H1)≤1|Eϕ(uR(t;x))-Eϕ(u(t;x))|≤2∫ΩI{supn∈ℕ∥u(t;xn)∥H1>R}P(dw)+2∫ΩI{∥u(t;x)∥H1>R}P(dw)⟶0,asR→∞,
where the inequality follows by the consistency of u(t;x) and u(R)(t;x), when ∥u(t;x)∥H1≤R, and the limit follows by (A.21). Therefore,
(223)∥P(t,xn,·)-P(t,x,·)∥TV≤∥P(t,xn,·)-PR(t,xn,·)∥TV+∥PR(t,xn,·)-PR(t,x,·)∥TV+∥PR(t,x,·)-P(t,x,·)∥TV⟶0,
as n→∞.


## 5.1. Existence

In this section, we will establish the existence of invariant measure for (2). Analogously to [24], we extend the Wiener process W(t) to ℝ by setting
(127)W(t)≔W1(t),t≤0,
where W1(t) is another H-valued Wiener process satisfying conditions in Lemma 2 and being independent of W(t). For any τ≥0, we consider the following equation:
(128)duτ+[Auτ+B(uτ,uτ)+f(uτ)]dt=dW,hhhhhhhhhhhhon[0,T]×D,uτ(-τ)=0.
By Theorem 7, we know that there exists unique solution. In order to obtain the invariant measure, we should show that the family of laws {ℒ(uτ(0))}τ≥0 is tight. Since H1+δ⊂H1 is compact, for any δ>0, we only need to show that  {ℒ(uτ(0))}τ≥0 is bounded in probability in H1+δ. As we know,
(129)WA(t)=∫-∞te-(t-s)AdW(s),t∈ℝ
is the mild solution of (8) with the following initial condition:
(130)WA(0)=∫-∞0esAdW(s).
Making the classical change of variable vτ(t)=uτ(t)-WA(t), (128) is equivalent to
(131)dvτ(t)dt=Avτ(t)+B(vτ(t)WA(t),vτ(t)++WA(t),vτ(t)+WA(t))+f(vτ(t)+WA(t)),
with initial condition
(132)vτ(-τ)=-WA(-τ).
In order to get the invariant measure of (131), it is enough to show that  vτ(0) is bounded in probability in H1+δ, for some δ>0. That is what we have to do in Theorem 8 below.Theorem 8.
With conditions in Lemma2, when ϑ>1/4, there exists an invariant measure for (2).Proof.
Multiplying (131) by vτ and integrating on D, we get
(133)12ddt∥vτ(t)∥H2+∥vτ(t)∥H12+〈f(vτ(t)+WA(t)),vτ(t)〉=〈B(vτ(t)+WA(t),vτ(t)+WA(t)),vτ(t)〉.
For the third term on the left hand side of (133), we deduce that
(134)〈f(vτ(t)+WA(t)),vτ(t)〉=ϑ〈|vτ(t)+WA(t)|2(vτ(t)+WA(t)),vτ(t)+WA(t)〉-ϑ〈|vτ(t)+WA(t)|2(vτ(t)+WA(t)),WA(t)〉=ϑ∥vτ(t)+WA(t)∥L44-ϑ〈∥vτ(t)+WA(t)∥2(vτ(t)+WA(t)),WA(t)〉≥ϑ[∥vτ(t)∥L4-∥WA(t)∥L4]4-ϑ〈∥vτ(t)+WA(t)∥2(vτ(t)+WA(t)),WA(t)〉≥ϑ∥vτ(t)∥L44-4ϑ∥vτ(t)∥L43∥WA(t)∥L4-4ϑ∥vτ(t)∥L41∥WA(t)∥L43-ϑ〈∥vτ(t)+WA(t)∥2(vτ(t)+WA(t)),WA(t)〉.
Substituting (134) into (133), we have
(135)12ddt∥vτ(t)∥H2+∥vτ(t)∥H12+ϑ∥vτ(t)∥L44≤4ϑ∥vτ(t)∥L43∥WA(t)∥L4+4ϑ∥vτ(t)∥L4∥WA(t)∥L43+ϑ〈∥vτ(t)+WA(t)∥2(vτ(t)+WA(t)),WA(t)〉+〈B(vτ(t)+WA(t),vτ(t)+WA(t)),vτ(t)〉.
For the third term on the right hand side of (135), we get by the Young inequality that
(136)ϑ〈∥vτ(t)+WA(t)∥2(vτ(t)+WA(t)),WA(t)〉≤ɛ∥vτ(t)∥L44+C∥WA(t)∥L44.
For the last term on the right hand side of (135),
(137)〈B(vτ(t)+WA(t),vτ(t)+WA(t)),vτ(t)〉=〈(vτ(t)·∇)vτ(t),vτ(t)〉+〈(WA(t)·∇)vτ(t),vτ(t)〉+〈(vτ(t)·∇)WA(t),vτ(t)〉+〈(WA(t)·∇)WA(t),vτ(t)〉=r1+r2+r3+r4.
Sincevτ(t) is vector field, we denote it by vτ(t)=(vτ1(t),vτ2(t)), where vτi(t) is real valued function, i=1,2. For r1, we have
(138)r1=〈vτ1(t)∂1vτ1(t)+vτ2(t)∂2vτ1(t),vτ1(t)〉+〈vτ1(t)∂1vτ2(t)+vτ2(t)∂2vτ2(t),vτ2(t)〉=〈vτ2(t)∂2vτ1(t),vτ1(t)〉+〈vτ1(t)∂1vτ2(t),vτ2(t)〉≤-12〈∂2vτ2(t),(vτ1(t))2〉-12〈∂1vτ1(t),(vτ2(t))2〉≤14|∂1vτ1(t)|H2+14|vτ2(t)|L44+14|∂2vτ2(t)|H2+14|vτ1(t)|L44≤14∥vτ(t)∥H12+14∥vτ(t)∥L44.
Similarly for r2,
(139)r2=〈WA,1(t)∂1vτ1(t)+WA,2(t)∂2vτ1(t),vτ1(t)〉+〈WA,1(t)∂1vτ2(t)+WA,2(t)∂2vτ2(t),vτ2(t)〉=-〈∂1WA,1(t),(vτ1(t))2〉-〈∂2WA,2(t),(vτ1(t))2〉-〈∂1WA,1(t),(vτ2(t))2〉-〈∂2WA,2(t),(vτ2(t))2〉≤ɛ∥vτ(t)∥L44+C∥WA(t)∥H12.
Analogously to r1, we deduce that
(140)r3=〈vτ1(t)∂1WA,1(t)+vτ2(t)∂2WA,1(t),vτ1(t)〉+〈vτ1(t)∂1WA,2(t)+vτ2(t)∂2WA,2(t),vτ2(t)〉≤ɛ∥vτ(t)∥L44+C∥WA(t)∥H12.
For r4, we have
(141)r4=〈WA,1(t)∂1WA,1(t)+WA,2(t)∂2WA,1(t),vτ1(t)〉+〈WA,1(t)∂1WA,2(t)+WA,2(t)∂2WA,2(t),vτ2(t)〉≤ɛ|vτ1(t)|H2+C|WA,1(t)∂1WA,1(t)|H2+ɛ|vτ1(t)|H2+C|WA,2(t)∂2WA,1(t)|H2+ɛ|vτ2(t)|H2+C|WA,1(t)∂1WA,2(t)|H2+ɛ|vτ2(t)|H2+C|WA,2(t)∂2WA,2(t)|H2≤ɛ|vτ1(t)|H2+|WA,1(t)|L42·|WA,1(t)|W1,42+C|WA,2(t)|L42·|WA,1(t)|W1,42+ɛ|vτ2(t)|H2+C|WA,1(t)|L42·|WA,2(t)|W1,42+C|WA,2(t)|L42·|WA,2(t)|W1,42≤ɛ∥vτ(t)∥H2+C∥WA(t)∥L42·∥WA(t)∥W1,42.
Since {A1/2WA(t)}t∈ℝ is a Gaussian process, we infer that
(142)E(|A1/2WA(t)|4)≤C[E(|A1/2WA(t)|2)]2.
Then, with the proof of Lemma 2, we know that ∥WA(t)∥W1,42 is continuous with respect to t. By (137)–(141), we have
(143)〈B(vτ(t)+WA(t),vτ(t)+WA(t)),vτ(t)〉≤(14+ɛ)∥vτ(t)∥H12+(14+2ɛ)∥vτ(t)∥L44+C(∥WA(t)∥H12+∥WA(t)∥L42∥WA(t)∥W1,42).
By (135), (136), and (143), we arrive at
(144)12ddt∥vτ(t)∥H2+∥vτ(t)∥H12+ϑ∥vτ(t)∥L44≤(14+3ɛ)∥vτ(t)∥L44+(14+ɛ)∥vτ(t)∥H12+C∥WA(t)∥W1,44+C.
It is equivalent to
(145)12ddt∥vτ(t)∥H2+(34-ɛ)∥vτ(t)∥H12+(ϑ-14-3ɛ)∥vτ(t)∥L44≤C(1+∥WA(t)∥W1,44).
Since ϑ>1/4, let ɛ be small enough, such that
(146)34-ɛ>0;ϑ-14-3ɛ>0.
Then, the above estimates can be changed into
(147)ddt∥vτ(t)∥H2+α1∥vτ(t)∥H12+Cν∥vτ(t)∥L44≤C(1+∥WA(t)∥W1,44).
By the Gronwall inequality, we get
(148)∥vτ(t)∥H2≤∥WA(-τ)∥H2e-α1(τ+t)+C∫-τt(1+∥WA(s)∥W1,44)eα1(s-t)ds≤∥WA(-τ)∥H2e-α1(τ+t)+C∫-∞0(1+∥WA(s)∥W1,44)eα1(s-t)ds.
Similarly to the argument of [26], we will prove that ∥WA(t)∥W1,4 has at most polynomial growth, when t→-∞ a.s. So, we conclude that
(149)sup0≤τ;t≤T∥vτ(t)∥H2<∞.a.s.
Multiplying eδt on both sides of (147) and integrating with respect to t, we have
(150)∫-τteδs∥vτ(s)∥H12ds≤e-δτ∥WA(-τ)∥H2+α1∫-τteδs∥vτ(s)∥H2ds+C∫-τt(1+∥WA(t)∥W1,44)eδsds≤e-δτ∥WA(-τ)∥H2+α1∫-∞0eδs∥vτ(s)∥H2ds+C∫-∞0(1+∥WA(t)∥W1,44)eδsds.
As
(151)∫-∞0(1+∥WA(t)∥W1,44)eδsds<∞,
by (149), we have
(152)sup0≤τ;t≤T∫-τteδs∥vτ(s)∥H12ds<∞.a.s.
By Theorem 7, we know that for problem (131) there exists unique mild solution, which has the following:
(153)vτ(0)=eτAWA(-τ)+∫-τ0‍etAB((vτ(t)+WA(t),vτ(t)+WA(t)))dt+∫-τ0etAf(vτ(t)+WA(t))dt.
Then, for any ζ∈(0,θ)∩(0,1/4), where the θ is the parameter in Lemma 2,
(154)∥A(1+ζ)/2vτ(0)∥H≤∥eτAA(1+ζ)/2WA(-λ)∥H+∫-τ0∥A(1+ζ)/2etAB((vτ(t)+WA(t),vτ(t)+WA(t))∥Hdt+∫-τ0∥A(1+ζ)/2etAf(vτ(t)+WA(t))∥Hdt.
Since
(155)B((vτ(t)+WA(t),vτ(t)+WA(t)))=(vτ(t)·∇)vτ(t)+(vτ(t)·∇)WA(t)+(WA(t)·∇)vτ(t)+(WA(t)·∇)WA(t),
then,
(156)∥A(1+ζ)/2etAB((vτ(t)+WA(t),vτ(t)+WA(t)))∥H≤∥A(1+ζ)/2etA[vτ(t)·∇]vτ(t)∥H+∥A(1+ζ)/2etA[vτ(t)·∇]WA(t)∥H+∥A(1+ζ)/2etA[WA(t)·∇]vτ(t)∥H+∥A(1+ζ)/2etA[WA(t)·∇]WA(t)∥H=z1+z2+z3+z4.
For z1, we have
(157)z1≤|A(1+ζ)/2etA[vτ1(t)∂1vτ1(t)]|H+|A(1+ζ)/2etA[vτ2(t)∂2vτ1(t)]|H+|A(1+ζ)/2etA[vτ1(t)∂1vτ2(t)]|H+|A(1+ζ)/2etA[vτ2(t)∂2vτ2(t)]|H=z1,1+z1,2+z1,3+z1,4.
In the following, we use Theorem 6.13 in chapter two of [27] to estimate them respectively as follows:
(158)z1,1=12|etAA(1+ζ)/2∂1(vτ1)2|H≤12|etAA(3+2ζ)/4(vτ1)2|H1/2≤C|t|-(3+2ζ)/4eδt|(vτ1)2|H1/2≤C|t|-(3+2ζ)/4eδt|2vτ1(t)A1/4vτ1(t)+R4|H,
the last inequality follows by Theorem A.8 in [25], where δ>0,  R4=A1/4(vτ1)2-2vτ1A1/4vτ1, and |R4|H≤C|A1/8vτ1(t)|L42≤C|vτ1(t)|H12. So, by Hölder inequality and interpolation inequality, we have
(159)z1,1≤C|t|-(3+2ζ)/4eδt∥vτ(t)∥H12.
For z1,2, we have
(160)z1,2=|A(1+ζ)/2etA[vτ2(t)∂2vτ1(t)]|H≤|A(1+ζ)/2etA[A1/4(vτ2(t)A1/4vτ1(t))]|H+|A(1+ζ)/2etA[A1/4vτ1(t)A1/4vτ2(t)]|H+|A(1+ζ)/2etAR5|H,
where
(161)R5=A1/4[vτ2(t)A1/4vτ1(t)]-[A1/4vτ1(t),A1/4vτ2(t)]-vτ2(t)A1/2vτ1(t).
Analogously to estimating z1,1, we have
(162)z1,2≤C|t|-(3+2ζ)/4eδt|vτ2(t)|L4|vτ1(t)|W1/2,4+C|t|-(1+ζ)/2eδt|vτ1(t)|W1/2,4|vτ2(t)|W1/2,4≤C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)eδt∥vτ(t)∥W1/2,42≤C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)eδt∥vτ(t)∥H12.
Similarly, we can get the same estimates for z1,3 and z1,4. Therefore,
(163)z1≤C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)eδt∥vτ(t)∥H12.
Analogously to estimating z1, we can get for z2,z3, and z4 that
(164)z2≤C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)×eδt(∥WA(t)∥H12+∥vτ(t)∥H12),z3≤C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)×eδt(∥WA(t)∥H12+∥vτ(t)∥H12),z4≤C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)×eδt∥WA(t)∥H12.
So, by (163)–(164) and (156), we get
(165)∥A(1+ζ)/2etAB((vτ(t)+WA(t),vτ(t)+WA(t)))∥H≤C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)×eδt(∥WA(t)∥H12+∥vτ(t)∥H12).
For the third term on the right hand side of (154), we obtain
(166)∥A(1+ζ)/2etAf(vτ(t)+WA(t))∥H≤C|t|-(1+ζ)/2eδt(∥vτ(t)∥L63+∥WA(t)∥L63)≤C|t|-(1+ζ)/2×eδt(∥vτ(t)∥H·∥vτ(t)∥H12+∥WA(t)∥H13)≤C|t|-(1+ζ)/2eδt×eδt(∥vτ(t)∥H12+∥WA(t)∥H12),
since ∥vτ(t)∥H and eδt∥WA(t)∥H12 are bounded for t,τ∈(-∞,T], the last inequality follows. For the first term on the right hand side of (154), we have
(167)∥eτAA(1+ζ)/2WA(-τ)∥H≤e-δτ∥A(1+ζ)/2WA(-τ)∥H.
Similar to [26], we can prove that ∥A(1+ζ)/2WA(-τ)∥H has at most polynomial growth when τ→∞. For the reader convenience, we sketch a proof. By Lemma 2, we know that W(t)-W(s) is a D(Aθ/2) valued Brownian motion, for s≤t≤0. So, by the law of iterated logarithm, we have
(168)wn≔supn≤s≤t≤n+1∥W(t)-W(s)∥Hθ|t-s|1/2+(ζ-θ)/4<∞,a.s.n∈ℤ.
Obviously, wn is a i.i.d sequence. By the law of large numbers, there exists an integer-valued random variable n0(w)>0, when n≥n0(w), we have
(169)w-nn≤w-n+⋯+w-1n≤Ew0+1<∞.
This implies that
(170)w-n≤C0(w)n,
for all n>0. In other words,
(171)∥W(t)-W(s)∥Hθ≤C0(w)|[s]|·|t-s|1/2+(ζ-θ)/4,
when s≤t≤[s]+1. By the law of iterated logarithm, we have
(172)∥W(t)∥Hθ≤C1(w)|t|,t∈(-∞,0],
for some positive random variable. By Theorem 5.14 in [23], we know that
(173)WA(t)=∫-∞tAe-(t-s)A(W(t)-W(s))ds.
So, we have that
(174)∥A(1+ζ)/2WA(t)∥H≤∫-∞t∥A1+1/2+ζ/2e-(t-s)A(W(t)-W(s))∥Hds=∫-∞t∥A1+1/2+(ζ-θ)/2e-(t-s)A[Aθ/2(W(t)-W(s))]∥Hds≤∫-∞te-δ(t-s)|t-s|1+1/2+(ζ-θ)/2∥W(t)-W(s)∥Hθds≤∫[t]-1te-δ(t-s)|t-s|1+(ζ-θ)/4·∥W(t)-W(s)∥Hθ|t-s|1/2+(ζ-θ)/4+∫-∞[t]-1e-δ(t-s)|t-s|1+(ζ-θ)/4·C1(w)(|t|+|s|)|t-s|1/2+(ζ-θ)/4≤∫[t]-1te-δ(t-s)|t-s|1+(ζ-θ)/4·C0(w)|[s]|ds+∫-∞[t]-1e-δ(t-s)C1(w)(|t|+|s|)≤(C0(w)+C1(w))(|t|+1),
since s≤[t]-1, the fourth inequality follows. By (167) and (174), we know that
(175)supτ≥0∥eτAA(1+ζ)/2WA(-τ)∥H<∞,a.s.
If we let ζ=1/2<θ, repeating the argument of (174), we can see that ∥WA(t)∥W1,4 also has at most polynomial growth, when t→-∞ a.s., since we have the Sobolev embedding H3/2⊂W1,4. Consider the second term on the right hand side of (154), by (165),
(176)∫-τ0∥A(1+ζ)/2etAB((vτ(t)+WA(t),vτ(t)+WA(t)))∥Hdt≤∫-τ0C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)×eδt(∥WA(t)∥H12+∥vτ(t)∥H12)dt≤∫-10C(|t|-(3+2ζ)/4+|t|-(1+ζ)/2)×eδt(∥WA(t)∥H12+∥vτ(t)∥H12)dt+∫-∞-1Ceδt(∥WA(t)∥H12+∥vτ(t)∥H12)dt<∞,
where the last inequality follows by (152). Analogously, we can prove that
(177)∫-τ0∥A(1+ζ)/2etAf(vτ(t)+WA(t))∥Hdt≤∫-τ0C(|t|-(1+ζ)/2)×eδt(∥vτ(t)∥H∥vτ(t)∥H12+∥WA(t)∥H13)<∞,
where we used (149) and (152) for the last inequality. By (154) and (175)–(177), we get
(178)∥A(1+ζ)/2vτ(0)∥H≤ξ(w),a.s.,
for some positive random variable ξ(w). As H1+δ⊂H1 is compact, by Prohorov Theorem, we know that the family of laws for (vτ(0))τ≥0 taking values in H1 is tight. Since vτ(0)=uτ(0)-WA(0), then so does the law of (uτ(0))τ≥0 taking values in the same space. For t≥0, set
(179)(Ptf)(x)=Ef(u(t,.;0,x)),
where f∈Cb(H01). Following the arguments in [24], for all t0<s<t and all ut0∈H01, by proving
(180)E(f(u(t;t0,ut0))∣ℱs)=Pt-s(u(s;t0,ut0)),
we can show that u is a Markov process. Here, ℱs is the σ-algebra generated by W(r) for r≤s. So, (Pt)t≥0 is the Markov semigroup. Define a dual semigroup Pt* in the space P(H01) of probability measures on H01 as follows:
(181)∫H01fd(Pt*μ)=∫H01Ptfdμ.
Let μτ be the law of uτ(0), which is the solution of (2) with initial condition u(-τ)=0. Then, we have
(182)μτ(f)=Ef(u-τ(0))=Ef(u(τ,·;0,0))=(Pτf)(0)=∫H01Pτfdδ0=∫H01fd(Pτ*δ0),
where we use the fact that u(τ,·;0,0) and uτ(0) have the same law, the second equality follows. Therefore,
(183)Pτ1*μτ=μτ+τ1.
Since (μτ)τ≥0 is tight, then by Prokhorov theorem, we know that (μτ)τ≥0 is relatively compact. We can choose a subsequence of (μτ)τ≥0 denoted by (μτn)n∈ℕ such that for μ∈P(Hσ),
(184)∫H01(Ptf)(x)μ(dx)=limn→∞∫H01(Ptf)(x)μτn(dx)=limn→∞∫H01f(x)Pt*μτn(dx)=limn→∞∫H01f(x)μτn+t(dx)=∫H01f(x)μ(dx).


## 5.2. Uniqueness

The main result of this part is as follows.Theorem 9.
Assumeθ>1/2 in Lemma 2 and ϑ>1/4; then,(i)
the stochastic Burgers equation (2) has a unique invariant measure μ;(ii)
for allu0∈H01 and all Borel measurable functions φ, H01→ℝ, such that ∫H01‍|φ|dμ<∞,
(185)limT→∞1T∫0Tφ(u(t;u0))dt=∫H01φdμa.s.;(iii)
for every Borel measureμ* on H01, one has that
(186)∥Pt*μ*-μ∥TV⟶0ast⟶∞,where ∥·∥TV stands for the total variation of a measure. In particularly, one has that
(187)Pt*μ*(B)⟶μ(B),ast⟶∞,
for every Borel set B∈ℬ(H01)(the Borel σ-algebra of H01).In order to prove Theorem9, we only need Theorem 10 below, see [28, Theorem 4.2.1]. We define P(t,x,·),t>0, x∈H01, to be the transition probability measure that is,
(188)P(t,x,B)=Pt*δx(B)=P(u(t;x)∈B)
for B∈ℬ(H01).Theorem 10.
Assume that the probability measuresP(t,x,·),t>0,x∈H01, are all equivalent, in the sense that they are mutually absolutely continuous. Then, Theorem 9 holds true.In the following, we will prove the irreducibility and the strong Feller property inH01 to get the equivalence of the measure P(t,x,·). For the two notations, we outline them below. For y∈H01,ɛ>0, let
(189)B(y,ɛ)={x∈H01;∥x-y∥H1<ɛ}.(I)
For anyx,y∈H01, such that for all ɛ>0,
(190)P(t,x,B(y,ɛ))>0for eacht>0.(S)
For allO∈ℬ(H01), every t>0, and all xn,x∈H01 such that xn→x in H01, it holds that
(191)P(t,xn,O)⟶P(t,x,O).Before checking the condition (I), we need Lemma11 below. For x∈H01 and ϕ:[0,T]→H01, set
(192)u(t,x,ϕ)=v(t,x,ϕ)+ϕ(t),
where v(t,x,ϕ) is solution of the following equation:
(193)dvdt+Av+B(v+ϕ,v+ϕ)+f(v+ϕ)=0,
for t∈[0,T], with initial condition v(0)=x. As it is proved in previously this equation has a unique solution as follows:
(194)v∈C([0,T];H01),
when x∈H01 and ϕ∈C([0,T];H01).Lemma 11.
DefineΨ(ϕ)=u(·,x,ϕ); then, (i)
the mapping(195)Ψ:C0([0,T];H3/2)⟶C([0,T];H01)is continuous, whereC0([0,T];B)≔{h∈C([0,T];B);h(0)=0} for Banach space B;(ii)
for everyx,y∈H3/2 and T>0 there exists z¯∈C0([0,T];H3/2) such that u(T,x,z¯)=y.Proof.
(i) is proved by(A.30) in the Appendix. To prove (ii), let x,y,∈H3/2 and T>0, define u¯ as
(196)u¯(t)=e-tAx,t∈[0,t0],u¯(t)=e-(T-t)Ay,t∈[t1,T],u¯(t)=u¯(t0)+t-t0t1-t0(u¯(t1)-u¯(t0)),hhhhhhhhhhhhhhhhhhhhhht∈(t0,t1).
Obviously, u¯(t)∈C([0,T];H3/2). Define v¯ as the solution of the following equation:
(197)ddtv¯+Av¯+B(u¯,u¯)+f(u¯)=0,
with initial condition v¯(0)=x; then v¯∈C([0,T];H3/2). Set z¯=u¯-v¯; then it satisfies all the requirements of the lemma.Proposition 12.
With conditions in Theorem9, the irreducibility property (I) is satisfied.Proof.
Letx∈H3/2 and z¯ be the same as (ii) in Lemma 11. By the above lemma, we have that for ɛ>0, we can find δ>0, such that
(198)∥z-z¯∥C0([0,T];H3/2)<δ
implies that
(199)∥u(·,x,z)-u(·,x,z¯)∥C([0,T];H1)<ɛ.
If θ>1/2 in Lemma 2, and denote z and z¯ the corresponding Ornstein-Uhlenbeck process satisfying conditions in the lemma, then z,z¯∈C([0,T];H3/2). Choose δ1>0 such that δ1<δ and
(200)z∈Uδ1=:{z∈C0([0,T];H3/2);∥z-z¯∥C([0,T];H3/2)<δ1}.
Then, for z∈Uδ1, we have that
(201)∥u(T,,x,z)-y∥H1<ɛ.
Recall now that the solution u of the stochastic Burgers equation is equal to Ψ(z), z being the Ornstein-Uhlenbeck process. Then, it remains to show that
(202)P{z(·,w)∈Uδ1}>0.
But this is obviously true. So far, we have proved that for for all t>0, forallx,y∈H3/2, for all ɛ>0,
(203)P(t,x,B(y,ɛ))>0.
Next, we will prove for all x0∈H01,y0∈H3/2, the above inequality also holds. Indeed, for 0<h<t, by Chapman-Kolmogorov equation, we have
(204)P(t,x0,B(y0,ɛ))=∫H01P(t-h,x0,dy)P(h,y,B(y0,ɛ))=∫H3/2P(t-h,x0,dy)P(h,y,B(y0,ɛ))>0.
Since P(t-h,x0,H3/2)=1, we will extend (204) to the case for all x0∈H01,y0∈H01. If this is not true, there exists t0>0,  x0,y0∈H01, ɛ>0 such that
(205)P(t0,x0,B(y0,ɛ))=0.
Then, we can choose y1∈H3/2,  ɛ1>0 such that B(y1,ɛ1)⊂B(y0,ɛ). By (204), we have
(206)P(t0,x0,B(y1,ɛ1))>0,
which is contrary to (205).In this part, it is time to check the condition (S).We will first obtain the strong Feller property inH01 for modified Burgers equation (208) below, then let R→∞ to check the condition (S).FixR>0, let KR:[0,∞[→[0,∞[ satisfy KR∈C1(ℝ+) such that |KR|≤1,  |KR′|≤2 and
(207)KR=1,ifx<R,KR=0,ifx≥R+1.
Consider the following equation:
(208)duR(t)+AuR(t)dt+KR(∥uR(t)∥H12)B(uR(t),uR(t))dt+KR(∥uR(t)∥H12)f(uR)(t)=dW(t).Proposition 13.
There exists a unique mild solutionuR(·,w)∈C([0,T];H01) for (208) which is Markov process with the Feller property in H01, that is for every R>0, t>0, there exists a constant L=L(t,R)>0 such that
(209)|Pt(R)ϕ(x)-Pt(R)ϕ(y)|≤L∥x-y∥H1
holds for all x,y∈H01, and all ϕ∈Cb(H01)≤1, where Pt(R)ϕ(x)≔∫H01ϕ(y)PR(t,x,dy),PR(t,x,·) is the transition probabilities corresponding to (204).Proof.
The proof of existence and uniqueness is similar to Section2. Let ϕ1=ϕ2 in (A.28), by the Gronwall inequality, we know that uR is Lipschitz continuous with respect to initial value. Using the method in Proposition 4.3.3 in [24], we can prove that the solution is a Markov process. To prove the Fell property, we first consider the following Galerkin approximations of (208). Let Pn be the orthogonal projection in  H defined as Pnx=∑j=1n‍〈x,ej〉ej,x∈H. Clearly, Hn≔PnH for every n. Consider the equation in Hn as follows:
(210)dun(R)(t)+Aun(R)(t)dt+KR(∥un(R)(t)∥H12)PnB(un(R)(t),un(R)(t))+KR(∥un(R)(t)∥H12)f(un(R))(t)=dW(t),
with initial condition un(R)(0)=Pnu0. This is a finite-dimensional equation with globally Lipschitz nonlinear functions, so it has a unique progressively measurable solution with P-a.e. trajectory un(R)(·,w)∈C([0,T];Hn), which is also a Markov process in Hn with associated semigroup Pn,t(R) defined as
(211)Pn,t(R)ϕ(x)=Eϕ(un(R)(t;x)),
for all x∈Hn and ϕ∈Cb(Hn). For every R>0,t>0, we can prove that there exists a constant L=L(t,R)>0 such that
(212)|Pn,t(R)ϕ(x)-Pn,t(R)ϕ(y)|≤L∥x-y∥H1
hold for all n∈ℕ,x,y∈Hn, and all ϕ∈Cb(Hn) with ∥ϕ∥H1≤1. Indeed, the following remarkable formula holds true for the differential in x of Pn,t(R)ϕ [29]:
(213)DxPn,t(R)ϕ(x)·h=1tE(ϕ(un(R)(t;x))∫0t〈(PnQQ*Pn)-1/2Dxun(R)(s;x)·h,dβn(s)(PnQQ*Pn)-1/2Dxun(R)〉ϕ(un(R)(t;x))∫0t〈(PnQQ*Pn)-1/2Dxun(R)(s;x)),
for all h∈Hn, where βn is a n-dimensional standard Wiener process with incremental covariance PnQ and Q is the covariance operator of W(t). Obviously, Q is nonnegative, adjoint, Hilbert-Schmidt operator with inverse. Since the eigenvalues αn of the Stokes operator A, in 2-space dimension, behave like n, let θ=1/2+ɛ for some ɛ>0, in Lemma 2, we have D(A)⊂ℛ(Q)⊂D(A3/4), where ℛ(Q) is the image of Q. Therefore,
(214)|DxPn,t(R)ϕ(x)·h|≤1tE(∫0t∥(PnQQ*Pn)-1/2Dxun(R)(s;x)·h∥H2ds)1/2.
Since for y∈Hn(215)∥(PnQQ*Pn)-1/2y∥H2=〈(PnQQ*Pn)-1y,y〉=〈(APnQQ*PnA)-1Ay,Ay〉≤C∥y∥H22,
it follows that
(216)|DxPn,t(R)ϕ(x)·h|≤1tCE(∫0t∥Dxun(R)(s;x)·h∥H22ds)1/2≤1tC(R)∥h∥H1,
where the last inequality follows by the Estimate 4 of the Appendix (note that C(R) is independent of x∈Hn and n∈ℕ). Indeed, un(R)(t,x) is given by vn(t,x)+Pnz(t), where z is the Ornstein-Uhlenbeck process, and vn is the solution of (A.2). Therefore,
(217)|Pn,t(R)ϕ(x)-Pn,t(R)ϕ(y)|≤sup∥h∥H1≤1,k∈Hn|DxPn,t(R)ϕ(k)·h|·∥x-y∥H1≤1tC(R)∥x-y∥H1.
In the following step, we will let n→∞ to get the Fell property for (208). Let x∈H01 and ϕ∈Cb(H01) be given. From the Appendix, Remark A.1, we know that un(R)(t) converges to u(R)(t) strongly in L2(0,T;H01), p-a.s.. By the boundedness and continuous of ϕ as well as Lebesgue dominated convergence theorem, we have
(218)E∫0T|ϕ(un(R)(,;x))-ϕ(u(R)(,;x))|dt⟶0,
which implies that for some subsequence nk,
(219)Eϕ(unk(R)(,;x))⟶Eϕ(u(R)(,;x)),
for a.e.  t∈[0,T]. Take x,y∈H01, by the previous argument, we can find a subsequence nk such that the previous almost sure convergence in t∈[0,T] holds true both x and y.
Thus, from (212), we have
(220)|Pt(R)ϕ(x)-Pt(R)ϕ(y)|≤L∥x-y∥H1,
for a.e. t∈[0,T]. As u(R)(t;x) has continuous trajectories with values in H01, the above inequality holds for all t∈[0,T].Proposition 14.
Under conditions of Theorem9, (S) holds true.Proof.
Taket>0,xn,x∈H01 satisfying xn→x in H1. For every R>0, we have that
(221)∥PR(t,xn,·)-PR(t,x,·)∥TV=sup∥ϕ∥Cb(H1)≤1|Pt(R)ϕ(xn)-Pt(R)ϕ(x)|≤L∥xn-x∥H1⟶0,
as n→∞ by Proposition 13. Then,
(222)∥PR(t,xn,·)-P(t,xn,·)∥TV+∥PR(t,x,·)-P(t,x,·)∥TV=sup∥ϕ∥Cb(H1)≤1|Pt(R)ϕ(xn)-Ptϕ(xn)|+sup∥ϕ∥Cb(H1)≤1|Pt(R)ϕ(x)-Ptϕ(x)|=sup∥ϕ∥Cb(H1)≤1|Eϕ(uR(t;xn))-Eϕ(u(t;xn))|+sup∥ϕ∥Cb(H1)≤1|Eϕ(uR(t;x))-Eϕ(u(t;x))|≤2∫ΩI{supn∈ℕ∥u(t;xn)∥H1>R}P(dw)+2∫ΩI{∥u(t;x)∥H1>R}P(dw)⟶0,asR→∞,
where the inequality follows by the consistency of u(t;x) and u(R)(t;x), when ∥u(t;x)∥H1≤R, and the limit follows by (A.21). Therefore,
(223)∥P(t,xn,·)-P(t,x,·)∥TV≤∥P(t,xn,·)-PR(t,xn,·)∥TV+∥PR(t,xn,·)-PR(t,x,·)∥TV+∥PR(t,x,·)-P(t,x,·)∥TV⟶0,
as n→∞.


## 6. Example

Our theory can be applied to stochastic reaction diffusion equations or stochastic real valued Ginzburg Landau equation in high dimensions as follows:(224)∂u∂t-Δu+|u|2u-u=dW,on[0,T]×D,u(t,x)=0,t∈[0,T],x∈∂D,u(0,x)=u0(x),x∈D,
where u(t,x)=(u1(t,x),u2(t,x)) is the velocity field, Δ denotes the Laplace operator, W stands for the Q-Wiener process, and D is a regular bounded open domain of ℝ2.



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*Source: 267328-2013-12-31.xml*