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Stochastic wave equations with dissipative damping
Stochastic Processes and their Applications 117 (2007) 1001–1013 www.elsevier.com/locate/spa
Stochastic wave equations with dissipative damping Viorel Barbu a , Giuseppe Da Prato b , Luciano Tubaro c,∗ a University Al. I. Cuza, 700506, Iasi, Romania b Scuola Normale Superiore, 56126, Pisa, Italy c Department of Mathematics, University of Trento, Italy
Received 11 April 2006; received in revised form 23 October 2006; accepted 25 November 2006 Available online 14 December 2006
Abstract We prove existence and (in some special case) uniqueness of an invariant measure for the transition semigroup associated with the stochastic wave equations with nonlinear dissipative damping. c 2006 Elsevier B.V. All rights reserved.
MSC: 47D007; 35K90 Keywords: Stochastic wave equations; Dissipative damping; White noise; Invariant measure
1. Introduction We are here concerned with the following problem in a bounded open set O ⊂ Rn with regular boundary ∂O: p d X˙ (t, ξ ) = (∆ξ X (t, ξ ) − X˙ (t, ξ ) − U 0 ( X˙ (t, ξ )))dt + QdW (t, ξ ), (1.1) X (t, ξ ) = 0 in ∂O, X (0, ·) = x ∈ H01 (O), X˙ (0, ·) = y ∈ L 2 (O), where U is a regular convex function from H into R fulfilling suitable growth conditions (see Hypothesis 1.1 below), W (t) is a cylindrical Wiener process in H and Q ∈ L(H ) is symmetric, nonnegative and of trace class. We set H = L 2 (O) (norm | · |2 , inner product h·, ·i2 ) and ∗ Corresponding author. Tel.: +39 0461 881518; fax: +39 0461 881624.
E-mail address: [email protected] (L. Tubaro). c 2006 Elsevier B.V. All rights reserved. 0304-4149/$ - see front matter doi:10.1016/j.spa.2006.11.006
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V = H01 (O) (norm |A1/2 · |2 = |∇ · |2 ). The norm of L p (O), 1 ≤ p < ∞ will be denoted by | · | p . We write problem (1.1) as an abstract system setting Y (t) = X˙ (t), dX (t) = Y (t)dt, p (1.2) dY (t) = −(AX (t) + Y (t) + U 0 (Y (t)))dt + QdW (t) X (0) = x, Y (0) = y, where A is the realization of the Laplace operator with Dirichlet boundary conditions in L 2 (O), that is Ax = −∆ξ x,
∀x ∈ H 2 (O) ∩ H01 (O).
It is well known that A is self-adjoint and A−1 is compact. Let us denote by {ek } a complete orthonormal system of eigenfunctions of A and by {αk } the corresponding eigenvalues. We shall choose for simplicity Q as a diagonal operator with respect to {ek }, i.e. of the form Qek = λk ek ,
k ∈ N,
where λk > 0 for all k ∈ N and we shall assume that Tr(AQ) < +∞
(1.3)
and ∞ X
λk kek k2∞ : = κ < +∞.
(1.4)
k=1
We notice that when O is an n-dimensional rectangle there exists c > 0 such that kek k∞ ≤ c for all k ∈ N, so that (1.4) is a consequence of (1.3). Stochastic wave equations of the form (Klein–Gordon equations) p d X˙ (t, ξ ) = (∆ξ X (t, ξ ) − X˙ (t, ξ ) − g(X (t, ξ )))dt + QdW (t, ξ ), (1.5) X (t, ξ ) = 0 in ∂O, X (0, ·) = x ∈ H01 (O), X˙ (0, ·) = y ∈ L 2 (O), were studied in [5–8,15,16,19] and [3]. For instance in [3] there were proved, under suitable assumptions on g, existence and uniqueness of an invariant measure for the transition semigroup associated with this equation as well as the corresponding Kolmogorov equation. Nonlinear stochastic equations of the form (1.1), with nonlinearity on the damping, were first studied by Pardoux [17]. However, the existence of an invariant measure for the corresponding transition semigroup associated with (1.1) seems to be studied here for the first time. We note however that a finite-dimensional version of the problem considered here was previously studied in [4]. Finally, concerning the function U , we shall assume that the following hypothesis holds. Hypothesis 1.1. U : R → R is a convex C 2 function which satisfies the following conditions. (i) There exist γ > 0, ω > 0 such that U 0 (r )r ≥ ω|r |γ +2 ,
∀r ∈ R.
(1.6)
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(ii) There exist η1 , η2 > 0 and C1 , C2 ∈ R such that |U 0 (r )| ≤ η1 |r |γ +1 + C1 ,
∀r ∈ R
(1.7)
and |U 00 (r )| ≤ η2 |r |γ + C2 ,
∀r ∈ R.
(1.8)
γ +2 γ +1
and assume also that U (0) = 0. We set = The main result of this paper Theorem 3.4 amounts to saying that under Hypothesis 1.1 the stochastic flow t → (X (t), Y (t)) associated with Eq. (1.1) has an invariant measure. The uniqueness of the invariant measure is proven in the special case where γ , n satisfy condition (4.1) (Theorem 4.1). The uniqueness of invariant measure in the general case remains an open and apparently a difficult problem if one takes into account that the usual techniques relying on Malliavin calculus (see [4]) or coupling seem to be inapplicable in the present situation. In this context the works [11–14] should also be cited. γ∗
2. Existence and uniqueness of solutions We can rewrite problem (1.2) in the space V × H as dZ = AZ dt − F (Z )dt + BdW (t) x , Z (0) = y
(2.1)
where X (t) 0 1 Z (t) = , A = Y (t) −A 0 0 0 p F (Z ) = , B= . Y (t) + U 0 (Y (t)) Q
It is well known (see e.g. [18, p. 220]) that A is the infinitesimal generator of a strongly continuous group in V × H given by cos(A1/2 t) A−1/2 sin(A1/2 t) et A = . −A1/2 sin(A1/2 t) cos(A1/2 t) (t) Definition 2.1. An adapted stochastic process Z (t) = YX (t) with values in V × H is called a mild solution to system (1.2) if for any T > 0 we have (i) X ∈ C W ([0, T ]; V ), Y ∈ C W ([0, T ]; H ), ∗ (ii) U 0 (Y ) ∈ C W ([0, T ]; L γ (O)), (iii) for all t ≥ 0, Z t X (t) = x + Y (s)ds, 0 Y (t) = −A1/2 sin(A1/2 t)x + cos(A1/2 t)y Z t − cos(A1/2 (t − s))(Y (s) + U 0 (Y (s)))ds 0 Z t p + cos(A1/2 (t − s)) QdW (s). 0
(2.2)
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The suffix W in the space above means that the processes considered are adapted and mean square integrable. Remark 2.2. Theorem 2.3 below can be obtained from Pardoux existence results for secondorder stochastic equations of hyperbolic type; see [17, Theorem 2.1]. However for later purposes we shall consider below a slightly different version and give a direct proof via Yosida approximations. The main result of this section is the following. Theorem 2.3. Let x ∈ V and y ∈ H . Then there is a unique mild solution e e x x X (t) replacing is another mild solution of (1.2), with e(t) e y y we have Y
e(t)|2 ≤ |x − e E|X (t) − e X (t)|2V + E|Y (t) − Y x |2V + |y − e y|2H . H Proof. For any ε > 0 we consider the approximating problem dX ε (t) = Yε (t)dt, p dY (t) = −(AX ε (t) + Yε (t) + βε (Y (t)))dt + QdW (t) ε X ε (0) = x, Yε (0) = y,
X (t) Y (t)
of (1.2). If
(2.3)
(2.4)
where βε is the Yosida approximation of the mapping U 0 : 1 (r − (1 + εU 0 )−1 (r )) = U 0 (1 + εU 0 )−1 (r ), ε We can rewrite Eq. (2.4) in the space V × H as dt − Fε (Z ε )dt + BdW (t) dZ ε = AZ ε x , Z ε (0) = y βε (r ) =
ε > 0.
(2.5)
where Z ε (t) =
X ε (t) , Yε (t)
Fε (Z ε ) =
0 , βε (Yε (t))
0 . B= p Q
It is well known that the linear operator A generates a strongly continuous semigroup of contractions on V × H while the mapping Fε : V × H → V × H is Lipschitz continuous and monotone. Then by standard existence results, see e.g. [10, p. 81], there is a unique solution X ε ∈ C W ([0, T ]; V ),
Yε ∈ C W ([0, T ]; H )
of (2.5). Moreover, using the Itˆo formula for function φ(x, y) = 12 (hAx, xi2 + |y|22 ), we see that Z t E|∇ X ε (t)|22 + E|Yε (t)|22 + 2E [hβε (Yε (s)), Yε (s)i2 + |Yε (s)|22 ]ds 0
=
|∇x|22
+ |y|22
+ t Tr Q.
(2.6)
We infer that E|∇ X ε (t)|22
+ E|Yε (t)|22
where C is independent of ε.
t
Z + 2E 0
γ +2
[|Yε (s)|γ +2 + |Yε (s)|22 ]ds ≤ C(1 + t),
(2.7)
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Then on a subsequence of {X ε , Yε }, still denoted {X ε , Yε }, we have as ε → 0 (i) X ε → X weak star in L ∞ (0, T, L 2 (Ω , V )), (ii) Yε → Y weak star in L ∞ (0, T, L 2 (Ω , H )), (iii) Yε → Y weakly in L γ +2 (0, T ; L γ +2 (Ω , L γ +2 (O))) ∗ ∗ ∗ (iv) βε (Yε ) → η weakly in L γ ([0, T ]; L γ (Ω , L γ (O))).
(2.8)
It follows that Z t Y (s)ds, X (t) = x + 0 Z t 1/2 1/2 1/2 Y (t) = −A sin(A t)x + cos(A t)y + cos(A1/2 (t − s))(Y (s) + η(s))ds (2.9) 0 Z t p + cos(A1/2 (t − s)) QdW (s). 0
It is clear that the process (X, Y ) is adapted. To prove that it satisfies (1.2) we have to prove that η(t, ξ, ω) = U 0 (Y )(t, ξ, ω),
a.e. (t, ξ, ω) ∈ [0, T ] × O × Ω .
(2.10)
Taking into account (2.8), in order to prove (2.2) it suffices to show, see e.g. [2, page 42], T
Z
Z
lim sup E ε→0
0
O
βε (Yε )Yε dξ dt ≤ E
By (2.4) we see, via Itˆo’s formula, that Z 2 2 E|∇ X ε (T )|2 + E|Yε (T )|2 + 2E
T
T
Z
O
0
O
0
Z
Z
ηY dξ dt.
(2.11)
βε (Yε )Yε dsdξ + E
T
Z 0
|Yε (s)|22 ds
= |∇x|22 + |y|22 + T Tr(Q).
(2.12)
Similarly by (2.9) E|∇ X (T )|22 + E|Y (T )|22 + 2E
T
Z
Z
0
O
ηY dsdξ + E
T
Z 0
|Y (s)|22 ds
= |∇x|22 + |y|22 + T Tr(Q).
(2.13)
Since by virtue of (2.8) and (2.9) X ε (T ) → X (T ) Yε (T ) → Y (T )
weakly in L 2W (Ω , V ), weakly in
L 2W (Ω ,
H ),
(2.14) (2.15)
we have lim inf(E|∇ X ε (T )|22 + E|Yε (T )|22 ) ≥ E|∇ X (T )|22 + E|Y (T )|22 ε→0
and by (2.12) and (2.13) we get (2.10) as claimed. This completes the proof of existence. Inequality (2.3) and in particular the uniqueness follow from the Itˆo formula applied to X (t) − e X (t) and to the function φ(x, y) = 12 (hAx, xi2 + |y|22 ).
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3. Existence of invariant measures In order to prove the existence of an invariant measure we shall choose several Lyapunov functions φ to obtain estimates for the expectation of suitable norms of X (t), Y (t) using Itˆo’s formula 1 (3.1) dφ(X, Y ) = hφx (X, Y ), dX i2 + hφ y (X, Y ), dY i2 + Tr[Qφ yy (X, Y )]dt. 2 With the help of these functions we shall prove suitable a priori estimates on the solution of (1.2). Then existence will follow from the Krylov–Bogoliubov theorem (see [9], Proposition 11.3). We first choose φ(x, y) =
1 (|Ax|22 + hAy, yi2 ). 2
(3.2)
Then we have φx (x, y) = A2 x,
φ y (x, y) = Ay,
φ yy (x, y) = A
and so p 1 2 0 dφ(X, Y ) = hA X, Y i2 − hAY, AX + Y + U (Y )i2 + Tr(Q A) dt + hY, QdW (t)i2 . 2 It follows that Z p 1 1/2 2 00 2 dφ(X, Y ) = −|A Y |2 − U (Y )|∇Y | dξ + Tr(Q A) dt + hY, QdW (t)i, 2 O which yields E|AX (t)|22 + E|∇Y (t)|22 + 2E Z tZ + 2E 0
O
t
Z 0
|∇Y (s)|22 ds
U 00 (Y (s))|∇Y (s)|2 dξ ds = |Ax|22 + |∇ y|22 + t Tr(Q A).
(3.3)
As an immediate consequence of (3.3) we find the following result: Lemma 3.1. We have Z t E |∇Y (s)|22 ds ≤ |Ax|22 + |∇ y|22 + t Tr(Q A).
(3.4)
0
Now we choose Z 1 U (y(ξ ))dξ. φ1 (x, y) = (hAx, xi2 + hAx, yi2 ) + 2 O Then we have (φ1 )x (x, y) = Ax + Ay,
(φ1 ) y (x, y) = Ax + U 0 (y),
(3.5)
(φ1 ) yy (x, y) = U 00 (y)
and so dφ(X, Y ) = hAX + AY, Y i2 − hAX + U 0 (Y ), AX + Y + U 0 (Y )i2 dt
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p 1 + Tr(QU 00 (Y ))dt + hY, QdW (t)i2 2 Z 1 0 0 2 00 U (Y (ξ ))Y (ξ )dξ + Tr(QU (Y )) dt = hAY, Y i2 − |AX + U (Y )|2 − 2 O p + hY, QdW (t)i2 . We note that U 0 (y)y ≥ U (y); hence integrating over t and taking expectation yields Z tZ Z t 0 2 |AX (s) + U (Y (s))|2 ds + E U (Y (s, ξ ))dξ ds E[φ1 (X (t), Y (t))] + E 0
0
≤ φ1 (x, y) + E
t
Z 0
|∇Y (s)|22 ds +
t
Z
O
Tr(QU 00 (Y (s)))ds.
Lemma 3.2. There exists a constant κ1 such that for all t ≥ 0 Z tZ Z t 0 2 |AX (s) + U (Y (s))|2 ds + E U (Y (s, ξ ))dξ ds ≤ κ1 (1 + t). E 0
(3.6)
0
O
0
(3.7)
Proof. Since 1 1 (|∇x|22 − |∇x|2 |∇ y|2 ) ≥ − |∇ y|22 , 2 2 we deduce by (3.6) that Z t Z tZ 0 2 E |AX (s) + U (Y (s))|2 ds + E U (Y (s, ξ ))dξ ds φ1 (x, y) ≥
0
O
0
≤ φ1 (x, y) + E
t
Z 0
|∇Y (s)|22 ds +
t
Z 0
1 Tr(QU 00 (Y (s)))ds + E|∇Y (t)|22 . 2
(3.8)
Now by (3.3) we have that Z t 2 E|∇Y (t)|2 + E |∇Y (s)|22 ds ≤ 2φ(x, y) + t Tr(Q A) 0
and so, substituting in (3.8), we obtain that Z t Z tZ U (Y (s, ξ ))dξ ds E |AX (s) + U 0 (Y (s))|22 ds + E 0
0
≤ 2φ(x, y) + φ1 (x, y) +
t
Z 0
O
t Tr(QY 2 (s))ds + Tr(Q A). 2
(3.9)
To estimate the last integral in (3.9) we notice that, by (1.4) and (1.8), Z Z ∞ X 00 2 γ Tr(QU (Y (s))) = λk ek (η2 |Y | + C2 )dξ ≤ κ (η2 |Y |γ + C2 )dξ. k=1
So, (3.7) follows.
O
O
Corollary 3.3. There exists a constant κ2 such that for all t ≥ 0 Z t γ∗ E |AX (s)|γ ∗ ds ≤ κ2 (1 + t). 0
(3.10)
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Proof. By (3.7) it follows that Z t γ∗ E |U 0 (Y (s))|γ ∗ ds ≤ κ1 (1 + t). 0
Consequently, for suitable constants c1 , c2 , Z t Z t Z t γ∗ γ∗ γ∗ 0 |U 0 (Y (s))|γ ∗ ds |AX (s) + U (Y (s))|γ ∗ ds + c1 E |AX (s)|γ ∗ ds ≤ c1 E E 0 0 0 Z t Z t γ∗ |U 0 (Y (s))|γ ∗ ds |AX (s) + U 0 (Y (s))|22 ds + c1 E ≤ c2 E 0
0
≤ κ1 (1 + t).
We now prove the existence of an invariant measure for the Markov process X (t, x, y) , t ≥ 0, x ∈ H, y ∈ V. t 7→ Y (t, x, y) (t, x, y) We denote by πt (x, y, ·, ·) the law of YX (t, x, y) in V × H . We recall that the transition semigroup Pt is given by Z ϕ(x1 , y1 )πt (x, y, dx1 , dy1 ), (3.11) Pt ϕ(x, y) = E[ϕ(X (t, x, y), Y (t, x, y))] = V ×H
for all (x, y) ∈ V × H and ϕ ∈ Cb (V × H ). Theorem 3.4. There exists an invariant measure µ for Pt . Proof. Fix x0 in D(A) and y0 in V . For any R > 0 define n o γ∗ K R = (x, y) ∈ V × H : |Ax|γ ∗ + |∇ y|22 ≤ R . By the Agmon–Douglis–Nirenberg theorem, see e.g. [1], combined with the Sobolev embedding theorem, K R is a compact subset of V × H . We have Z πt (x0 , y0 , K Rc ) = πt (x0 , y0 , dx, dy) γ∗ [|Ax|γ ∗ +|∇ y|22 >R]
Z 1 γ∗ ≤ (|Ax|γ ∗ + |∇ y|22 )πt (x0 , y0 , dx, dy) R V ×H 1 γ∗ = E(|AX (t, x0 , y0 )|γ ∗ + |∇Y (t, x0 , y0 )|22 ). R Consequently Z Z t 1 1 t γ∗ c πs (x0 , y0 , K R )ds ≤ E(|AX (s, x0 , y0 )|γ ∗ + |∇Y (s, x0 , y0 )|22 )ds t 0 tR 0 and by Lemma 3.1 and Corollary 3.3 it follows that there exists a constant M > 0 such that Z 1 t M πs (x0 , y0 , K Rc )ds ≤ . t 0 R
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Therefore the family of measures Z t 1 πs (x0 , y0 , ·, ·)ds t 0 t≥0 is tight in (V × H, B(V × H )) and by the Krylov–Bogoliubov theorem there exists an invariant measure for Pt . ∗
Corollary 3.5. The support of µ is in W 2,γ (O) × V , i.e. Z γ∗ (kxkW 2,γ ∗ (O ) + kyk2H 1 (O ) )dµ(x, y) < ∞.
(3.12)
0
V ×H
Moreover, we have for λ > 0 and sufficiently small, Z 2 2 |y|22 eλ(kxk +|y|2 ) dµ(x, y) < ∞,
(3.13)
V ×H
and (3.13) remains true for any invariant measure µ for Pt . (Here k · k = k · k H 1 (O ) .) 0
Proof. Estimate (3.12) follows by estimates (3.4) and (3.10) taking into account that, by the Agmon–Douglis–Nirenberg theorem, W 2,γ (O) ⊂ {x ∈ V : Ax ∈ L γ (O)}. ∗
∗
In order to prove (3.13) we consider the Lyapunov function ψ(x, y) = eλ(kxk
2 +|y|2 ) 2
and notice that by Itˆo’s formula, dψ(X (t), Y (t)) = ψ(2λhAX (t), Y (t)i2 − hY (t), AX (t) + Y (t) + U 0 (Y (t))i2 )dt p 1 00 + Tr Qψ yy dt + 2λh QdW, Y i2 ψ. 2 This yields γ +2
dψ(X (t), Y (t)) + ψ(2λ|Y (t)|22 + 2λω|Y (t)|γ +2 ) dt ≤ λψ(Tr Q + λ|Q 1/2 Y |22 ) dt p + 2λh QdW, Y i2 ψ and so for λ > 0 sufficiently small we get Z t 2 2 E |Y (s)|22 eλ(kX (s)k +|Y (s)|2 ) ds ≤ C1 t + C2 , 0
which implies the desired relation (3.13) for any invariant measure µ for Pt .
4. Uniqueness of the invariant measure We shall prove uniqueness of µ if γ is related to dimension n of the space by condition (4.1) below. 0≤γ ≤2 We have
if n = 1,
0≤γ <2
if n = 2,
γ =1
if n = 3.
(4.1)
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Theorem 4.1. Assume that Hypothesis (1.1) holds with γ , n as in (4.1). Then there is a unique invariant measure µ for Pt . Proof. We set Z 1 = Dx X (t, x, y)h, Z 2 = Dx Y (t, x, y)h. We have Z 10 = Z 2 , Z 1 (0) = h,
Z 20 + AZ 1 + Z 2 + U 00 (Y (t))Z 2 = 0
(4.2)
Z 2 (0) = 0.
Consider the Lyapunov function 1 (kz 1 k2 + |z 2 |22 ) + λhz 1 , z 2 i2 . 2 (Here kzk = |∇z|2 .) We have ϕ(z 1 , z 2 ) =
d ϕ(Z 1 (t), Z 2 (t)) + λkZ 1 (t)k2 + (1 − λ)|Z 2 (t)|22 + λhZ 1 , Z 2 i2 dt + λhU 00 (Y )Z 2 , Z 1 i2 ≤ 0, a.e. t > 0.
(4.3)
We are going to show that |D Pt ϕ(X, Y )| ≤ CkDϕk0 e−δt
∀t ≥ 0, ϕ ∈ Cb1 (V × H )
and this implies the uniqueness of the invariant measure by standard arguments. Indeed if ζ is another invariant measure for Pt , we have Z Z Pt ϕ(x, y) − ϕ(x , y )dζ (x , y ) |Pt ϕ(x, y) − Pt ϕ(x1 , y1 )|dζ (x1 , y1 ) 1 1 1 1 ≤ V ×H V ×H Z ≤ CkDϕk0 e−δt (kx − x1 k + |y − y1 |)dζ (x1 , y1 ). V ×H
Hence lim Pt ϕ(x, y) =
t→∞
Z
ϕ(x1 , y1 )dζ (x1 , y1 ) V ×H
and this yields Z Z ϕ(x1 , ζ1 )dζ (x1 , ζ1 ) = V ×H
ϕ(x1 , ζ1 )dµ(x1 , ζ1 ), V ×H
i.e., ζ = µ as claimed. We note the estimate (see (2.6) and (1.6)) Z t γ +2 2 2 EkX (t)k + E|Y (t)|2 + 2E (|Y (s)|22 + ω|Y (s)|γ +2 )ds ≤ t Tr Q + kxk2 + |y|22 . (4.4) 0
Now we come back to solution (Z 1 , Z 2 ) to (4.2). By (4.3) we have for 0 < λ ≤ d λ 3λ 2 ϕ(Z 1 (t), Z 2 (t)) + kZ 1 (t)k + 1 − |Z 2 (t)|22 dt 2 2
1 2
≤ λkZ 1 (t)kkU 00 (Y (t))Z 2 (t)k−1 . (Here k · k−1 is the norm of
H −1 (O).)
kU 00 (Y )Z 2 k−1 ≤ kU 00 (Y )Z 2 kq ∗
(4.5)
We have via the Sobolev embedding theorem Z 1/q ∗ q∗ γ q∗ ≤ η1 |Z 2 (t)| |Y (t)| dξ O
(4.6)
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where q1∗ = 1 − p1∗ and p ∗ = ∞ if n = 1, p ∗ ≥ 2 if n = 2, p ∗ = 6 if n = 3. Taking into account (1.8) we have by (4.6) that γ
γ
kU 00 (Y )Z 2 k−1 ≤ η1 |Z 2 |2 |Y | 2γ q ∗ ≤ η1 |Z 2 |2 |Y |γ +2 .
(4.7)
2−q ∗
Substituting (4.7) into (4.5) we get λ 3λ d γ |Z 2 (t)|22 ≤ λη1 kZ 1 (t)k|Z 2 (t)|2 |Y (t)|γ +2 , ϕ(Z 1 (t), Z 2 (t)) + kZ 1 (t)k2 + 1 − dt 2 2 ∀t ≥ 0. Recalling that 1−λ 1+λ (kz 1 k2 + |z 2 |2 ) ≤ ϕ(z 1 , z 2 ) ≤ (kz 1 k2 + |z 2 |2 ) 2 2 we obtain that d λ γ ϕ(Z 1 (t), Z 2 (t)) + ϕ(Z 1 (t), Z 2 (t)) ≤ 4η1 λϕ(Z 1 (t), Z 2 (t))|Y (t)|γ +2 , dt 4
∀t ≥ 0.
Integrating the latter we get ϕ(Z 1 (t), Z 2 (t)) ≤ e
− 41 λt+4η1 λ
Rt 0
γ
|Y (s)|γ +2 ds
ϕ(Z 1 (0), Z 2 (0))
∀t ≥ 0, P-a.s.
(4.8)
It remains to estimate the expectation of the right side of (4.8). To this end we shall proceed as in [3]. Namely, consider the process Z t ν γ 2 2 φ(t) = (kX (t)k + |Y (t)|2 ) + 4η1 λ |Y (s)|γ +2 ds. 2 0 By the Itˆo formula, ν γ dφ(t) = −ν|Y (t)|22 + 4η1 λ|Y (t)|γ +2 − νhU 0 (Y (t)), Y (t)i2 + Tr Q dt 2 p + hνY (t), QdW (t)i2 . This yields p γ +2 ν ν γ +2 dφ(t) ≤ −ν|Y (t)|22 + Tr Q + (4η1 λ) 2 − |Y (t)|γ +2 dt + hνY (t), QdW (t)i2 . 2 2 Next we apply Itˆo’s formula to eφ . We get p d(eφ ) ≤ eφ dφ + ν 2 eφ | QY |2 dt and therefore ν p γ +2 ν γ +2 d(eφ ) ≤ ν 2 eφ | QY |22 dt + eφ Tr Q + (4η1 λ) 2 − |Y (t)|γ +2 dt 2 2 p + φ ν hY (t), QdW (t)i2 . Finally, taking ν ≤ 3λ, with λ sufficiently small, we get Eeφ(t) ≤ Eeφ(0) + Cλ(λkQk
γ +2 2
γ
γ +2
+ Tr Q + λ 2 η1 2 )
t
Z 0
Eeφ(s) ds
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V. Barbu et al. / Stochastic Processes and their Applications 117 (2007) 1001–1013
where C is independent of λ and kQk is the norm of Q in L 2 (O). This yields Eφ(t) ≤ eCλ(λkQk
γ +2 2 +Tr
γ
γ +2 2 )t
Q+λ 2 η1
eν(kxk
2 +|y|2 ) 2
.
For ν ≥ 2λ and λ small this yields 4η1 λ
Ee
Rt 0
γ
|Y (s)|γ +2 ds
≤e
ν(kxk2 +|y|22 ) Cλ(λkQk
e
γ +2 2 +Tr
γ +2 2 )t
γ
Q+λ 2 η1
.
(4.9)
Substituting (4.9) into (4.8) we see that Eϕ(Z 1 (t), Z 2 (t)) ≤ e
ν(kxk2 +|y|22 ) (C(λkQk
e
γ +2 2 +Tr
γ
γ +2 2 )− 1 )λt 4
Q+λ 2 η1
ϕ(Z 1 (0), Z 2 (0)).
Assume that λkQk
γ +2 2
γ +2
γ
+ Tr Q + λ 2 η1 2 <
1 . 4C
(4.10)
Then there is a δ > 0 such that Eϕ(Z 1 (t), Z 2 (t)) ≤ eν(kxk
2 +|y|2 ) 2
e−δt ϕ(Z 1 (0), Z 2 (0)).
(4.11)
Now, let µ be any invariant measure for Pt . Recalling that (see Corollary 3.5) we have Z 2 2 eν(kxk +|y|2 ) dµ(x, y) < ∞ V ×H
then we infer by (4.11) that |D Pt ϕ(x, y)| ≤ CkDϕk0 e−δt K (x, y),
∀(x, y) ∈ V × H
where K ∈ L 1 (V × H ; dµ) and so we conclude as above about the uniqueness of invariant measure µ as in the proof of Theorem 4.1. We notice that condition (4.10) can be reduced via the transformation X 7→ ε X , with ε appropriately chosen, to the condition: λ sufficiently small. Indeed by this transformation Eq. (1.1) reduces to an equation of the same type with ε Q instead of Q and εU 0 ( εy ) instead of U 0 . Then condition (4.10) reduces to λε
γ +2 2
kQk
γ +2 2
γ +2
γ
+ εTr Q + ε −γ η1 2 λ 2 ≤
This completes the proof.
1 . 4C
Acknowledgements The first author was partially supported by the CEEX research project 09-D11 of the Romanian Ministry of Education and Research. The second and third authors were partially supported by the Italian National Project MURST “Equazioni di Kolmogorov.” References [1] [2] [3] [4] [5]
S.A. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, 1965. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1975. V. Barbu, G. Da Prato, The stochastic nonlinear damped wave equation, Appl. Math. Optim. 46 (2002) 125–141. V. Barbu, G. Da Prato, L. Tubaro, Stochastic second order equations with nonlinear damping, 2005. Preprint. R. Carmona, D. Nualart, Random non-linear wave equation: Smoothness of solutions, Probab. Theory Related Fields 95 (1993) 87–102.
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[6] H. Crauel, A. Debussche, F. Flandoli, Random attractors, J. Dynam. Differential Equations 9 (1997) 307–341. [7] S. Cerrai, M. Freidlin, On the Smoluchowski–Kramers approximation for a system with an infinite number of degrees of freedom, 2004. Preprint SNS. [8] R. Dalang, N. Frangos, The stochastic wave equation in two spatial dimensions, Ann. Probab. 26 (I) (1998) 187–212. [9] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. [10] G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systems, in: London Mathematical Society Lecture Notes, vol. 229, Cambridge University Press, 1996. [11] M. Hairer, J. Mattingly, Ergodicity of the degenerate stochastic 2-D Navier–Stokes equations, 2004. Preprint. [12] S. Kuksin, A. Shirikyan, Stochastic dissipative PDEs and Gibbs measures, Commun. Math. Phys. 213 (2) (2000) 291–330. [13] N. Masmoudi, L. Sang Yang, Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs, Commun. Math. Phys. 227 (3) (2002) 461–481. [14] J. Mattingly, Ergodicity of 2-D Navier–Stokes equations with random forcing and large viscosity, Commun. Math. Phys. 206 (2) (1999) 273–288. [15] A. Millet, P.L. Morien, On a nonlinear stochastic wave equation in the plane: Existence and uniqueness of the solution, Ann. Appl. Probab. 11 (2001) 922–951. [16] A. Millet, M. Sanz-Sol´e, Approximation and support theorem for a wave equation in two space dimensions, Bernoulli 6 (2000) 887–915. [17] E. Pardoux, Equations aux deriv´ees partielles stochastiques nonlin´eaires monotones, Th`ese, Universit´e Paris XI, 1975. [18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. [19] S. Peszat, J. Zabczyk, Nonlinear stochastic wave and heat equations, Probab. Theory Related Fields 116 (2000) 421–443.
Stochastic wave equations with dissipative damping Viorel Barbu a , Giuseppe Da Prato b , Luciano Tubaro c,∗ a University Al. I. Cuza, 700506, Iasi, Romania b Scuola Normale Superiore, 56126, Pisa, Italy c Department of Mathematics, University of Trento, Italy
Received 11 April 2006; received in revised form 23 October 2006; accepted 25 November 2006 Available online 14 December 2006
Abstract We prove existence and (in some special case) uniqueness of an invariant measure for the transition semigroup associated with the stochastic wave equations with nonlinear dissipative damping. c 2006 Elsevier B.V. All rights reserved.
MSC: 47D007; 35K90 Keywords: Stochastic wave equations; Dissipative damping; White noise; Invariant measure
1. Introduction We are here concerned with the following problem in a bounded open set O ⊂ Rn with regular boundary ∂O: p d X˙ (t, ξ ) = (∆ξ X (t, ξ ) − X˙ (t, ξ ) − U 0 ( X˙ (t, ξ )))dt + QdW (t, ξ ), (1.1) X (t, ξ ) = 0 in ∂O, X (0, ·) = x ∈ H01 (O), X˙ (0, ·) = y ∈ L 2 (O), where U is a regular convex function from H into R fulfilling suitable growth conditions (see Hypothesis 1.1 below), W (t) is a cylindrical Wiener process in H and Q ∈ L(H ) is symmetric, nonnegative and of trace class. We set H = L 2 (O) (norm | · |2 , inner product h·, ·i2 ) and ∗ Corresponding author. Tel.: +39 0461 881518; fax: +39 0461 881624.
E-mail address: [email protected] (L. Tubaro). c 2006 Elsevier B.V. All rights reserved. 0304-4149/$ - see front matter doi:10.1016/j.spa.2006.11.006
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V. Barbu et al. / Stochastic Processes and their Applications 117 (2007) 1001–1013
V = H01 (O) (norm |A1/2 · |2 = |∇ · |2 ). The norm of L p (O), 1 ≤ p < ∞ will be denoted by | · | p . We write problem (1.1) as an abstract system setting Y (t) = X˙ (t), dX (t) = Y (t)dt, p (1.2) dY (t) = −(AX (t) + Y (t) + U 0 (Y (t)))dt + QdW (t) X (0) = x, Y (0) = y, where A is the realization of the Laplace operator with Dirichlet boundary conditions in L 2 (O), that is Ax = −∆ξ x,
∀x ∈ H 2 (O) ∩ H01 (O).
It is well known that A is self-adjoint and A−1 is compact. Let us denote by {ek } a complete orthonormal system of eigenfunctions of A and by {αk } the corresponding eigenvalues. We shall choose for simplicity Q as a diagonal operator with respect to {ek }, i.e. of the form Qek = λk ek ,
k ∈ N,
where λk > 0 for all k ∈ N and we shall assume that Tr(AQ) < +∞
(1.3)
and ∞ X
λk kek k2∞ : = κ < +∞.
(1.4)
k=1
We notice that when O is an n-dimensional rectangle there exists c > 0 such that kek k∞ ≤ c for all k ∈ N, so that (1.4) is a consequence of (1.3). Stochastic wave equations of the form (Klein–Gordon equations) p d X˙ (t, ξ ) = (∆ξ X (t, ξ ) − X˙ (t, ξ ) − g(X (t, ξ )))dt + QdW (t, ξ ), (1.5) X (t, ξ ) = 0 in ∂O, X (0, ·) = x ∈ H01 (O), X˙ (0, ·) = y ∈ L 2 (O), were studied in [5–8,15,16,19] and [3]. For instance in [3] there were proved, under suitable assumptions on g, existence and uniqueness of an invariant measure for the transition semigroup associated with this equation as well as the corresponding Kolmogorov equation. Nonlinear stochastic equations of the form (1.1), with nonlinearity on the damping, were first studied by Pardoux [17]. However, the existence of an invariant measure for the corresponding transition semigroup associated with (1.1) seems to be studied here for the first time. We note however that a finite-dimensional version of the problem considered here was previously studied in [4]. Finally, concerning the function U , we shall assume that the following hypothesis holds. Hypothesis 1.1. U : R → R is a convex C 2 function which satisfies the following conditions. (i) There exist γ > 0, ω > 0 such that U 0 (r )r ≥ ω|r |γ +2 ,
∀r ∈ R.
(1.6)
V. Barbu et al. / Stochastic Processes and their Applications 117 (2007) 1001–1013
1003
(ii) There exist η1 , η2 > 0 and C1 , C2 ∈ R such that |U 0 (r )| ≤ η1 |r |γ +1 + C1 ,
∀r ∈ R
(1.7)
and |U 00 (r )| ≤ η2 |r |γ + C2 ,
∀r ∈ R.
(1.8)
γ +2 γ +1
and assume also that U (0) = 0. We set = The main result of this paper Theorem 3.4 amounts to saying that under Hypothesis 1.1 the stochastic flow t → (X (t), Y (t)) associated with Eq. (1.1) has an invariant measure. The uniqueness of the invariant measure is proven in the special case where γ , n satisfy condition (4.1) (Theorem 4.1). The uniqueness of invariant measure in the general case remains an open and apparently a difficult problem if one takes into account that the usual techniques relying on Malliavin calculus (see [4]) or coupling seem to be inapplicable in the present situation. In this context the works [11–14] should also be cited. γ∗
2. Existence and uniqueness of solutions We can rewrite problem (1.2) in the space V × H as dZ = AZ dt − F (Z )dt + BdW (t) x , Z (0) = y
(2.1)
where X (t) 0 1 Z (t) = , A = Y (t) −A 0 0 0 p F (Z ) = , B= . Y (t) + U 0 (Y (t)) Q
It is well known (see e.g. [18, p. 220]) that A is the infinitesimal generator of a strongly continuous group in V × H given by cos(A1/2 t) A−1/2 sin(A1/2 t) et A = . −A1/2 sin(A1/2 t) cos(A1/2 t) (t) Definition 2.1. An adapted stochastic process Z (t) = YX (t) with values in V × H is called a mild solution to system (1.2) if for any T > 0 we have (i) X ∈ C W ([0, T ]; V ), Y ∈ C W ([0, T ]; H ), ∗ (ii) U 0 (Y ) ∈ C W ([0, T ]; L γ (O)), (iii) for all t ≥ 0, Z t X (t) = x + Y (s)ds, 0 Y (t) = −A1/2 sin(A1/2 t)x + cos(A1/2 t)y Z t − cos(A1/2 (t − s))(Y (s) + U 0 (Y (s)))ds 0 Z t p + cos(A1/2 (t − s)) QdW (s). 0
(2.2)
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The suffix W in the space above means that the processes considered are adapted and mean square integrable. Remark 2.2. Theorem 2.3 below can be obtained from Pardoux existence results for secondorder stochastic equations of hyperbolic type; see [17, Theorem 2.1]. However for later purposes we shall consider below a slightly different version and give a direct proof via Yosida approximations. The main result of this section is the following. Theorem 2.3. Let x ∈ V and y ∈ H . Then there is a unique mild solution e e x x X (t) replacing is another mild solution of (1.2), with e(t) e y y we have Y
e(t)|2 ≤ |x − e E|X (t) − e X (t)|2V + E|Y (t) − Y x |2V + |y − e y|2H . H Proof. For any ε > 0 we consider the approximating problem dX ε (t) = Yε (t)dt, p dY (t) = −(AX ε (t) + Yε (t) + βε (Y (t)))dt + QdW (t) ε X ε (0) = x, Yε (0) = y,
X (t) Y (t)
of (1.2). If
(2.3)
(2.4)
where βε is the Yosida approximation of the mapping U 0 : 1 (r − (1 + εU 0 )−1 (r )) = U 0 (1 + εU 0 )−1 (r ), ε We can rewrite Eq. (2.4) in the space V × H as dt − Fε (Z ε )dt + BdW (t) dZ ε = AZ ε x , Z ε (0) = y βε (r ) =
ε > 0.
(2.5)
where Z ε (t) =
X ε (t) , Yε (t)
Fε (Z ε ) =
0 , βε (Yε (t))
0 . B= p Q
It is well known that the linear operator A generates a strongly continuous semigroup of contractions on V × H while the mapping Fε : V × H → V × H is Lipschitz continuous and monotone. Then by standard existence results, see e.g. [10, p. 81], there is a unique solution X ε ∈ C W ([0, T ]; V ),
Yε ∈ C W ([0, T ]; H )
of (2.5). Moreover, using the Itˆo formula for function φ(x, y) = 12 (hAx, xi2 + |y|22 ), we see that Z t E|∇ X ε (t)|22 + E|Yε (t)|22 + 2E [hβε (Yε (s)), Yε (s)i2 + |Yε (s)|22 ]ds 0
=
|∇x|22
+ |y|22
+ t Tr Q.
(2.6)
We infer that E|∇ X ε (t)|22
+ E|Yε (t)|22
where C is independent of ε.
t
Z + 2E 0
γ +2
[|Yε (s)|γ +2 + |Yε (s)|22 ]ds ≤ C(1 + t),
(2.7)
V. Barbu et al. / Stochastic Processes and their Applications 117 (2007) 1001–1013
1005
Then on a subsequence of {X ε , Yε }, still denoted {X ε , Yε }, we have as ε → 0 (i) X ε → X weak star in L ∞ (0, T, L 2 (Ω , V )), (ii) Yε → Y weak star in L ∞ (0, T, L 2 (Ω , H )), (iii) Yε → Y weakly in L γ +2 (0, T ; L γ +2 (Ω , L γ +2 (O))) ∗ ∗ ∗ (iv) βε (Yε ) → η weakly in L γ ([0, T ]; L γ (Ω , L γ (O))).
(2.8)
It follows that Z t Y (s)ds, X (t) = x + 0 Z t 1/2 1/2 1/2 Y (t) = −A sin(A t)x + cos(A t)y + cos(A1/2 (t − s))(Y (s) + η(s))ds (2.9) 0 Z t p + cos(A1/2 (t − s)) QdW (s). 0
It is clear that the process (X, Y ) is adapted. To prove that it satisfies (1.2) we have to prove that η(t, ξ, ω) = U 0 (Y )(t, ξ, ω),
a.e. (t, ξ, ω) ∈ [0, T ] × O × Ω .
(2.10)
Taking into account (2.8), in order to prove (2.2) it suffices to show, see e.g. [2, page 42], T
Z
Z
lim sup E ε→0
0
O
βε (Yε )Yε dξ dt ≤ E
By (2.4) we see, via Itˆo’s formula, that Z 2 2 E|∇ X ε (T )|2 + E|Yε (T )|2 + 2E
T
T
Z
O
0
O
0
Z
Z
ηY dξ dt.
(2.11)
βε (Yε )Yε dsdξ + E
T
Z 0
|Yε (s)|22 ds
= |∇x|22 + |y|22 + T Tr(Q).
(2.12)
Similarly by (2.9) E|∇ X (T )|22 + E|Y (T )|22 + 2E
T
Z
Z
0
O
ηY dsdξ + E
T
Z 0
|Y (s)|22 ds
= |∇x|22 + |y|22 + T Tr(Q).
(2.13)
Since by virtue of (2.8) and (2.9) X ε (T ) → X (T ) Yε (T ) → Y (T )
weakly in L 2W (Ω , V ), weakly in
L 2W (Ω ,
H ),
(2.14) (2.15)
we have lim inf(E|∇ X ε (T )|22 + E|Yε (T )|22 ) ≥ E|∇ X (T )|22 + E|Y (T )|22 ε→0
and by (2.12) and (2.13) we get (2.10) as claimed. This completes the proof of existence. Inequality (2.3) and in particular the uniqueness follow from the Itˆo formula applied to X (t) − e X (t) and to the function φ(x, y) = 12 (hAx, xi2 + |y|22 ).
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3. Existence of invariant measures In order to prove the existence of an invariant measure we shall choose several Lyapunov functions φ to obtain estimates for the expectation of suitable norms of X (t), Y (t) using Itˆo’s formula 1 (3.1) dφ(X, Y ) = hφx (X, Y ), dX i2 + hφ y (X, Y ), dY i2 + Tr[Qφ yy (X, Y )]dt. 2 With the help of these functions we shall prove suitable a priori estimates on the solution of (1.2). Then existence will follow from the Krylov–Bogoliubov theorem (see [9], Proposition 11.3). We first choose φ(x, y) =
1 (|Ax|22 + hAy, yi2 ). 2
(3.2)
Then we have φx (x, y) = A2 x,
φ y (x, y) = Ay,
φ yy (x, y) = A
and so p 1 2 0 dφ(X, Y ) = hA X, Y i2 − hAY, AX + Y + U (Y )i2 + Tr(Q A) dt + hY, QdW (t)i2 . 2 It follows that Z p 1 1/2 2 00 2 dφ(X, Y ) = −|A Y |2 − U (Y )|∇Y | dξ + Tr(Q A) dt + hY, QdW (t)i, 2 O which yields E|AX (t)|22 + E|∇Y (t)|22 + 2E Z tZ + 2E 0
O
t
Z 0
|∇Y (s)|22 ds
U 00 (Y (s))|∇Y (s)|2 dξ ds = |Ax|22 + |∇ y|22 + t Tr(Q A).
(3.3)
As an immediate consequence of (3.3) we find the following result: Lemma 3.1. We have Z t E |∇Y (s)|22 ds ≤ |Ax|22 + |∇ y|22 + t Tr(Q A).
(3.4)
0
Now we choose Z 1 U (y(ξ ))dξ. φ1 (x, y) = (hAx, xi2 + hAx, yi2 ) + 2 O Then we have (φ1 )x (x, y) = Ax + Ay,
(φ1 ) y (x, y) = Ax + U 0 (y),
(3.5)
(φ1 ) yy (x, y) = U 00 (y)
and so dφ(X, Y ) = hAX + AY, Y i2 − hAX + U 0 (Y ), AX + Y + U 0 (Y )i2 dt
V. Barbu et al. / Stochastic Processes and their Applications 117 (2007) 1001–1013
1007
p 1 + Tr(QU 00 (Y ))dt + hY, QdW (t)i2 2 Z 1 0 0 2 00 U (Y (ξ ))Y (ξ )dξ + Tr(QU (Y )) dt = hAY, Y i2 − |AX + U (Y )|2 − 2 O p + hY, QdW (t)i2 . We note that U 0 (y)y ≥ U (y); hence integrating over t and taking expectation yields Z tZ Z t 0 2 |AX (s) + U (Y (s))|2 ds + E U (Y (s, ξ ))dξ ds E[φ1 (X (t), Y (t))] + E 0
0
≤ φ1 (x, y) + E
t
Z 0
|∇Y (s)|22 ds +
t
Z
O
Tr(QU 00 (Y (s)))ds.
Lemma 3.2. There exists a constant κ1 such that for all t ≥ 0 Z tZ Z t 0 2 |AX (s) + U (Y (s))|2 ds + E U (Y (s, ξ ))dξ ds ≤ κ1 (1 + t). E 0
(3.6)
0
O
0
(3.7)
Proof. Since 1 1 (|∇x|22 − |∇x|2 |∇ y|2 ) ≥ − |∇ y|22 , 2 2 we deduce by (3.6) that Z t Z tZ 0 2 E |AX (s) + U (Y (s))|2 ds + E U (Y (s, ξ ))dξ ds φ1 (x, y) ≥
0
O
0
≤ φ1 (x, y) + E
t
Z 0
|∇Y (s)|22 ds +
t
Z 0
1 Tr(QU 00 (Y (s)))ds + E|∇Y (t)|22 . 2
(3.8)
Now by (3.3) we have that Z t 2 E|∇Y (t)|2 + E |∇Y (s)|22 ds ≤ 2φ(x, y) + t Tr(Q A) 0
and so, substituting in (3.8), we obtain that Z t Z tZ U (Y (s, ξ ))dξ ds E |AX (s) + U 0 (Y (s))|22 ds + E 0
0
≤ 2φ(x, y) + φ1 (x, y) +
t
Z 0
O
t Tr(QY 2 (s))ds + Tr(Q A). 2
(3.9)
To estimate the last integral in (3.9) we notice that, by (1.4) and (1.8), Z Z ∞ X 00 2 γ Tr(QU (Y (s))) = λk ek (η2 |Y | + C2 )dξ ≤ κ (η2 |Y |γ + C2 )dξ. k=1
So, (3.7) follows.
O
O
Corollary 3.3. There exists a constant κ2 such that for all t ≥ 0 Z t γ∗ E |AX (s)|γ ∗ ds ≤ κ2 (1 + t). 0
(3.10)
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V. Barbu et al. / Stochastic Processes and their Applications 117 (2007) 1001–1013
Proof. By (3.7) it follows that Z t γ∗ E |U 0 (Y (s))|γ ∗ ds ≤ κ1 (1 + t). 0
Consequently, for suitable constants c1 , c2 , Z t Z t Z t γ∗ γ∗ γ∗ 0 |U 0 (Y (s))|γ ∗ ds |AX (s) + U (Y (s))|γ ∗ ds + c1 E |AX (s)|γ ∗ ds ≤ c1 E E 0 0 0 Z t Z t γ∗ |U 0 (Y (s))|γ ∗ ds |AX (s) + U 0 (Y (s))|22 ds + c1 E ≤ c2 E 0
0
≤ κ1 (1 + t).
We now prove the existence of an invariant measure for the Markov process X (t, x, y) , t ≥ 0, x ∈ H, y ∈ V. t 7→ Y (t, x, y) (t, x, y) We denote by πt (x, y, ·, ·) the law of YX (t, x, y) in V × H . We recall that the transition semigroup Pt is given by Z ϕ(x1 , y1 )πt (x, y, dx1 , dy1 ), (3.11) Pt ϕ(x, y) = E[ϕ(X (t, x, y), Y (t, x, y))] = V ×H
for all (x, y) ∈ V × H and ϕ ∈ Cb (V × H ). Theorem 3.4. There exists an invariant measure µ for Pt . Proof. Fix x0 in D(A) and y0 in V . For any R > 0 define n o γ∗ K R = (x, y) ∈ V × H : |Ax|γ ∗ + |∇ y|22 ≤ R . By the Agmon–Douglis–Nirenberg theorem, see e.g. [1], combined with the Sobolev embedding theorem, K R is a compact subset of V × H . We have Z πt (x0 , y0 , K Rc ) = πt (x0 , y0 , dx, dy) γ∗ [|Ax|γ ∗ +|∇ y|22 >R]
Z 1 γ∗ ≤ (|Ax|γ ∗ + |∇ y|22 )πt (x0 , y0 , dx, dy) R V ×H 1 γ∗ = E(|AX (t, x0 , y0 )|γ ∗ + |∇Y (t, x0 , y0 )|22 ). R Consequently Z Z t 1 1 t γ∗ c πs (x0 , y0 , K R )ds ≤ E(|AX (s, x0 , y0 )|γ ∗ + |∇Y (s, x0 , y0 )|22 )ds t 0 tR 0 and by Lemma 3.1 and Corollary 3.3 it follows that there exists a constant M > 0 such that Z 1 t M πs (x0 , y0 , K Rc )ds ≤ . t 0 R
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Therefore the family of measures Z t 1 πs (x0 , y0 , ·, ·)ds t 0 t≥0 is tight in (V × H, B(V × H )) and by the Krylov–Bogoliubov theorem there exists an invariant measure for Pt . ∗
Corollary 3.5. The support of µ is in W 2,γ (O) × V , i.e. Z γ∗ (kxkW 2,γ ∗ (O ) + kyk2H 1 (O ) )dµ(x, y) < ∞.
(3.12)
0
V ×H
Moreover, we have for λ > 0 and sufficiently small, Z 2 2 |y|22 eλ(kxk +|y|2 ) dµ(x, y) < ∞,
(3.13)
V ×H
and (3.13) remains true for any invariant measure µ for Pt . (Here k · k = k · k H 1 (O ) .) 0
Proof. Estimate (3.12) follows by estimates (3.4) and (3.10) taking into account that, by the Agmon–Douglis–Nirenberg theorem, W 2,γ (O) ⊂ {x ∈ V : Ax ∈ L γ (O)}. ∗
∗
In order to prove (3.13) we consider the Lyapunov function ψ(x, y) = eλ(kxk
2 +|y|2 ) 2
and notice that by Itˆo’s formula, dψ(X (t), Y (t)) = ψ(2λhAX (t), Y (t)i2 − hY (t), AX (t) + Y (t) + U 0 (Y (t))i2 )dt p 1 00 + Tr Qψ yy dt + 2λh QdW, Y i2 ψ. 2 This yields γ +2
dψ(X (t), Y (t)) + ψ(2λ|Y (t)|22 + 2λω|Y (t)|γ +2 ) dt ≤ λψ(Tr Q + λ|Q 1/2 Y |22 ) dt p + 2λh QdW, Y i2 ψ and so for λ > 0 sufficiently small we get Z t 2 2 E |Y (s)|22 eλ(kX (s)k +|Y (s)|2 ) ds ≤ C1 t + C2 , 0
which implies the desired relation (3.13) for any invariant measure µ for Pt .
4. Uniqueness of the invariant measure We shall prove uniqueness of µ if γ is related to dimension n of the space by condition (4.1) below. 0≤γ ≤2 We have
if n = 1,
0≤γ <2
if n = 2,
γ =1
if n = 3.
(4.1)
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Theorem 4.1. Assume that Hypothesis (1.1) holds with γ , n as in (4.1). Then there is a unique invariant measure µ for Pt . Proof. We set Z 1 = Dx X (t, x, y)h, Z 2 = Dx Y (t, x, y)h. We have Z 10 = Z 2 , Z 1 (0) = h,
Z 20 + AZ 1 + Z 2 + U 00 (Y (t))Z 2 = 0
(4.2)
Z 2 (0) = 0.
Consider the Lyapunov function 1 (kz 1 k2 + |z 2 |22 ) + λhz 1 , z 2 i2 . 2 (Here kzk = |∇z|2 .) We have ϕ(z 1 , z 2 ) =
d ϕ(Z 1 (t), Z 2 (t)) + λkZ 1 (t)k2 + (1 − λ)|Z 2 (t)|22 + λhZ 1 , Z 2 i2 dt + λhU 00 (Y )Z 2 , Z 1 i2 ≤ 0, a.e. t > 0.
(4.3)
We are going to show that |D Pt ϕ(X, Y )| ≤ CkDϕk0 e−δt
∀t ≥ 0, ϕ ∈ Cb1 (V × H )
and this implies the uniqueness of the invariant measure by standard arguments. Indeed if ζ is another invariant measure for Pt , we have Z Z Pt ϕ(x, y) − ϕ(x , y )dζ (x , y ) |Pt ϕ(x, y) − Pt ϕ(x1 , y1 )|dζ (x1 , y1 ) 1 1 1 1 ≤ V ×H V ×H Z ≤ CkDϕk0 e−δt (kx − x1 k + |y − y1 |)dζ (x1 , y1 ). V ×H
Hence lim Pt ϕ(x, y) =
t→∞
Z
ϕ(x1 , y1 )dζ (x1 , y1 ) V ×H
and this yields Z Z ϕ(x1 , ζ1 )dζ (x1 , ζ1 ) = V ×H
ϕ(x1 , ζ1 )dµ(x1 , ζ1 ), V ×H
i.e., ζ = µ as claimed. We note the estimate (see (2.6) and (1.6)) Z t γ +2 2 2 EkX (t)k + E|Y (t)|2 + 2E (|Y (s)|22 + ω|Y (s)|γ +2 )ds ≤ t Tr Q + kxk2 + |y|22 . (4.4) 0
Now we come back to solution (Z 1 , Z 2 ) to (4.2). By (4.3) we have for 0 < λ ≤ d λ 3λ 2 ϕ(Z 1 (t), Z 2 (t)) + kZ 1 (t)k + 1 − |Z 2 (t)|22 dt 2 2
1 2
≤ λkZ 1 (t)kkU 00 (Y (t))Z 2 (t)k−1 . (Here k · k−1 is the norm of
H −1 (O).)
kU 00 (Y )Z 2 k−1 ≤ kU 00 (Y )Z 2 kq ∗
(4.5)
We have via the Sobolev embedding theorem Z 1/q ∗ q∗ γ q∗ ≤ η1 |Z 2 (t)| |Y (t)| dξ O
(4.6)
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where q1∗ = 1 − p1∗ and p ∗ = ∞ if n = 1, p ∗ ≥ 2 if n = 2, p ∗ = 6 if n = 3. Taking into account (1.8) we have by (4.6) that γ
γ
kU 00 (Y )Z 2 k−1 ≤ η1 |Z 2 |2 |Y | 2γ q ∗ ≤ η1 |Z 2 |2 |Y |γ +2 .
(4.7)
2−q ∗
Substituting (4.7) into (4.5) we get λ 3λ d γ |Z 2 (t)|22 ≤ λη1 kZ 1 (t)k|Z 2 (t)|2 |Y (t)|γ +2 , ϕ(Z 1 (t), Z 2 (t)) + kZ 1 (t)k2 + 1 − dt 2 2 ∀t ≥ 0. Recalling that 1−λ 1+λ (kz 1 k2 + |z 2 |2 ) ≤ ϕ(z 1 , z 2 ) ≤ (kz 1 k2 + |z 2 |2 ) 2 2 we obtain that d λ γ ϕ(Z 1 (t), Z 2 (t)) + ϕ(Z 1 (t), Z 2 (t)) ≤ 4η1 λϕ(Z 1 (t), Z 2 (t))|Y (t)|γ +2 , dt 4
∀t ≥ 0.
Integrating the latter we get ϕ(Z 1 (t), Z 2 (t)) ≤ e
− 41 λt+4η1 λ
Rt 0
γ
|Y (s)|γ +2 ds
ϕ(Z 1 (0), Z 2 (0))
∀t ≥ 0, P-a.s.
(4.8)
It remains to estimate the expectation of the right side of (4.8). To this end we shall proceed as in [3]. Namely, consider the process Z t ν γ 2 2 φ(t) = (kX (t)k + |Y (t)|2 ) + 4η1 λ |Y (s)|γ +2 ds. 2 0 By the Itˆo formula, ν γ dφ(t) = −ν|Y (t)|22 + 4η1 λ|Y (t)|γ +2 − νhU 0 (Y (t)), Y (t)i2 + Tr Q dt 2 p + hνY (t), QdW (t)i2 . This yields p γ +2 ν ν γ +2 dφ(t) ≤ −ν|Y (t)|22 + Tr Q + (4η1 λ) 2 − |Y (t)|γ +2 dt + hνY (t), QdW (t)i2 . 2 2 Next we apply Itˆo’s formula to eφ . We get p d(eφ ) ≤ eφ dφ + ν 2 eφ | QY |2 dt and therefore ν p γ +2 ν γ +2 d(eφ ) ≤ ν 2 eφ | QY |22 dt + eφ Tr Q + (4η1 λ) 2 − |Y (t)|γ +2 dt 2 2 p + φ ν hY (t), QdW (t)i2 . Finally, taking ν ≤ 3λ, with λ sufficiently small, we get Eeφ(t) ≤ Eeφ(0) + Cλ(λkQk
γ +2 2
γ
γ +2
+ Tr Q + λ 2 η1 2 )
t
Z 0
Eeφ(s) ds
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where C is independent of λ and kQk is the norm of Q in L 2 (O). This yields Eφ(t) ≤ eCλ(λkQk
γ +2 2 +Tr
γ
γ +2 2 )t
Q+λ 2 η1
eν(kxk
2 +|y|2 ) 2
.
For ν ≥ 2λ and λ small this yields 4η1 λ
Ee
Rt 0
γ
|Y (s)|γ +2 ds
≤e
ν(kxk2 +|y|22 ) Cλ(λkQk
e
γ +2 2 +Tr
γ +2 2 )t
γ
Q+λ 2 η1
.
(4.9)
Substituting (4.9) into (4.8) we see that Eϕ(Z 1 (t), Z 2 (t)) ≤ e
ν(kxk2 +|y|22 ) (C(λkQk
e
γ +2 2 +Tr
γ
γ +2 2 )− 1 )λt 4
Q+λ 2 η1
ϕ(Z 1 (0), Z 2 (0)).
Assume that λkQk
γ +2 2
γ +2
γ
+ Tr Q + λ 2 η1 2 <
1 . 4C
(4.10)
Then there is a δ > 0 such that Eϕ(Z 1 (t), Z 2 (t)) ≤ eν(kxk
2 +|y|2 ) 2
e−δt ϕ(Z 1 (0), Z 2 (0)).
(4.11)
Now, let µ be any invariant measure for Pt . Recalling that (see Corollary 3.5) we have Z 2 2 eν(kxk +|y|2 ) dµ(x, y) < ∞ V ×H
then we infer by (4.11) that |D Pt ϕ(x, y)| ≤ CkDϕk0 e−δt K (x, y),
∀(x, y) ∈ V × H
where K ∈ L 1 (V × H ; dµ) and so we conclude as above about the uniqueness of invariant measure µ as in the proof of Theorem 4.1. We notice that condition (4.10) can be reduced via the transformation X 7→ ε X , with ε appropriately chosen, to the condition: λ sufficiently small. Indeed by this transformation Eq. (1.1) reduces to an equation of the same type with ε Q instead of Q and εU 0 ( εy ) instead of U 0 . Then condition (4.10) reduces to λε
γ +2 2
kQk
γ +2 2
γ +2
γ
+ εTr Q + ε −γ η1 2 λ 2 ≤
This completes the proof.
1 . 4C
Acknowledgements The first author was partially supported by the CEEX research project 09-D11 of the Romanian Ministry of Education and Research. The second and third authors were partially supported by the Italian National Project MURST “Equazioni di Kolmogorov.” References [1] [2] [3] [4] [5]
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