Uniform exponential dichotomy of stochastic cocycles

Stochastic Processes and their Applications 120 (2010) 1920–1928 www.elsevier.com/locate/spa

Uniform exponential dichotomy of stochastic cocycles Diana Stoica Politehnica University of Timis¸oara, The Faculty of Engineering of Hunedoara, Revolutiei Str., No. 5, 331128, Hunedoara, Romania Received 12 August 2009; received in revised form 28 April 2010; accepted 23 May 2010 Available online 1 June 2010

Abstract The aim of this paper is to give a generalization of the well-known theorem of Perron for uniform exponential dichotomy in mean square for stochastic cocycles in Hilbert spaces. c 2010 Elsevier B.V. All rights reserved.

MSC: Primary 60H10; 60H15; Secondary 60H20 Keywords: Stochastic semiflow; Stochastic cocycles; Uniform exponential dichotomy

1. Introduction In the qualitative theory of evolution equations, exponential dichotomy is one of the most important asymptotic properties. In the last few years, it has been treated from various perspectives. The notion of exponential dichotomy for linear differential equations was introduced in a paper [11] by Perron in 1930, which was concerned with the problem of conditional stability of a deterministic differential equation x 0 (t) = A(t)x(t) and its connection with the existence of bounded solutions of the perturbed equation x 0 (t) = A(t)x(t) + α(t). The nonuniform asymptotic behavior of linear evolutionary processes in terms of Lyapunov equations is presents in [5]. In [1,2], Arnold present the random dynamical system generated by random differential equations and stochastic differential equations. A sufficient condition for stochastic Ito systems to be exponentially dichotomous is proves in [3]. E-mail address: [email protected]. c 2010 Elsevier B.V. All rights reserved. 0304-4149/$ - see front matter doi:10.1016/j.spa.2010.05.016

D. Stoica / Stochastic Processes and their Applications 120 (2010) 1920–1928

1921

The problem of the existence of stochastic semiflows for semilinear stochastic evolution equations is a non-trivial one, mainly due to the well-established fact that finite-dimensional methods for constructing (even continuous) stochastic flow break down in the infinitedimensional setting of semilinear stochastic evolution equations (see [7,8,14]). For linear stochastic evolution equations with finite-dimensional noise, a stochastic semiflow (i.e. a random evolution operator) was obtained in [4]. In [10], the existence of perfect differentiable cocycles generated by mild solutions of a large class of semilinear stochastic evolution equations (sees) and stochastic partial differential equations (spdes) is proved. In [15,16] the author presents the uniform exponential stability and instability for stochastic cocycles. In this paper we consider the general case of stochastic cocycles over stochastic semiflows. The main objective is to give a generalization of the classical Perron result for uniform exponential dichotomy in mean square of stochastic cocycles in Hilbert spaces. 2. Stochastic cocycles
Let Ω , F, {Ft }t≥0 , P be a standard filtered probability space, i.e. (Ω , F, P) is a probability space, {Ft }t≥0 is an increasing family of σ -algebras, F0 contains all P-null sets of F and \ Ft = Fs , for every t ≥ 0. s≥t

Definition 2.1. A stochastic semiflow on Ω is a measurable random field ϕ : R+ × Ω → Ω satisfying the following properties: (f1) ϕ(0, ω) = ω, (f2) ϕ(t + s, ω) = ϕ(t, ϕ(s, ω)) 2 × Ω. for all (t, s, ω) ∈ R+

Example 2.1. Let X be a real separable Hilbert space and let Ω be the space of all continuous paths ω : R+ → X , such that ω(0) = 0 with the compact open topology. Let Ft , for t ≥ 0, be the σ -algebra generated by the set {ω → ω(u) ∈ X with u ≤ t} and let
F be the associated Borel σ -algebra to Ω . If P is a Wiener measure on Ω then Ω , F, {Ft }t≥0 , P is a filtered probability space with the Wiener motion W (t, ω) = ω(t) for all (t, ω) ∈ R+ × Ω . Then ϕ : R+ × Ω → Ω defined by ϕ(t, ω)(τ ) = ω(t + τ ) − ω(t) is a stochastic semiflow on Ω . Indeed, ϕ(0, ω)(s) = ω(s), and ϕ(t + s, ω)(τ ) = ω(t + s + τ ) − ω(t + s) = ϕ(s, ω)(t + τ ) − ϕ(s, ω)(t) = ϕ(t, ϕ(s, ω))(τ ). Let X be a real separable Hilbert space and let L(X ) be the set of all linear bounded operators from X to X . Definition 2.2. A mapping Φ : R+ × Ω → L(X ) with the properties

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D. Stoica / Stochastic Processes and their Applications 120 (2010) 1920–1928

(c1) Φ(0, ω) = I (the identity operator on X ), (c2) Φ(t + s, ω) = Φ(t, ϕ(s, ω))Φ(s, ω), 2 × Ω , is called a stochastic cocycle on X over the stochastic semiflow for all (t, s, ω) ∈ R+ ϕ : R+ × Ω → Ω . A stochastic cocycle Φ : R+ × Ω → L(X ) with the property
(c3) for every (ω, x) ∈ Ω × X the mapping t → E kΦ(t, ω)xk2 is measurable, is called strongly measurable in mean square. A stochastic cocycle Φ : R+ × Ω → L(X ) with the property (c4) there are M ≥ 1 and λ > 0 such that EkΦ(t, ω)k2 ≤ Meλ(t−s) EkΦ(s, ω)k2 , for all t ≥ s ≥ 0, and ω ∈ Ω , is called a stochastic cocycle with uniform exponential growth in mean square. Example 2.2. Let X be a real separable Hilbert space and let {W (t)}t≥0 be an X -valued Brownian motion with a separable covariance Hilbert space H and defined on the canonical
complete filtered Wiener space Ω , F, {Ft }t≥0 , P introduced in Example 2.1. Let L(H, X ) be the Banach space of all bounded linear operators from H to X . Denote by L 2 (H, X ) ⊂ L(H, X ), the Hilbert–Schmidt operators S : H → X , with the norm !1/2 ∞ X kSk = |S( f k )|2 , k=1

where |.| is the norm on H , and { f k , k ≥ 1} is a complete orthonormal system on H . Consider the linear stochastic evolution equation of the form  du(t, x, .) = Au(t, x, .)dt + Bu(t, x, .)dW (t), t > 0; u(0, x, ω) = x ∈ X, where A : D(A) ⊂ X → X is the infinitesimal operator of a strongly continuous semigroup of bounded linear operators T (t) : X → X, t ≥ 0, and B : X → L 2 (H, X ) is a bounded linear operator. Suppose that B can be extended toPa bounded linear operator B : X → L(X ), which 2 can be denoted by the same, and the series ∞ k=1 kBk k L(X ) converge, where Bk : X → X are bounded linear operators defined by Bk (x) = B(x)( f k ), x ∈ X, k ≥ 1. A mild solution of this equation is given (see [10,7]) by the family of {Ft }t≥0 -adapted processes u(., x, .) : R+ × Ω → X, x ∈ X , satisfying the following stochastic integral equation: Z t u(t, x, .) = T (t)x + T (t − τ )Bu(τ, x, .)dW (τ ). 0

Then the mapping Φ : R+ × Ω → L(X ) defined by Φ(t, ω)x = u(t, x, ω), is a stochastic cocycle over the stochastic semiflow ϕ : R+ × Ω → Ω defined in Example 2.1. (see [10]). Example 2.3. Let G be the Banach space of X -valued stochastic processes α with the norm !1/2 kαk = sup Ekα(t)k2 t≥0

.

D. Stoica / Stochastic Processes and their Applications 120 (2010) 1920–1928

1923

If Φ : R+ × Ω → L(X ) is the stochastic cocycle over the stochastic semiflow ϕ : R+ × Ω → Ω then Φα : R+ × Ω → L(X ) defined by Z t Φα (t, ω)x = Φ(t, ω)x + Φ(t − τ, ϕ(τ, ω))α(τ )dτ, (1) 0

is also a stochastic cocycle over the stochastic semiflow ϕ, which is called an α-shifted stochastic cocycle of Φ. In particular, if Φ is the stochastic cocycle defined in Example 2.2 then Φα is the stochastic cocycle associated to the stochastic equation (see [10])  du(t, x, .) = (Au(t, x, .) + α(t)) dt + Bu(t, x, .)dW (t), t > 0; (2) u(0, x, ω) = x ∈ X. 3. Uniform exponential dichotomy in mean square Let Φ : R+ × Ω → L(X ) be a stochastic cocycle on the real Hilbert space over the stochastic semiflow ϕ : R+ × Ω → Ω . In what follows, we denote by X 1 the linear subspace of all x ∈ X with the property EkΦ(0, ω)xk2 < ∞, and this is called the stable manifold. Let X 2 be the complementary subspace of X 1 , i.e. we have X = X 1 ⊕ X 2. Then we denote by P1 the projection on X 2 (that is, ker P1 = X 2 ) and by P2 = I − P1 , the projection on X 1 . Definition 3.1. The stochastic cocycle Φ : R+ × Ω → L(X ) is said to be (1) uniformly dichotomic in mean square if the constants N1 , N2 ≥ 1 exist, such that: EkΦ(t, ω)xk2 ≤ N1 EkΦ(s, ω)xk2 ,

∀ t ≥ s ≥ 0, ∀ x ∈ X 1

(3)

EkΦ(s, ω)xk ≤ N2 EkΦ(t, ω)xk ,

∀ t ≥ s ≥ 0, ∀ x ∈ X 2

(4)

2

2

(2) uniformly exponentially dichotomic in mean square if the constants N1 , N2 ≥ 1, ν1 , ν2 ≥ 0 exist, such that: EkΦ(t, ω)xk2 ≤ N1 e−ν1 (t−s) EkΦ(s, ω)xk2 , EkΦ(s, ω)xk ≤ N2 e 2

−ν2 (t−s)

EkΦ(t, ω)xk , 2

∀ t ≥ s ≥ 0, ∀ x ∈ X 1

(5)

∀ t ≥ s ≥ 0, ∀ x ∈ X 2 .

(6)

Lemma 3.1. Let Φ : R+ × Ω → L(X ) be a stochastic cocycle with uniform exponential growth in mean square, and with the property that, for each stochastic process α(t) ∈ G the α-shifted stochastic cocycle Φα is bounded in mean square. Then there is a constant K > 0 such that for every α(t) ∈ G there exists a unique x ∈ X 2 such that EkΦα (t, ω)xk2 ≤ K sup Ekα(t)k2 , t≥0

for all (t, ω) ∈ R+ × Ω .

(7)

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D. Stoica / Stochastic Processes and their Applications 120 (2010) 1920–1928

Proof. Suppose that for an arbitrary stochastic process α(t) ∈ G the α-shifted stochastic cocycle Φα satisfies EkΦα (t, ω)xk2 < ∞, for all (t, x, ω) ∈ R+ × X × Ω . Let xk = Pk x, for all x ∈ X . Then from relation Φα (t, ω)x = Φ(t, ω)x1 + Φα (t, ω)x2 , for all (t, ω) ∈ R+ × Ω , it results that Φα (t, ω)x, x ∈ X is an α-shifted stochastic cocycle of Φ, and from the fact that Φ(t, ω)x1 is a stochastic cocycle bounded in mean square, it results that Φα (t, ω)x2 is an α-shifted stochastic cocycle with the properties sup EkΦα (t, ω)x2 k2 < ∞ and t≥0

Φα (0, ω)x2 = x − P1 x = (I − P1 )x = P2 x ∈ X 2 . Thus for all t ≥ 0 and α(t) ∈ G, there exists x ∈ X 2 such that Φα (t, ω)x is an α-shifted stochastic cocycle of Φ with the property sup EkΦα (t, ω)xk2 < ∞. t≥0

Assume that, for α(t) ∈ G, t ≥ 0, there exists x20 , x200 ∈ X 2 such that sup EkΦα (t, ω)x20 k2 < ∞, t≥0

sup EkΦα (t, ω)x200 k2 < ∞, t≥0

for all (t, ω) ∈ R+ × Ω . Then we have EkΦα (t, ω)x20 − Φα (t, ω)x200 k2 ≤ 2EkΦ(t, ω)x20 − Φ(t, ω)x200 k2 Z t +2 EkΦ(t − τ, ω)x20 − Φ(t − τ, ω)x200 k2 Ekα(τ )k2 dτ 0   Z t ≤ 2 Meλt Ekx20 − x200 k2 + N M eλ(t−τ ) dτ Ekx20 − x200 k2 0   N ≤ 2Meλt 1 + Ekx20 − x200 k2 < ∞ λ where M, λ are constants from Definition 3.1, and N = supt≥0 kα(t)k2 . Thus we obtain x20 − x200 ∈ X 1 ∩ X 2 = 0, and hence x20 = x200 . Let us consider the space G 1 of all α-shifted stochastic cocycles of Φ, α(t) ∈ G, with the condition Φα (0, ω)x ∈ X 2 bounded in mean square. Let C : G → G 1 be the linear operator defined by Cα = Φα (t, ω)x, for all (t, ω) ∈ R+ × Ω . To establish property (7) it is enough to show by the closed graph theorem that C is a closed operator. Let {αn } be a sequence of G; thus there exists a limit of this in G, such that Ekαn (t) − α(t)k2 → 0,

n → ∞, ∀t ≥ 0.

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D. Stoica / Stochastic Processes and their Applications 120 (2010) 1920–1928

Then there exists y ∈ G 1 such that EkΦαn (t, ω)x − y(t)k2 → 0,

n → ∞,

for all (t, ω) ∈ R+ × Ω . It follows that there exist a subsequence {αn k } of {αn }, and xk ∈ X 2 such that EkΦαnk (t, ω)xk − y(t)k2 → 0,

k → ∞,

in the space G 1 , for all (t, ω) ∈ R+ × Ω . From Eky(t) − Φα (t, ω)xk2   ≤ 2 Eky(t) − Φαnk (t, ω)xk k2 + EkΦαnk (t, ω)xk − Φα (t, ω)xk2 ≤ 2Eky(t) − Φαnk (t, ω)xk k2 + 4EkΦ(t, ω)xk − Φ(t, ω)xk2 Z t +4 EkΦ(t − τ, ω)xk − Φ(t − τ, ω)xk2 Ekαn k (τ ) − α(τ )k2 dτ 0

λt

≤ 2Eky(t) − Φαnk (t, ω)xk k + 4Me Ekxk − xk 2

2

1 1 + sup Ekαn k − αk2 λ t≥0

!

for k → ∞ we obtain that y(t) = Φα (t, ω)x, for all t ≥ 0, with a probability of 1. Hence supt≥0 EkΦα (t, ω)xk2 < ∞, and thus we have Cα(t) = Φα (t, ω)x = y(t),

∀ t ≥ 0. 

The main result of this paper is the next theorem. In the papers [12,13] the authors proves the connections between some “Perron-type” conditions and the exponential dichotomy in deterministic case, and in [6,9] is presents the asymptotic behavior of skew-product semiflows in Banach spaces. Theorem 3.1. If the stochastic cocycle Φ : R+ × Ω → L(X ) satisfies the hypothesis from Lemma 3.1, then it is uniformly exponential dichotomic in mean square. Proof. Let x ∈ X and α(t) =

χ(t)

( EkΦ (t,ω)xk2 )1/2

 1, χ (t) = 1 − (t − t0 − τ ),  0,

Φ(t, ω)x where

t ∈ [0, t0 + τ ]; t ∈ [t0 + τ, t0 + τ + 1]; t ≥ t0 + τ

for all ω ∈ Ω . Obviously supt≥0 α(t) < ∞, and the α-shifted stochastic cocycle Φα (t, ω)x of Φ is from G. It results, from Lemma 3.1, that there exists a unique x ∈ X 2 such that !1/2  1/2 2 2 EkΦα (t, ω)xk ≤ K sup Ekα(t)k ≤ K, t≥0

for all (t, ω) ∈ R+ × Ω . In particular, if t = t0 + τ then

1/2 Z t0 +τ  1/2 

2

EkΦα (t, ω)xk = E Φ(t0 + τ, ω)x + Φ(t0 + τ − s, ω)xα(s)ds

0

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D. Stoica / Stochastic Processes and their Applications 120 (2010) 1920–1928 t0 +τ

  Z ≤ 2EkΦ(t0 + τ, ω)xk2 1 + 0

Thus we have Z t0 +τ 0

1
1/2 EkΦ(s, ω)xk2

ds ≤ K

1 ds EkΦ(s, ω)xk2 1

EkΦ(t0 + τ, ω)xk2

1/2

≤ K.


1/2 .

(8)

Let us consider the function Z t 1 δ(t) =
1/2 ds. 0 EkΦ(s, ω)xk2 Since the second moment of the stochastic cocycle satisfies a system of ordinary linear differential equations [see [7]], this function is continuously differentiable. Then, from (8), it results that δ 0 (t0 + τ ) 1 ≥ . δ(t0 + τ ) K Integrating this inequality on [1, τ ], we have δ(t0 + τ ) ≥ e K (τ −1) δ(t0 + 1), 1

∀τ ≥ 1.

(9)

If t ∈ [t0 , t0 + 1], from uniform exponential growth in mean square of stochastic cocycle Φ we have EkΦ(t, ω)xk2 ≤ Meλ(t−t0 ) EkΦ(t0 , ω)xk2 ,

(10)

for all (ω, x) ∈ Ω × X . By integration on [t0 , t0 + 1], we obtain   − λ2 Z t0 +1 2 1 − e 1 1 δ(t0 + 1) =
1/2 ds ≥
1/2 . 1/2 2 λM t0 EkΦ(s, ω)xk EkΦ(t0 , ω)xk2 For τ ≥ 1, and from relation (7) we have  1/2 1 1 K K EkΦ(t0 + τ, ω)xk2 = 0 ≤ ≤ e− K (τ −1) δ (t0 + τ ) δ(t0 + τ ) δ(t0 + 1)  1/2 1 M 1/2 K  e− K (τ −1) EkΦ(t0 , ω)xk2 ≤  . 2 2 −λ 1 − e λ Denoting N = 

M 1/2 K e1/K (1−e−2/λ )

2 λ

results in

EkΦ(t0 + τ, ω)xk2

1/2

τ

≤ N e− K



EkΦ(t0 , ω)xk2

1/2

.

(11)

From relation (10), if t = t0 + τ, τ ≤ 1 we have EkΦ(t0 + τ, ω)xk2 ≤ Meλτ EkΦ(t0 , ω)xk2 ,

(12)

for all (ω, x) ∈ Ω × X . For t = t0 + τ , from relations (11) and (12) we obtain EkΦ(t, ω)xk2 ≤ N1 e−ν1 (t−t0 ) EkΦ(t0 , ω)xk2 ,

∀t ≥ t0 ≥ 0, ∀x ∈ X 1

(13)

D. Stoica / Stochastic Processes and their Applications 120 (2010) 1920–1928

1927

where N1 = max(N 2 , M), and ν1 = 1/K . Let Φ : R+ × Ω → L(X ) be a stochastic cocycle with uniformly exponential growth in mean χ(t) square and Φ(0, ω)x = x ∈ X 2 . If we take the function α(t) = − Φ(t, ω)x, it ( EkΦ (t,ω)xk2 )1/2 results that α(t) is from G. Thus we have Z t Z t χ (τ ) Φ(t − τ, ω)xα(τ )dτ = −
1/2 Φ(t − τ, ω)Φ(τ, ω)xdτ 0 0 EkΦ(τ, ω)xk2 Z t χ (τ ) = −Φ(t, ω)x
1/2 dτ 0 EkΦ(τ, ω)xk2 Z ∞ Z ∞ χ (τ ) χ (τ ) = Φ(t, ω)x
1/2 dτ − Φ(t, ω)x
1/2 dτ. t 0 EkΦ(τ, ω)xk2 EkΦ(τ, ω)xk2 Passing to the norm in G, and using R t the uniform exponential growth in mean square of stochastic cocycle Φ, we have that 0 EkΦ(t − τ, ω)xk2 Ekα(τ )k2 dτ < ∞. Thus for this α, from Lemma 3.1, it results that there exists a unique x ∈ X 2 such that

!1/2

Z ∞

 1/2 χ (τ )

2 dτ ≤ K, EkΦα (t, ω)xk ≤ E Φ(t, ω)x


1/2

t EkΦ(τ, ω)xk2 for all (t, ω) ∈ R+ × Ω . Then, for all τ ≥ 0 and t ≥ 0, we have Z ∞ χ (τ ) K .
1/2 dτ ≤ 2 (EkΦ(t, ω)xk)1/2 t EkΦ(τ, ω)xk Because the integral of the left side is monotone increasing for τ ≥ 0 and is bounded, it results that it has a limit for τ → ∞, but Z t0 +τ Z ∞ 1 χ (s) ds =
1/2
1/2 ds 2 t t EkΦ(s, ω)xk EkΦ(s, ω)xk2 Z t0 +τ +1 χ (τ ) +
1/2 ds. t0 +τ EkΦ(s, ω)xk2 Since the second integral is convergent, it tends to zero for τ → ∞ and we get Z ∞ 1 K .
1/2 ds ≤ (EkΦ(t, ω)xk)1/2 t EkΦ(s, ω)xk2

(14)

If we consider the function Z ∞ 1 δ(t) =
1/2 ds, t EkΦ(s, ω)xk2 then it follows that 1 δ 0 (t) ≤ − δ(t). K Integrating this inequality on [t0 , t] results in δ(t) ≤ e− K (t−t0 ) δ(t0 ). 1

(15)

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D. Stoica / Stochastic Processes and their Applications 120 (2010) 1920–1928

Since the stochastic cocycle Φ has uniform exponential growth in mean square, there exist the positive constants M, λ such that EkΦ(τ, ω)xk2 ≤ Meλ(τ −t) EkΦ(t, ω)xk2 ,

(16)

for all (ω, x) ∈ Ω × X and τ ≥ t. Thus we obtain  1/2  1/2 Z δ(t) EkΦ(t, ω)xk2 = EkΦ(t, ω)xk2 t

Z

∞

1 EkΦ(s, ω)xk2


1/2 ds

∞

1 −λ(s−t) e ds = L , M t where L is a positive constant. From the relations (14) and (15) it results that  1/2  1/2 L L 1 (t−t0 ) L 1 EkΦ(t, ω)xk2 ≥ ≥ eK ≥ e K (t−t0 ) EkΦ(t0 , ω)xk2 . δ(t) δ(t0 ) K ≥

Denoting K 2 =

L2 ,ν K2 2

=

2 K,

we have

EkΦ(s, ω)xk2 ≤ N2 e−ν2 (t−s) EkΦ(t, ω)xk2 ,

∀ t ≥ s ≥ 0, ∀ x ∈ X 2 .

(17)

It results from inequalities (13) and (17) that, the stochastic cocycle Φ : R+ × Ω → L(X ) is uniformly exponentially dichotomic in mean square.  Acknowledgement The author would like to express her gratitude to the anonymous referee for his/her helpful remarks and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

[11] [12] [13] [14] [15] [16]

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