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M. Chakir, S. EL Mourchid

PII: DOI: Reference:

S0022-247X(17)30996-4 https://doi.org/10.1016/j.jmaa.2017.11.003 YJMAA 21801

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Journal of Mathematical Analysis and Applications

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23 September 2017

Please cite this article in press as: M. Chakir, S. EL Mourchid, Strong mixing Gaussian measures for chaotic semigroups, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2017.11.003

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Strong mixing Gaussian measures for chaotic semigroups

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M. Chakir1, S. EL Mourchid2

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Abstract

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In this paper we will be concerned with the problem of the existence of an invariant mixing measure considering its connection with the chaotic behavior of linear semigroups on separable Banach spaces. We ﬁrst prove an identity characterizing invariant Gaussian measure involving its covariance operator and the inﬁnitesimal generator of the semigroup . This gives an answer to a question raised by Rudnicki in his inspiring review paper [35]. Under suitable conditions, we use the proved identity to give an invariant mixing Gaussian measure as distribution of a Wiener integral.

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1 Introduction It is well known that the essence of deterministic chaos is the sensitive dependence on initial conditions. The last property means that small variations of the initial state may produce large variations in the long term behavior of the studied dynamical system. As a consequence, the evolution becomes unpredictable despite the deterministic description of the dynamic. It is commonly believed that chaos can occur only within nonlinear phenomena but models arising from biology, physics and other ﬁelds show that linear inﬁnite dimensional systems can exhibit chaotic behavior exactly like nonlinear ones, see for instance [1,2,10,20,22,23,27,34] and references therein. For a systematic study of linear chaos see the seminal paper [17] and the survey book [24]. In almost all cases, the deﬁnition of a chaotic system introduced by Devaney [19], is proved to be the most convenient. One of the features of the chaotic dynamic, following this deﬁnition, is the so called topological transitivity. This property is a sign of complexity preventing to reduce the study to simple subsystems. Let us formulate this concept for a linear C0 -semigroup T(·) on a Banach space E. We say that T(·) is topologically transitive if for any two nonempty open subsets U, V of E there exists t > 0 such that Tt (U) ∩ V = . Furthermore, if E is assumed to be separable then this is 1

Département de Mathematiques, Faculté des Sciences Agadir BP 8106 - Cité Dakhla Agadir, Maroc. e-mail: [email protected] 2 Département de Mathematiques, Faculté des Sciences Agadir BP 8106 - Cité Dakhla Agadir, Maroc. e-mail: [email protected]

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equivalent to T(·) is hypercyclic, that is the existence of a vector x ∈ E, with a dense orbit, {Tt x ; t ≥ 0}, in E. It is an open problem to recognize when the property of hypercyclicity is robust under perturbation. In fact it is a very unstable property as formulated by Desch and Schappacher in the title of their paper [18]. Here we mention that, in the sense of Kato-Trotter convergence, neither the hypercyclic semigroups nor the non-hypercyclic semigroups form an open set, see [18]. Due to this fact, the study of the hypercyclicity becomes very hard in some cases. The ﬁrst criteria for hypercyclicity of C0 -semigroups is given by Desch, Schappacher and Webb in [17], see also other versions in [11] and [12]. The second required condition in the Devaney’s deﬁnition is the denseness of the set of periodic points, E per := {x ∈ E ; there exists t > 0 such that T(t )x = x}. Hence every nonempty open set of E contains an unstable periodic point. Here we point out that the two requirements, hypercyclicity and denseness of periodic points, yield the sensitive dependence on initial conditions as it was proved by Banks et all in [3]. Finally a C0 -semigroup T(·) on E is called chaotic if it is hypercyclic and the set of periodic points, E per , is dense in E. A very useful spectral criterion, due to Desch, Schappacher and Webb, says that a very large set of eigenvectors of the generator associated to eigenvalues in an open set intersecting the imaginary axis leads to chaos for the semigroup, (see [17] or (Theorem 7.30 pp 197, [24]). This criterion is the continuous version of a result proved in the discrete case in [15]. In fact the chaotic behavior of the semigroup is essentially due to the imaginary eigenvalues of the generator as shown in [21] or (Theorem 7.32 pp 199, [24]). Furthermore, the generator of a chaotic semigroup must have an inﬁnite set of imaginary eigenvalues, (Proposition 7.18 pp 191, [24]). It is always hopeful to conﬁrm that despite the erratic behavior of a chaotic system, it is likely to be statistically predictable. But in order to perform statistics in the phase space where the orbits live, one needs a measure on the topological structure. This measure is usually required to be preserved during the dynamic with interesting ergodic properties and to give positive values to nonempty open sets. Precisely one needs an answer to the following question (Problem 3, [35]). Let T(·) be a linear semigroup. Assume that this semigroup is chaotic. Does there exist (or when exists) an invariant measure with respect to T(·) having interesting ergodic properties? In [29], Murillo-Arcilla and Peris showed that the continuous version of the frequent hypercyclicity criterion, proved by Mangino and Peris in [28], is sufﬁcient to construct an invariant strongly mixing probability measure with full support. As a corollary they showed that this is possible if the imaginary eigenvectors ﬁeld for the generator has C2 or weakly C1 smoothness. To illustrate this, they give several examples ranging from birth and death models to Black and Scholes equations. The method of the construction is new and it is general for all semigroup satisfying the frequent hypercyclicity criterion. We believe that the idea is quite old and it goes back independently to Rudnicki [33], and to Brunovský and Komornik [6]. It aims to show that the semigroup (Tt ) is homeomorphic to a translation semigroup S t f (·) = f (· + t ) on a well deﬁned space of functions Y. Then one tries to give the measure as a the image by the homeomorphism of a measure ν on Y. Here we recall that the constructed measure in [33] and [6] is Gaussian as it is the induced measure of a Gaussian process with 2

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sample paths in Y, while in [29] it is not. Further explanations and a recent application of this method to a model of population dynamic can be found in [34] and [31]. In general, the direct construction of a Gaussian measure on E with interesting ergodic properties is not an easy task. For this purpose, one uses a basic tool named the covariance operator. This method has been adopted in the case of linear semigroups by Bayart in [5] for a general Banach space and by the second named author for weighted Lp spaces in [13]. The covariance operator of the Gaussian measure is always given under the form KK ∗ , where K is a gamma radonifying operator. For a note on this class of operators one can see [7]. Hereafter, in the setting of separable Lp (T) spaces, we will be able to construct an invariant mixing Gaussian measure induced by a given Gaussian process (Z x )x∈T , (Theorem.3.2). We will see that in the case of a chaotic translation semigroup, (Z x )x∈T is just the well known Ornstein-Uhlenbeck process (Remark.4.2). For the convenience of the reader we recall in an appendix the main deﬁnitions and basic results used in this work.

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2 Invariant Gaussian measures and linear chaos

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Probably, Lasota is the ﬁrst author who examined the chaotic behavior of a linear inﬁnite dimensional system using ergodic theory [25, 26]. In ﬁnite dimensional spaces strong mixing properties of dynamical can be successfully studied using Frobenius-Perron operators, while its application becomes difﬁcult in the inﬁnite dimensional setting. In this case Lasota [25] applied the well known Krylov-Bogoliubov theorem to show the existence of an ergodic invariant measure and he also proved that this measure is continuous, i.e., the set of periodic points has zero measure. This is of course not enough to deduce in general the chaotic behavior of the system. Let us ﬁrst formalize the ergodicity which is the ﬁrst level of irregularity that could be displayed by a dynamical system with an invariant measure,(see [16] pages (58 to 64)). A probability measure μ, deﬁned on the σ algebra B of Borel subsets of E, is called invariant under a C0 -semigroup (Tt ) if μ(Tt−1 F) = μ(F) for each t ≥ 0 and each Borel subset F of E. A Borel subset F is said to be invariant with respect to (Tt ) if Tt−1 F = F for all t ≥ 0. If the measure μ(F) of any invariant subset F equals 0 or 1, then the system ((Tt ), μ) or simply the measure μ is called ergodic. The important property of an ergodic system is shown by Birkhoff’s ergodic theorem: μ is ergodic if and only if for each f ∈ L1 (E, B, μ), 1 lim T→+∞ T 15 16 17 18

T 0

f (Tt x)d t =

X

f (z)d μ(z), μ almost surely in E.

If f = 1F , is the characteristic function of a Borelian set F, then, almost surely, the orbits spend as much time in F as signiﬁcant is the value μ(F). It is a well known fact that if μ is a full support measure, then μ preserving ergodic transformation has wandering orbits. In the setting of a C0 -semigroup, we have precisely the following result [35].

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2.1 Proposition. let E be a separable Banach space, and μ a probability Borel measure on E with full support. If μ is ergodic with respect to the semigroup (Tt )t ≥0 , then the orbit {Tt x, t ≥ 0} is dense in E, μ almost surely. We note that, in general, it is not an easy task to check ergodicity of μ, and we often prove that it is strongly mixing which is a higher level of irregularity of a chaotic dynamic. In this case the system becomes memoryless and the Borelian sets or events are asymptotically independent. This property can be formalized in the following way. 2.2 Deﬁnition. (Tt ) is called a strongly mixing semigroup if for all Borelian sets A, B lim μ(Tt−1 A ∩ B) = μ(A)μ(B).

t →∞ 8

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μ will be called a strong mixing measure. In the sequel we will deal with the special class of Gaussian measures (see the appendix) since it is known how to characterize the invariance and the mixing property in terms of the semigroup and the covariance operator, ([32] and [4]). For the convenience of the reader, a proof of the invariance is given in Theorem 4.5. 2.3 Theorem. Let (Tt ) be a C0 -semigroup on a separable Banach space E. If μ is a centered Gaussian measure with covariance operator Q, then μ is invariant if and only if Tt QTt∗ = Q, for all t > 0, μ is strong mixing if and only if lim 〈x ∗ , Tt Qy ∗ 〉 = 0, for all x ∗ , y ∗ ∈ E∗ . t →∞

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3 Main result In the following we characterize the invariance of a centered Gaussian measure μ by means of its covariance operator and the generator of the semigroup, which is a question raised by Rudnicki in [35]. One used to prove the invariance of the measure by showing that the covariance operator Q has the form KK ∗ , where K : H → E is a gamma radonifying operator from a Hilbert space H. Furthermore, K satisﬁes the identity Tt K = KGt for all t and (Gt ) is a semigroup of unitary operators on H. (see [4, 13, 14]) 3.1 Proposition. A centered Gaussian measure μ, on E with covariance operator Q, is invariant if and only if AQ + QA∗ = 0 holds on D(A∗ ). Proof. First assume that μ is invariant, that is, Q = Tt QTt∗ , for all t ≥ 0, and let x ∗ ∈ D(A∗ ). We know that (A∗ , D(A∗ )) is the weak ∗ generator of the adjoint semigroup: 1 lim〈 (Tt∗ − I)x ∗ , x〉 = 〈A∗ x ∗ , x〉, for all x ∈ E. t →0 t 4

Therefore the family ( 1t (Tt∗ − I)x ∗ )0

To show that w = −QA∗ x ∗ , let us take y ∗ arbitrarily in E∗ . We have 〈y ∗ , w〉 = lim 〈y ∗ , Q k→∞

= lim 〈 k→∞

1 t nk

1 t nk

(x ∗ − T ∗ (t nk )x ∗ )〉,

(x ∗ − T ∗ (t nk )x ∗ ), Qy ∗ 〉,

= 〈−A∗ x ∗ , Qy ∗ 〉, = 〈y ∗ , −QA∗ x ∗ 〉. Finally, for all null sequence (t n ) there exists a subsequence t nk such that

1 tnk

(T(t nk )Qx ∗ −

Qx ∗ ) is convergent to −QA∗ x ∗ . Taking into account the strong continuity of the semigroup (Tt ), we deduce that Qx ∗ ∈ D(A) and AQx ∗ = −QA∗ x ∗ . To prove the converse, let t > 0, h = 0 and x ∗ ∈ D(A∗ ). We have 1 1 (Tt +h QTt∗+h x ∗ − Tt QTt∗ x ∗ ) = Tt +h Q [Tt∗+h − Tt∗ ]x ∗ h h 1 + [Tt +h − Tt ]QTt∗ x ∗ . h We know that h1 [Tt∗+h − Tt∗ ]x ∗ converges in the weak ∗ -topology to A∗ Tt∗ x ∗ and , as above, by the argument of compactness of Q, we can show that the ﬁrst term in the righthand side converges in norm to Tt QA∗ Tt∗ x ∗ . Since Tt∗ x ∗ belongs to D(A∗ ), and then QTt∗ x ∗ will be in D(A), the second term tends, as h goes to zero, to Tt AQTt∗ x ∗ . Therefore, for all x ∗ ∈ D(A∗ ), we obtain 1 (Tt +h QTt∗ x ∗ − Tt QTt∗ x ∗ ) = Tt [QA∗ + AQ]Tt∗ x ∗ = 0. h→0 h lim

1 2 3

4 5 6 7

This implies that Tt QTt∗ = Q, on D(A∗ ). Hence, by the symmetric property of Q, for all y ∗ ∈ E∗ , the two weak ∗ -continuous mapping 〈· , Qy ∗ 〉, and 〈· , Tt QTt∗ y ∗ 〉, coincide on the weak ∗ dense subspace D(A∗ ). Thus, Qy ∗ = Tt QTt∗ y ∗ holds for all y ∗ ∈ E∗ . Let E := Lp (T, d σ), 1 ≤ p < +∞, where (T, Σ, σ) is an arbitrary σ-ﬁnite measure space and assume that E is separable. Let (Tt ) be a C0 -semigroup on E, and A its generator with point spectrum, σp (A). The following criterion gives a sufﬁcient conditions on the imaginary eigenvectors ﬁeld to ensure the existence of a strong mixing Gaussian measure of full 5

1 2 3

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support. The method of the construction is different to that used in [5, 13]. Recall also that our conditions are weaker than the required ones in [29]. In the last paper the imaginary eigenvectors ﬁeld is assumed to be at least weakly C1 . 3.2 Theorem. Assume that σp (A) ∩ i R is contained in (i ω1 , i ω2 ) for some ω1 and ω2 such that −∞ ≤ ω1 < ω2 ≤ +∞, and there is a countable family of measurable functions (u j ) j ∈J u j : (ω1 , ω2 ) × T → C for every j ∈ J, satisfying the following conditions: j

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• a) For all j ∈ J, u s := u j (s, ·) ∈ ker (i s − A) for a.e s ∈ (ω1 , ω2 ),

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• b) (

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ω2 j 2 1 j ∈J ω1 |u s (·)| d s) 2

= v(·) ∈ E,

j

• c) lin{u s , s ∈ (ω1 , ω2 ) \ N, j ∈ J} is dense in E, for every subset N with zero Lebesgue measure. Then (Tt ) is strongly mixing with respect to a full support invariant Gaussian measure. Proof. Let {w j , j ∈ J} be a family of mutually independent two sided complex Brownian motions deﬁned on the same probability space (Ω, F , P). Note that by assumption b), for j a.e x ∈ T, and for all j ∈ J, the function s → u s (x) ∈ L2 (ω1 , ω2 ). Thus the Wiener integral, ω2 j j ω1 u s (x) d w s is a well deﬁned centered Gaussian random variable, (see Def. 4.6). Let us then consider for a.e. x ∈ T the following 1 ω2 j j Z x := u s (x) d w s , 2 j ∈J ω1 which is convergent in L2 (Ω, P), again via assumption b), and hence P almost surely. Since for all j , u j (·, ·) are measurable functions on (ω1 , ω2 ) × T, then P almost surely x → Z x is measurable in T. Using the independence property and Remark 4.7, one can calculate the covariance function of the Gaussian process, (Z x )x∈T . For a.e. x, y ∈ T one has ω2 1 ω2 j j 1 k k E(Z x Z y ) = E u s (x) d w s u s (y) d w s , 2 j ∈J ω1 2 k∈J ω1 ω2 ω2 1 j j E u s (x) d w s u sk (y) d w sk , = 2 j ,k∈J ω1 ω1 ω2 j j u s (x)u s (y) d s. = j ∈J ω1

In particular,

2

1 2

(E|Z x | ) =

ω2 j ∈J ω1

1 j |u s (x)|2 d s

2

1

1

By the Gaussian property there exists a constant c, such that (E|Z x |p ) p ≤ c(E|Z x |2 ) 2 . Us ing assumption b), we obtain that T E|Z x |p σ(d x) < ∞. We infer by Fubini theorem that 6

E T |Z x |p σ(d x) < ∞. Thus T |Z x |p σ(d x) < ∞, P almost surely. This means that the Gaussian process, (Z x )x∈T , has its paths almost surely in E. Now let us consider μ, its induced centered Gaussian measure deﬁned as, μ(B) = P{ω ∈ Ω, Z . (ω) ∈ B}, where B is a Borel set in E. To give the covariance operator Q, of the measure μ, we argue as in the proof of Example 3.11.14 page 148 in [9]. By deﬁnition for all f , g ∈ E∗ = Lq (T, d σ), where q is the conjugate of p. 〈g , Q f 〉 = 〈g , u〉 〈 f , u〉 μ(d u), E

g (x)Z x (ω) σ(d x) f (y)Z y (ω) σ(d y) P(d ω), T Ω T Z x (ω) Z y (ω) P(d ω) g (x) f (y) σ(d x) σ(d y), = T T Ω = E(Z x Z y ) g (x) f (y) σ(d x) σ(d y), T T ω2 j j u s (x)u s (y) d s g (x) f (y) σ(d x) σ(d y), = =

= =

T T j ∈J ω1 ω2

j ∈J ω1 ω2 j ∈J ω1

T

j

u s (x)g (x) σ(d x) j

T

j

u s (y) f (y) σ(d y) d s,

j

〈g , u s (·)〉 〈 f , u s (·)〉 d s. 1

1

Where we have used Fubini Theorem since |E(Z x Z y )| ≤ (E|Z x |2 ) 2 (E|Z y |2 ) 2 , and both of the precedent factors are in Lp (T, d σ). On the other hand the Gaussian measure, μ, has a full support since the operator Q, is positive deﬁnite as we shall prove. For all f ∈ E∗ , one has ω2 j |〈 f , u s (·)〉|2 d s ≥ 0. 〈f ,Qf 〉 = j ∈J ω1

Moreover, 〈 f , Q f 〉 = 0 implies that for all j ∈ J, there exists a zero Lebesgue measure subset j j N j , such that 〈 f , u s (·) >= 0, for all s ∉ N j . Then 〈 f , u s (·) >= 0, for all s ∉ N := N j . We conclude using assumption c), that f = 0. Now we use Proposition 3.1, to prove the invariance of the measure μ. For all t > 0, f , g ∈

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D(A∗ ) one has 〈g , (AQ + QA∗ ) f 〉 = 〈g , AQ f 〉 + 〈g , QA∗ f 〉, = 〈A∗ g , Q f 〉 + 〈g , QA∗ f 〉, ω2 ω2 ∗ j j j j 〈A g , u s 〉〈 f , u s 〉d s + 〈g , u s 〉〈A∗ f , u s 〉d s, = j ∈J ω1

= =

ω2 j ∈J ω1 ω2 j ∈J ω1

j ∈J ω1

j

j

〈g , Au s 〉〈 f , u s 〉d s + j

j

ω2

j ∈J ω1 ω2

〈g , i su s 〉〈 f , u s 〉d s +

j ∈J ω1

j

j

〈g , u s 〉〈 f , Au s 〉d s, j

j

〈g , u s 〉〈 f , i su s 〉d s,

= 0. Since D(A∗ ) is weak∗ dense in E∗ , then (AQ f + QA∗ ) f = 0, for all f ∈ D(A∗ ) and μ is (Tt ) invariant. Let us prove that μ is a strong mixing measure. For f , g ∈ Lq one has 〈g , Tt Q f 〉 = 〈Tt∗ g , Q f 〉, ω2 ∗ j j 〈Tt g , u s 〉< f , u s >d s, = = = j

j ∈J ω1 ω2 j ∈J ω1 ω2 j ∈J ω1

j

j

〈g , Tt u s 〉〈 f , u s 〉d s, j

j

e i t s 〈g , u s 〉〈 f , u s 〉d s.

j

2

s → 〈g , u s 〉〈 f , u s 〉 is in L1 (ω1 , ω2 ), since both factors lie in L2 (ω1 , ω2 ), then by RiemannLebesgue theorem’s lim 〈g , Tt Q f 〉 = 0. We conclude by using Theorem 2.3.

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4 Applications

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t →+∞

We start by treating the simple example of the translation semigroup on a weighted space E := Lp (I, ρ(x)d x) := { f , f : I → C measurable with I f ρ d x < ∞}. We give explicitly the Gaussian process inducing the strong mixing measure. Let us ﬁrst recall that ρ is an admissible weight measurable function in the following sense (i) ρ (x) > 0 for all x ∈ I, (ii) there exist constants M ≥ 1 and ω ∈ R such that ρ(x) ≤ M e ωt , for all x ∈ I and all t > 0. ρ(x + t ) 8

(4.1)

1

2 3

The translation semigroup Tt f (·) = f (· + t ) is then a well deﬁned C0 -semigroup on E. 4.1 Example. if full support.

I ρ(x)d x

< ∞ then there exists an invariant strong Gaussian measure with

Proof. One can take for all s ∈ R, x ∈ I, i sx 2e u s (x) := π(1 + s 2 ) It is clear that u s (·) ∈ E := Lp (I, ρ(x)d x) and Au s = i su s . In this example the imaginary eigen+∞ 1 vectors ﬁeld, s → u s (·), is continuous. The condition (ii) is satisﬁed since x → ( −∞ |u s (x)|2 d s) 2 = 2 ∈ Lp (I, ρ(x)d x). Let φ ∈ E∗ = Lq (I, ρ(x)d x), ( p1 + q1 = 1), such that 〈φ, u s 〉 = 0, 4 5 6

a.e.s ∈ (−∞, +∞).

+∞ Then on has −∞ e i sx φ(x)ρ(x) = 0, a.e, s ∈ (−∞, +∞). This means that the Fourier transform of the function φρ, which is in L1 (R, d x), vanishes almost everywhere, and hence everywhere by continuity. Thus φ = 0, and the condition (i i i ) is fulﬁlled. 4.2 Remark. The precedent result was already proved in [13] and it is in fact a characterization as shown in [5]. Like in the proof of Theorem 3.2, we deﬁne the Gaussian process 2 e i sx 1 d w s . Let us compute its covariance function to prove that is a complex Zx = R 2

π(1+s 2 )

Ornstein Uhlenbeck process. cov(Z x , Z y ) =

+∞ −∞ +∞

u s (x)u s (y)d s,

2 e i s(x−y) d s, 2 −∞ π(1 + s ) +∞

2 1 1 = 2( e i s(x−y) d s), π (1 + s 2 ) 2π −∞

=

= 2e −|x−y| . 7 8

On can say that for almost all function in E, the set of its translates and the realisations of the Ornstein Uhlenbeck process are statistically the same.

9

4.3 Example. Here we treat the perturbation of the generator of the translation u → ∂u by a ∂x multiplication operator. We will use a classical method to prove that the semigroup is homeomorphic to a translation semigroup on some weighted space, see([21, 36]). Let us consider the following equation ∂u = ∂u + h(x)u, ∂t ∂x (4.2) u(0, ·) = ϕ ∈ Lp (0, +∞), 9

where h : R+ → R is a bounded function and p ≥ 1. The solution semigroup (Tt ) is given by Tt (ϕ)(x) = e

x+t x

h(s)d s

ϕ(x + t ), x, t ∈ R+

+∞ x The semigroup T(·) is chaotic in E = Lp (R+) if and only if 0 exp(−p 0 h(s)d s)d x < ∞, and x hypercyclic if and only if, supx≥0 0 h(s)d s = ∞, [36]. To see this, just remark that e

x 0

h(s)d s

Tt ϕ(x) = e

x+t 0

h(s)d s

ϕ(x + t ), x, t ∈ R+

This formulae shows that the semigroup T(·) is isomorphic to the translation semigroup, S(·), x p −p 0 h(s)d s . The isomorphism is deﬁned on the weighted space, F = L (R+ , ρ), where ρ(x) = e x p p h(s)d s 0 as the multiplication operator M : L (R+ ) → L (R+ , ρ), Mϕ(x) = e ϕ(x), such that, M ◦ Tt = S t ◦ M, that is the following diagram commutes Tt

E −−−−→ ⏐ ⏐

M

E ⏐ ⏐

M

St

F −−−−→ F ·

1 2 3

Let μ be the induced measure by the Gaussian process, ξ· = M−1 Z · = e − 0 h(s)d s Z · , where (Z x )x∈R+is the Ornstein-Uhlenbeck process deﬁned in the previous example. Remark that 2 exp(i sx−0x h(τ)d τ) ξx = R d w s , which is exactly the Gaussian process given in the proof of 2 π(1+s )

4

Theorem 3.2.

5

Appendix

6 7 8 9 10 11 12 13 14 15 16 17

18 19 20

Gaussian measures or variables are customary studied on real Banach spaces, but since we will use spectral theory of strongly continuous semigroups we let, hereafter, E, be a complex Banach space. We additionally require that E is separable to support dense orbits {Tt x, t ≥ 0} where x is a vector in E and T(·) is a strongly continuous semigroup of linear operators on E. Note also that under this assumption the Borel sigma algebra of E, is the smallest one making all linear functional measurable. An E valued Gaussian variable, X(·), is deﬁned [8] by the natural way, for each x ∗ ∈ E, ξ(·) = 〈x ∗ , X(·)〉 is a complex Gaussian variable. Let us recall that this means ξ = ξ1 + i ξ2 , where ξ1 and ξ2 are independent and identically distributed real Gaussian variables. In the case where E(ξ1 ) = E(ξ1 ) = 0, ξ is called a centered complex variable. now we give the deﬁnition of a Gaussian measure on E and other related notions. Our main references in this subject are [9], [4], and [38]. 4.4 Deﬁnition. A centered Gaussian measure μ, on E is a probability measure such that each continuous linear functional x ∗ ∈ E∗ is a centered Gaussian variable when considered as a random variable on the probability space (E, μ). 10

It turns out that any Gaussian measure μ on E is the distribution of an E valued Gaussian random variable ξ on some probability space (Ω, F , P), deﬁned by μ(F) = P(ξ ∈ F), F ∈ F . Recall that if μ is a Gaussian measure on E, then the topological dual E∗ can be embedded into a subspace of L2 (E, μ). The covariance operator of μ is then the unique bounded conjugate linear operator Q : E∗ → E, such that for every (x ∗ , y ∗ ) ∈ E∗ × E∗ , ∗ ∗ 〈y , Qx 〉 = 〈y ∗ , x〉〈x ∗ , x〉 μ(d x) = 〈y ∗ , x ∗ 〉L2 (d μ) . E

Q is symmetric in the sense that 〈y ∗ , Qx ∗ 〉 = 〈x ∗ , Qy ∗ 〉 1 2

The important fact is that a centered Gaussian measure μ is uniquely determined by the associated covariance operator via its Fourier transform. See for example ([4] page 99) ˆ is the complex-valued function deﬁned on the dual space E∗ by 4.5 Theorem. μ −〈x ∗ ,Qx ∗ 〉 ∗ ∗ 4 ˆ ) = e i Re〈x ,x〉 d μ(x) = e . μ(x X

We shall give here a short justiﬁcation of the unusual factor 14 in the above formulae. If we denote by νx ∗ the distribution of the real Gaussian random variable Re〈x ∗ , ·〉 deﬁned on the σ2

ˆ ∗ ) = e − 2 where σ2 = Eμ |Re〈x ∗ , ·〉|2 which is also equal to probability space (E, μ), then μ(x Eμ |Im〈x ∗ , ·〉|2 , as Re〈x ∗ , ·〉 and Im〈x ∗ , ·〉 have the same distribution. Finally we conclude that ˆ ∗) = e μ(x 3 4 5

−Eμ |〈x ∗ ,·〉|2 4

=e

−〈x ∗ ,Qx ∗ 〉 4

.

It is not difﬁcult to prove that for each t ≥ 0, the covariance operator of the image measure ˆ and hence μt := μ(Tt−1 (·)) is the operator Tt QTt∗ . Then μ is invariant if and only if μˆt = μ, ∗ Q = Tt QTt , for all t ≥ 0. We start by introducing the complex-valued Wiener integral R g (s)d w s , where g ∈ L2 (R, C) and (w s )s∈R is the complex Brownian motion deﬁned as w s := w s1 + i w s2 ,

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for independent two sided Brownian motions (w s1 )s∈R and (w s2 )s∈R . 4.6 Deﬁnition. For g = g 1 + i g 2 ∈ L2 (R, C), the wiener integral Z = R g (s)d w s is the complex valued random variable Z = X + i Y, where X= Y=

9

R R

g 1 (s)d w s1 − g 2 (s)d w s1 +

R R

g 2 (s)d w s2 , g 1 (s)d w s2 .

4.7 Remark. One can show that for any g , h ∈ L2 (R, C), E( 11

R g (s)d w s R h(s)d w s ) = 2〈g , h〉L2 (R,C)

1

2 3

4 5

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