The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor

The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor.


1.
Introduction. Mean-square properties are of traditional interest in the investigation of stochastic systems in engineering and physics. This is quite natural since the Ito stochastic calculus is a mean-square calculus. At first sight, it is thus somewhat surprising that the classical theory of random dynamical systems and their spectra is a pathwise theory, although this can be justified by Doss-Sussman-like transformations between stochastic differential equations and path-wise random ordinary differential equations [1]. Such transformations, however, do not apply to mean-field stochastic differential equations, which include expectations of the solution in their coefficient functions [6].
Mean-square random dynamical systems based on deterministic two-parameter semi-groups from the theory of nonautonomous dynamical systems acting on a state space of random variables or random sets with the mean-square topology were introduced in [7]. These act like deterministic systems with the stochasticity built into the state spaces of mean-square random variables. A mean-square random attractor was defined as a nonautonomous pullback attractor for such systems from the theory of nonautonomous dynamical systems [9]. The main difficulty in applying the theory is the lack of useful characterisations of compact sets of such spaces of mean-square random variables.
In this paper, a theory of mean-square exponential dichotomies is presented for linear mean-field stochastic differential equations. (It also applies to classical linear stochastic differential equations). Although the corresponding mean-square random dynamical systems are essentially infinite-dimensional their dichotomy spectrum is given by the union of finitely many intervals. This is applied to analyse a nonlinear mean-field stochastic differential equation, for which it is shown that the trivial solution undergoes a mean-square bifurcation leading to a nontrivial mean-square attractor.
The paper is structured as follows. Section 2 contains the definition of a meansquare random dynamical system, and the notions of mean-square exponential dichotomy and mean-square dichotomy spectrum are introduced. Section 3 explains under which conditions, a mean-field stochastic differential equation generates a mean-square random dynamical system. In Section 4, the spectral theorem is established, which says that the mean-square spectrum of a linear mean-field stochastic differential equation consists of finitely many compact intervals. Finally, in the last section, it is shown that for a one-dimensional mean-field SDE of pitchfork-type, a stability change in the mean-square spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor.
2. Mean-square random dynamical systems. Consider the time set R, and define R 2 ≥ := (t, s) ∈ R 2 : t ≥ s . Let (Ω, F, {F t } t∈R , P) be a complete filtered probability space satisfying the usual hypothesis, i.e., {F t } t∈R is an increasing and right-continuous family of sub-σ-algebras of F, which contain all P-null sets. Essentially, F t represents the information about the randomness at time t ∈ R. Finally, define with the norm X ms := E|X| 2 , where | · | is the Euclidean norm on R d .

Definition 1.
A mean-square random dynamical system (MS-RDS for short) ϕ on the underlying phase space R d with the filtered probability space (Ω, F, {F t } t∈R , P) is a family of mappings for (t, s) ∈ R 2 ≥ , which satisfies: (1) Initial value condition. ϕ(s, s, X s ) = X s for all X s ∈ X s and s ∈ R.
Mean-square random dynamical systems are essentially deterministic with the stochasticity built into or hidden in the time-dependent state spaces.
A MS-RDS ϕ is called linear if for each (t, s) ∈ R 2 ≥ , the map ϕ t,s (·) := ϕ(t, s, ·) is a bounded linear operator. It will be denoted by Φ t,s , and Φ t,s (X) will conventionally be written Φ t,s X. A spectral theory for linear mean-square random dynamical systems can be established based on exponential dichotomies. Definition 2 (Mean-square exponential dichotomy). Let γ ∈ R. A linear meansquare random dynamical system Φ t,s : X s → X t is said to admit an exponential dichotomy with growth rate γ if there exist positive constants K, α and a timedependent decomposition A special case of exponential dichotomy, when the growth rate is equal to zero and the space of initial condition consists of the deterministic vectors in R d , is also investigated in [2,12], where a Perron-type condition for existence of this exponential dichotomy is established.
Definition 3 (Mean-square dichotomy spectrum). The mean-square dichotomy spectrum for a linear MS-RDS Φ is defined as Σ := γ ∈ R : Φ has no exponential dichotomy with growth rate γ .
The set ρ := R \ Σ is called the resolvent set of Φ.
The dichotomy spectrum was first introduced in [11] for nonautonomous differential equations. Dichotomy spectra for random dynamical systems have been discussed recently in [3,4,13].
3. Mean-field stochastic differential equations. Mean-field stochastic differential equations of the form were introduced in [6]. Here {W t } t∈R is a two-sided scalar Wiener process defined on a probability space (Ω, F, P), and F := (f, g) : (A2) For each R > 0, there exist a constant L R and a modulus continuity ω R such that Let {F t } t∈R be the natural filtration generated by {W t } t∈R , and define Given any initial condition X s ∈ X s , s ∈ R, a solution of (1) is a stochastic process {X t } t≥s with X t ∈ X t for t ≥ s, satisfying the stochastic integral equation It was shown in [6] that the SDE (1) has a unique solution and generates a MS-RDS {ϕ t,s } t≥s on the underlying phase space R d with a probability set-up (Ω,

4.
Mean-square dichotomy spectrum for linear mean-field stochastic differential equations. Consider a linear mean-field stochastic differential equation where A, B, C, D : R → R d×d are continuous bounded functions, which generates a linear mean-square random dynamical system Φ t,s .
Proposition 4 (Equations for the first and second moments). Let X s ∈ X s , and where the map π s = π 1 It is interesting to compare the above equations with the ordinary differential equations for the first moment and second moments of linear stochastic differential equations, see e.g. [5, Section 6.2].
The proofs of the following preparatory results are straightforward.
Lemma 7. Let γ ∈ R be such that that the linear MS-RDS Φ t,s : X s → X t generated by (1) admits an exponential dichotomy with the growth rate γ and a decomposition X t = U γ (t) ⊕ S γ (t). Then the subspace S γ (t) is uniquely determined, i.e., if the linear MS-RDS Φ t,s also admits an exponential dichotomy with the growth rate γ and another decomposition The subspaces S γ (t) of an exponential dichotomy with the growth rate γ are its stable subspaces. The following lemma provides an inclusion relation between these stable subspaces. Its proof follows directly from the definition of an exponential dichotomy.
Lemma 8. Let γ 1 < γ 2 be such that the linear MS-RDS admits an exponential dichotomy with the growth rates γ 1 and γ 2 . Then S γ1 (t) ⊂ S γ2 (t) for all t ∈ R.
One of the main results of this paper is the following characterisation of the dichotomy spectrum.
Theorem 9 (Spectral Theorem). Suppose that the coefficient functions in the linear mean-field stochastic differential equation (2) satisfy with some m > 0. Then the dichotomy spectrum Σ is the disjoint union of at most Proof. The proof is divided into several steps.
Step 2. It will be shown that for any t ∈ R, the set consists of at most d(d + 1) + 1 elements. Suppose the contrary, i.e., there exist n + 1 numbers Then by Lemma 8, S γ0 (t) S γ1 (t) · · · S γn (t).
Thus, there exist X 1 t , . . . , X n t such that By definition of the γ i , there exist K, α > 0 and complementary subspaces U γi (t) such that X t = U γi (t) ⊕ S γi (t) and and Φt ,t X t ms ≥ 1 K e (γi+α)(t−t) X t ms for X t ∈ U γi (t) and t ≥ t .
Since R Consequently, by Corollary 5, By definition of γ i and (6) Hence, it follows by (6) and (7) that X k t ∈ S γ k−1 (t), which leads to a contradiction.

It will be shown that
First, let γ ∈ I i be arbitrary. By the definition of I i , there exist K, α > 0 and a decomposition X t = U (t) ⊕ S i (t) and This implies that (γ − α, γ + α) ⊂ I i , so I i is open. It can be shown similarly that I i is connected. Hence I i = (a i , b i ). Combining this result and Step 1 gives ρ = (−∞, a 1 ) ∪ (b 1 , a 2 ) ∪ · · · ∪ (b n−1 , a n ) ∪ (b n , ∞), which implies that Σ = [a 1 , b 1 ] ∪ · · · ∪ [a n , b n ]. To conclude the proof, the filtration corresponding to the spectral intervals is constructed as follows: for t ∈ R, V 0 (t) := {0}, V n (t) := X t , and Due to Lemma 7, the definition of V i is independent of γ ∈ (b i , a i+1 ) for i ∈ {1, . . . , n − 1}. The strict inclusion V i V i+1 for i ∈ {0, . . . , n − 1} follows from the construction of the open interval (b i , a i+1 ) above. Finally, the dynamical characterisation of V i follows from the definition of (b i , a i+1 ) and the definition of exponential dichotomy. This completes the proof.

5.
Bifurcation of a mean-square random attractor. A mean-square random attractor was defined in [7] as the pullback attractor of the nonautonomous dynamical system formulated as a mean-square random dynamical system. Specifically, a family A = {A t } t∈R of nonempty compact subsets of X with A t ⊂ X t for each t ∈ R is called a pullback attractor if it pullback attracts all uniformly bounded families D = {D t } t∈R of subsets of {X t } t∈R , i.e., Uniformly bounded here means that there is an R > 0 such that X ms ≤ R for all X ∈ D t and t ∈ R.
The existence of pullback attractors follows from that of an absorbing family. A uniformly bounded family B = {B t } t∈R of nonempty closed subsets of {X t } t∈R is called a pullback absorbing family for a MS-RDS ϕ if for each t ∈ R and every uniformly bounded family D = {D t } t∈R of nonempty subsets of {X t } t∈R , there exists some T = T (t, D) ∈ R + such that Theorem 10. Suppose that a MS-RDS ϕ has a positively invariant pullback absorbing uniformly bounded family B = {B t } t∈R of nonempty closed subsets of {X t } t∈R and that the mappings ϕ(t, s, ·) : X s → X t are pullback compact (respectively, eventually or asymptotically compact) for all (t, s) ∈ R 2 ≥ . Then, ϕ has a unique global pullback attractor A = {A t } t∈R with its component sets determined by Consider the nonlinear mean-field SDE with real-valued parameters α, β. Note that the theory in Section 3 can be easily extended to include the second moment of the solution in the equation. This SDE has the steady state solutionX(t) ≡ 0. Linearising along this solution gives the bi-linear mean-field SDE Theorem 11. The dichotomy spectrum of the linear MS-RDS Φ generated by (10) is given by Proof. Taking the expectation of two sides of (10) yields that which implies that EΦ(t, s)Z s = e (α+β)(t−s) EZ s for (t, s) ∈ R 2 ≥ and Z s ∈ X s .
Theorem 12. The MS-RDS ϕ generated by (14) has a uniformly bounded positively invariant pullback absorbing family.
Proof. Let α be arbitrary and define B t = X ∈ X t : X ms ≤ |α| + 2 for t ∈ R .
Using (2α + 1)EX 2 Let D = {D t } t∈T be a uniformly bounded family of nonempty subsets of {X t } t∈R , i.e., D t ⊂ X t and there exists R > 0 such that X ms ≤ R for all X ∈ D t . Specifically, it will be shown that ϕ(t, s, D s ) ⊂ B t for t − s ≥ T , where T is defined by Pick X s ∈ D s arbitrarily and (t, s) ∈ R 2 ≥ with t − s ≥ T . Motivated by the differential inequality (17), consider the scalar systeṁ A direct computation yields From the definition of T in (18), it follows that min s≤u≤t y(u) ≤ |α| + 2. Furthermore, y = 0 and y = α + 3 2 are stationary points of the ODE (19). For this reason, min s≤u≤t y(u) ≤ |α| + 2 implies that y(t) ≤ |α| + 2. Then from (17), it follows that y(t) ≥ ϕ(t, s)X s 2 ms . This means that ϕ t,s X s 2 ms ≤ y(t) ≤ |α| + 2 , i.e., ϕ t,s X s ∈ B t for t − s ≥ T . Hence, {B t } t∈R is a pullback absorbing family for the MS-RDS ϕ. It is clear that this family is uniformly bounded and positively invariant for the MS-RDS ϕ.
Proof. Let α < −1 be arbitrary. Let D = {D t } t∈R be a uniformly bounded family of nonempty subsets of {X t } t∈R with X ms ≤ R for all X ∈ D t where R > 0. Let (X s ) s∈R be an arbitrary sequence with X s ∈ D s . The moment equations (15)-(16) can be written as which implies that lim s→−∞ ϕ(t, s, X s ) ms = 0. Thus {0} is the pullback attractor of (14) in this case.
Theorem 14. The MS-RDS ϕ generated by (14) has a nontrivial pullback attractor when −1 < α < − 1 2 . Remark 15. The idea for the proof is taken from [7, Subsection 4.1], but their result cannot be applied directly, since the Lipschitz constant of the nonlinear terms is at least 1, and α is not less than −4. Instead the uniform equicontinuity of the mapping t → EX t , and then the positivity of the second moment to obtain a better estimate are used.
Proof. In order to apply Theorem 10, it needs to be shown that ϕ is pullback asymptotically compact, i.e., given a uniformly bounded family D = {D t } t∈R of nonempty subsets of X t and sequences {t k } k∈N in (−∞, t) with t k → −∞ as k → ∞ and {X k } k∈N with X k ∈ D t k ⊂ X t k for each k ∈ N, then the subset {ϕ(t, t k , X k )} k∈N ⊂ X t is relatively compact. For this purpose, let ε > 0 be arbitrary. A finite cover of {ϕ(t, t k , X k )} k∈N with diameter less than ε will be constructed. Choose and fix s ∈ R with s < t such that 4e (2α+1)(t−s) (|α| + 2) 2 < ε 2 2 , and define Y k s := ϕ(s, t k , X k ). Using (20) it can be assumed without loss of generality that E(Y k s ) 2 ≤ |α| + 2 for k ∈ N.