A Sufficient and Necessary Condition of PS-ergodicity of Periodic Measures and Generated Ergodic Upper Expectations

This paper contains two parts. In the first part, we study the ergodicity of periodic measures of random dynamical systems on a separable Banach space. We obtain that the periodic measure of the continuous time skew-product dynamical system generated by a random periodic path is ergodic if and only if the underlying noise metric dynamical system at discrete time of integral multiples of the period is ergodic. For the Markov random dynamical system case, we prove that the periodic measure of a Markov semigroup is PS-ergodic if and only if the trace of the random periodic path at integral multiples of period either entirely lies on a Poincar\'{e} section or completely outside a Poincar\'{e} section almost surely. In the second part of this paper, we construct sublinear expectations from periodic measures and obtain the ergodicity of the sublinear expectations from the ergodicity of periodic measures. We give a number of examples including the ergodicity of the discrete time Wiener shift of Brownian motions. The latter result would have some independent interests.


Introduction
Ergodic theory is one of the most important observations in mathematics made in the last century with significance in many areas of physics such as statistical physics (c.f. [3], [23], [24], [25]). Concerning the spreading and irreducibility nature of dynamical systems, ergodicity is a fundamentally natural phenomenon to many stochastic systems (c.f. [5], [7], [8], [16]). The theory was substantially developed for (weakly) mixing invariant measures/stationary processes in the stationary regime. Many useful results were obtained especially in the Markov stochastic dynamical system case.
However, the classical theory excludes periodic cases, which is even the case in the theory of Markov chains. The ergodic theory of periodic random dynamical system was observed recently in [13]. The concept of periodic measure was introduced and its "equivalence" with random periodic processes was established. Moreover, the average of a periodic measure over one period is an invariant measure from which the ergodicity can be studied. It is defined as the ergodicity of a measure preserving canonical dynamical systems lifted from the invariant measure on the phase space. From the ergodic theory of periodic random dynamical systems in [13], the distinction between the stationary regime and random periodic regime is characterised by the spectral structure of Markov semigroups or their infinitesimal generators. In the stationary regime, Koopman-von Neumann theorem tells us that 0 is a simple and unique eigenvalue of the generator on the imaginary axis (c.f. [5]), while in the periodic regime, the infinitesimal generator has infinitely many equally placed eigenvalues (including 0), which are simple, and no other eigenvalues, on the imaginary axis ( [13]).
In this paper, we continue the study on the ergodicity of periodic measures and obtain some new results. First we study random periodic paths on a separable Banach space and associated periodic measures µ s = δ Y (s,θ −s ω) × dP on the product space (Ω,F) = (Ω × X, F ⊗ B(X)). Then {µ s } s≥0 is a periodic measure with the skew product (Θ t ) t≥0 . We prove in the first part of the paper that (Ω, F, P, (θ nτ ) n≥0 ) is ergodic if and only if (Ω,F , µ s , (Θ nτ ) n≥0 ) is ergodic, i.e. the dynamical system (Ω,F , {µ s } s∈R , (Θ t ) t≥0 ) is PS-ergodic. Note here there is no need of any other conditions on its random periodic paths Y apart from the existence and the definition of random periodic paths. The metric dynamical system (Ω, F, P, (θ nτ ) n≥0 ) being ergodic is stronger than the statement that (Ω, F, P, (θ t ) t≥0 ) is ergodic. They are not normally equivalent. We will give an example that (Ω, F, P, (θ t ) t≥0 ) is ergodic, but the discrete metric dynamical system (Ω, F, P, (θ nτ ) n≥0 ) is not ergodic. However, we will prove in this paper, for the canonical Wiener process and the Brownian shift, both the discrete dynamical systems and continuous time dynamical systems are ergodic. The result of the discrete dynamical systems of Wiener space is new. This means that our results can apply to stochastic differential equations and stochastic partial differential equations driven by Wiener processes. Suggested by fundamental results of [1], [9], [14], [18], [19] and [20], these equations can generate random dynamical systems, of which the noise metric dynamical system over a group of discrete time is ergodic according to our results here.
For the Markov random dynamical systems, the random periodic paths give rise to periodic measures (ρ s ) s∈R on the state space X. For each s, ρ s is an invariant measure with respect to discrete semigroup P (nτ ), n ∈ N. We will give a necessary and sufficient condition for the periodic measure (ρ s ) s∈R being PS-ergodic (i.e. for each s ∈ R, ρ s is ergodic as an invariant measure with respect to P (nτ ), n ∈ N), which says for any invariant set Γ such that P τ I Γ = I Γ ρ s − a.s., the section L ω s := {Y (s + kτ, ω), k ∈ Z} satisfies L ω s ⊂ Γ or L ω s ∩ Γ = ∅, P − a.s.. However, it is not known whether or not this result is true in the stationary case.
Sublinear expectation is used to model uncertainty and ambiguity of probabilities such as subjective probabilities due to heterogeneity of expectation formation process (c.f. [2], [6], [22]). An ergodic theory of sublinear expectation was developed recently by [12]. In the second part of this paper, we construct an ergodic sublinear expectation from an ergodic periodic measure as an upper expectation for the first time in literature. We prove that if a periodic measure is ergodic, then the generated sublinear expectation, which is invariant with respect to the skew product dynamical system or the Markov semigroup, is ergodic. As for the Birkhoff type ergodic theorem, i.e. the law of large number, we obtain the convergence in the quasi-sure sense when we apply the ergodic theory of upper expectations, whilst we can only obtain the convergence in the almost-sure sense by the ergodic theory of periodic measures ( [13]). This provides one of the justifications why we see construction of upper expectation and the investigation of its ergodicity, can provide useful new information. The point of view of upper expectations from periodic measures could also be interesting to the study of finance or coherent risk measure.
Definition 2.1. A random periodic path of period τ of the random dynamical system Φ : It is called a random periodic path with the minimal period τ if τ > 0 is the smallest number such that Consider a standard product measurable space (Ω,F ) = (Ω × X, F ⊗ B(X)) and the skewproduct of the metric dynamical system (Ω, F, P, (θ t ) t∈R ) and the cocycle Φ(t, ω) on X,Θ t : Recall P P (Ω × X) := {µ : probability measure on (Ω × X, F ⊗ B(X)) with marginal P on (Ω, F)} and P(X) = {ρ : probability measure on (X, B(X))}.
The following definition was given in [13].
is called a periodic probability measure of period τ on (Ω × X, F ⊗ B(X)) for the random dynamical system Φ if It is called a periodic measure with minimal period τ > 0 if τ is the smallest number such that (2.4) holds. It is an invariant measure if it also satisfies µ s = µ 0 for any s ∈ R, i.e. µ 0 is an invariant measure of Φ if µ 0 ∈ P P (Ω × X) and ) If a random dynamical system Φ : R + × Ω × X → X has a random periodic path Y : R × Ω → X, it has a periodic measure on (Ω × X, F ⊗ B(X)) µ : R → P P (Ω × X) given by where A ω is the ω-section of A. Moreover, the time average of the periodic measure defined bȳ is an invariant measure of Φ whose random factorisation is supported by L ω .
Remark 2.4. For a periodic path Y, it is easy to see that the factorization of µ s defined in In this section, we always assume the following condition and use the construction of periodic measure given in (2.6).
Condition P. There exists a random periodic path Y with period τ for the random dynamical system Φ.
Thoughout the paper, we adopt the standard definition of ergodicity of a measure preserving dynamical system, i.e. any invariant set of the dynamical system has either full measure or zero measure.
Proof. First, note by Theorem 2.3, that µ s defined in (2.6) is a periodic measure on (Ω,F ), sō Θ τ preserves measures µ s for each s ∈ R and (Θ t ) t≥0 preserves the measureμ.
Next, we will show that the dynamical system (Ω,F , µ s , (Θ n τ ) n≥0 ) is ergodic for any fixed s ∈ R. By definition of ergodicity, we need to show that for any A ∈F withΘ for all s ∈ R. Thus A s is an invariant set with respect to θ τ .
Let us show that the dynamical system (Ω,F ,μ, (Θ t ) t≥0 ) is ergodic, i.e. for any A ∈F with Θ −1 t A = A for all t ∈ R + , we need to prove thatμ(A) = 0 or 1. For such A, by what we just proved, we know that µ s (A) = 0 or 1 for all s ∈ R. On the other hand, since The following theorem shows that the converse of Theorem 2.5 also holds. Theorem 2.6. Assume Condition P. If (Ω,F , µ s , (Θ n τ ) n≥0 ) is ergodic for some s ∈ R, then (Ω, F, P, (θ n τ ) n≥0 ) is ergodic.
Remark 2.7. (i). From Theorem 2.5 and Theorem 2.6, we can conclude that (Ω,F , µ s , (Θ n τ ) n≥0 ) is ergodic for some s ∈ R implies that (Ω,F , µ s , (Θ n τ ) n≥0 ) is ergodic for all s ∈ R. This conclusion does not seem to be true in the phase space case, which we will consider in the next section.
(ii). It is easy to check that if the dynamical system (Ω, F, P, (θ n τ ) n≥0 ) is ergodic, then (Ω, F, P, (θ t ) t≥0 ) will be ergodic. The converse is not true in general. A counterexample can be found in Section 2.1.1.
(iii). It is well known that, for a canonical Wiener space (Ω, F, P ), the corresponding canonical dynamical system (Ω, F, P, (θ t ) t≥0 ) is ergodic, where θ t is the Brownian shift. In Section 2.1.2 we will prove the discrete dynamical system (Ω, F, P, (θ n τ ) n≥0 ) is also ergodic. Thus our results in this paper can apply to stochastic differential equations and stochastic partial differential equations driven by Brownian motions.
Next we will give these two examples of metric dynamical systems. where α is a fixed positive irrational number. Then it is easy to check that (Ω,F , (θ α t ) t∈R ) is a dynamical system. Let L be the Lebesgue measure on [0, 1) and P(Ω) be the set of probability measures on (Ω,F ). Define µ : R → P(Ω) by then µ is a periodic probability measure with period 1 on (Ω,F ). Letμ := 1 0 µ s ds. Proof.
Remark 2.9. It is easy to see that the dynamical system (Ω,F ,μ, ((θ α 1 ) n ) n∈N ) is not ergodic considering the first coordinate of the mapping. This provides an example that the continuous time dynamical system is ergodic, but its discretization may not be.

Example: the metric dynamical system of Brownian motions
A standard Brownian motion or Wiener process (W t ) t∈T (T = R + (one-sided time) or T = R (two-sided time)) in R m is a process with W 0 = 0 and stationary independent increments satisfying W t − W s ∼ N (0, |t − s|I). The corresponding measure P on (Ω, F), where Ω = C 0 (T, R m ) and F is the Borel σ-algebra on Ω, is called Wiener measure, the probability space (Ω, F, P) is called Wiener space. The corresponding canonical metric dynamical system Σ = (Ω, F, P, (θ t ) t∈T ) describes Brownian motion or (Gaussian) white noise as a metric dynamical system of random dynamical systems generated by stochastic differential equations or stochastic partial differential equations driven by Brownian motions.
Let Σ = (Ω, F, P, (θ t ) t∈T ) be one of the canonical dynamical system introduced above, with the canonical filtration F t and for two-sided time as the tail σ-algebras (T −∞ : remote past, T ∞ : remote future). Set and for a given τ > 0, define Proposition 2.10. Assume that I, I τ , T ∞ and T −∞ defined as above. Then Proof. Obviously, by the definitions of I, I τ , we have I ⊂ I τ .
(i) We just need to prove that I τ ⊂ T ∞ . For any A ∈ I τ , since (ii) Without loss of generality, we just need to prove I τ ⊂ T ∞ mod P. Similarly we can prove I τ ⊂ T −∞ mod P. To prove the desired result, for any A ∈ I τ , we set Then we know that P(A) = lim n→∞ P(B n ) from construction of finite dimensional distribution of Wiener measure. By the definition ofÃ we haveÃ ⊂ B n for all n, then P(Ã) ≤ lim n→∞ P(B n ) = P(A). Since we know that P(A) ≤ P(Ã), so we conclude P(Ã) = P(A) .
Next we prove the following claim. Claim ( * ): For all n ≥ 2, we have θ −1 τ A n ⊃ A n−1 ⊃ A n . Proof of Claim ( * ): The claim A n−1 ⊃ A n is obvious. Now for any ω ∈ A n−1 , there exists Thus θ τ ω ∈ A n and ω ∈ θ −1 τ A n . This means Claim ( * ) holds. Now we continue our proof. Since θ τ preserves probability P, then by Claim ( * ), we have It turns out that P(A n ) = P(Ã) = P(A) for all n ∈ N.
And also we have that for all k, n ∈ N.
Proof. By Proposition 2.10, if T ∞ is trival mod P, Σ and Σ τ are ergodic. Since these canonical dynamical systems Σ, Σ τ driven by a standard Brownian motion, then the tail σ-algebra T ∞ is trival mod P by Kolmogorov's zero-one law.
Remark 2.12. (i) It is noted that the ergodicity of Σ in Theorem 2.11 was known in literature (c.f. [1]). The main purpose of the Theorem is to prove Σ τ is ergodic in the discrete case. This result is new. But the continuous case Σ being ergodic is proved as a byproduct of the techniques we build here.
(ii) Proposition 2.10 and Theorem 2.11 hold also for Wiener process on a seperable Hilbert space where the Wiener measure was given in [4]. The proofs are exactly the same.

Ergodicity of canonical Markovian systems from periodic measures: necessary and sufficient conditions
Now we consider a Markovian cocycle random dynamical system Φ on a filtered dynamical system (Ω, F, P, (θ t ) t∈R , (F t s ) s≤t ), i.e. F t s ⊂ F, assuming for any s, t, u ∈ R, s ≤ t, θ −1 u F t s = F t+u s+u and for any t ∈ R + , Φ(t, ·) is measurable with respect to F t 0 . We also assume the random periodic path Y (s) is adapted, that is to say that for each s ∈ R, Y (s, ·) is measurable with respect to F s −∞ := ∨ r≤s F s r . Denote the transition probability of Markovian process Φ(t, ·)x on the polish space X with Borel σ-field B(X) by (c.f. [1], [5]) Denote L b (X) be the set of all real-valued bounded Borel measurable functions defined on X and P(X) be the set of all probability measures defined on (X, B(X)). For any t ≥ 0 and ρ ∈ P(X) we set and for any ϕ ∈ L b (X), define It is called a periodic measure with minimal period τ if τ > 0 is the smallest number such that (3.2) holds. It is called an invariant measure if it satisfies ρ s = ρ 0 for all s ∈ R, i.e. ρ 0 is an invariant measure for the Markovian semigroup P t if With a given Markovian semigroup P t , t ≥ 0, and an invariant measure ρ ∈ P(X), we will associate now, in the following unique way, a dynamical system (Ω * , F * , (θ * t ) t∈R , P µ ) on the space Ω * = X R of all X-valued functions.
Theorem 3.4. Assume the Markovian cocycle Φ : R + × Ω × X → X has an adapted random periodic path Y : R × Ω → X. Then the measure function ρ. : R → P(X) defined by which is the law of the random periodic path Y , is a periodic measure of the semigroup P t on (X, B(X)). Its time averageρ over a time interval of exactly one period defined bȳ is an invariant measure and satisfies that for any Γ ∈ B(X), t ∈ R, For these two sets, we consider Condition A. For any s ∈ R, Γ ∈ I τ s and P -almost all ω ∈ Ω, either L ω s ∩ Γ = ∅ or L ω s ⊆ Γ.

Ergodic canonical dynamical systems generated from periodic measure
Theorem 3.5. Assume that the random periodic path Y satisfies Condition A and the periodic measure ρ. : R → P(X) is given in Theorem 3.4. Then if the dynamical system (Ω, F, P, (θ n τ ) n≥1 ) is ergodic, the τ -periodic measure {ρ s } s∈R defined in (3.4) is PS-ergodic, and henceρ is ergodic.
Theorem 3.6. Assume Condition P. If the τ -periodic measure {ρ s } s∈R defined in (3.4) is PSergodic, then for any given s ∈ R, Γ ∈ I τ s , we have either for P -almost all ω, L ω s ∩ Γ = ∅ or for P -almost all ω, L ω s ⊆ Γ.
Corollary 3.8. If the dynamical system (Ω, F, P, (θ n τ ) n≥1 ) is ergodic, then Condition A and the τ -periodic measure {ρ s } s∈R being PS-ergodic are equivalent. In this case, the statement that P-a.s. either L ω s ∩ Γ = ∅ or L ω s ⊆ Γ and the statement that either P-a.s. L ω s ∩ Γ = ∅ or P-a.s. L ω s ⊆ Γ are equivalent.
Proof. This corollary can be easily obtained from Theorem 3.5, Theorem 3.6 and Remark 3.7.
3.2 Ergodicity of canonical dynamical system generated from invariant measure: sufficient condition Next we consider a stationary path Y : Ω → X of the Markovian cocycle Φ and the invariant measure µ defined in (2.17). Then we recall that the measure ρ ∈ P(X) defined by For these two sets, we consider Condition A ′ . For any Γ ∈ I and P -almost all ω ∈ Ω, L ω ∩ Γ = ∅ or L ω ⊆ Γ.
Theorem 3.9. Assume that the stationary path Y satisfies Condition A ′ and the invariant measure ρ ∈ P(X) given in (3.8). If the dynamical system (Ω, F, P, (θ t ) t∈R ) is ergodic, then ρ is ergodic.
To prove Theorem 3.9 we need the following lemma, which is of interests on its own right. We present it as a metric dynamical system. But it does not have to link with a random dynamical system. It is true for a continuous dynamical system of a probability space (measure space). Lemma 3.10. Assume that (Ω, F, P, (θ t ) t∈R ) is a dynamical system, then the following three statements are equivalent: Then it is easy to see that for all s ∈ R, Thus A ∞ is an invariant set. By the ergodicity assumption, we have For any T ∈ R, since we have But note as T → ∞, we have By continuity of measure, we have Thus the assertion (ii) is proved.
By the assertion (ii), we know that P (A) = 0 or 1.
Therefore the assertion (iii) is proved.
(iii) ⇒ (i). Assume A ∈ F and θ −1 t A = A for all t ∈ R. Then we have By assertion (iii), we have P (A) = 0 or 1.
Thus the assertion (i) is proved.
Next we will give the proof of Theorem 3.9.

Sublinear dynamical systems from periodic measures
In this section, we also assume Condition P. We will give a construction of upper expectations via the periodic measures µ. and ρ. defined in (2.6) and (3.4) respectively. Then we can study the ergodicity of the sublinear expectation dynamical system and sublinear canonical dynamical system generated by the upper expectations and Markov semigroup P t defined in (3.1).

Ergodic sublinear dynamical system on upper expectation space
Consider a sublinear expectationĒ on (Ω, L b (Ω)) defined bȳ where L b (Ω) := {ϕ :Ω → R|ϕ is measurable and bounded}, µ s is the periodic measure defined in (2.6), and Then by definition ofĒ, the periodic property of µ. and (4.2), we havē Recall the definition of ergodicity of a sublinear expectation dynamical system. We say a statement holdsĒ − q.s. if the statement is true on A c whereĒχ A = 0.
Proof. By (i) of Remark 2.7, we know that (Ω,F, µ s , (Θ n τ ) n≥1 ) is ergodic for each s ∈ R. For any ξ ∈ L 0 b (Ω), without any loss of generality, we can assume that ξ ≥ 0,Ē − q.s.. Let ξ τ : Applying Birkhoff's ergodic theorem (c.f. [5]) for ξ τ on (Ω,F , µ s , (Θ n τ ) n≥1 ), we have For any arbitrary T > 0, let n T = [ T τ ] be the maximal nonnegative integer less than or equal to T τ . Then n T τ ≤ T < (n T + 1)τ and However, Let A s be the µ s -null set such that From the Condition P and the assumption that the periodic measure {µ s } s∈R on the production space is PS-ergodic, we obtained the Birkhoff's law of large numbers with the convergence in the quasi-sure sense. This result is stronger than the Birkhoff ergodic type theorem in the almost sure sense that we can obtain from the ergodic theory of periodic measure ( [13]). This justifies the study of the construction of invariant sublinear expectations from periodic measures.
Next we also give two examples of ergodic sublinear dynamical system.
Proof. The proof is similar to that of Theorem 4.3. But in Theorem 4.3, we assumed that is a random periodic path. But its proof does not depend on this assumption. The key is µ s (Ã) = µ 0 (Ã) for any invariant setÃ. This identify for the current case is proved by a different method, see Proposition 2.8 and (2.16).

Ergodicity of sublinear canonical dynamical systems with respect to a Markovian semigroup
Next we will give the definitions of sublinear Markovian systems and their ergodicity ( [12], [21]).
Definition 4.7. ( [21]) We say that T : is a sublinear expectation defined on L b (X).
Here L b (X) is the B(X)-measurable real-valued function defined on X such that sup x∈X |φ(x)| < ∞.
It is clear that L 0 (F * ) is a linear subspace of L b (F * ).
Next we will consider the ergodic property of a sublinear expectationT on (X, L b (X)) defined byT where E ρs [ϕ] := X ϕ(x)ρ s (dx).
First we have the following property ofT .
Proposition 4.11. The sublinear expectationT is an invariant expectation under the Markovian semigroup P t , t ≥ 0.
Proof. By Theorem 3.25 in [12], we need to show that for any ϕ ∈ L b (X), if P t ϕ = ϕ (since P t is a linear operator, then P t (−ϕ) = −ϕ) and ϕ 2 has no mean uncertainty underT , then ϕ is constant,T -q.s. Firstly by Theorem 3.5, we know that {ρ s } s∈R is PS-ergodic. Then by P τ ϕ = ϕ, we have ϕ is constant, ρ s -a.s. for all s ∈ R. Let l s be that constant, i.e. ϕ = l s , ρ s -a.s.. Since P t ϕ = ϕ for all t ≥ 0 and by (4.4), we have