Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations
aa r X i v : . [ m a t h . P R ] A ug Random quasi-periodic paths and quasi-periodic measures ofstochastic differential equations
Chunrong Feng , Baoyou Qu , and Huaizhong Zhao Department of Mathematical Sciences, Loughborough University, LE11 3TU, UK Department of Mathematical Sciences, Shandong University, Jinan 250100, [email protected], [email protected], [email protected]
Abstract
In this paper, we define random quasi-periodic paths for random dynamical systemsand quasi-periodic measures for Markovian semigroups. We give a sufficient condition forthe existence and uniqueness of random quasi-periodic paths and quasi-periodic measuresfor stochastic differential equations and a sufficient condition for the density of the quasi-periodic measure to exist and to satisfy the Fokker-Planck equation. We obtain an invariantmeasure by considering lifted flow and semigroup on cylinder and the tightness of the averageof lifted quasi-periodic measures. We further prove that the invariant measure is unique.
Keywords: quasi-periodic measures, invariant measures, random dynamical systems, ran-dom quasi-periodic paths, Markovian random dynamical system, Markovian semigroup.
Quasi-periodic oscillation of a dynamical system is a motion given by a quasi-periodic function F such that F ( t ) = f ( t, t, · · · , t ) , (1.1)for some continuous function f ( t , t , · · · , t m ) , ( t , t , · · · , t m ) ∈ R m ( m ≥
2) which is periodicin t , t , · · · , t m with periods τ , τ , · · · , τ m respectively, where τ , τ , · · · , τ m are strictly positiveand their reciprocals are rationally linearly independent i.e. for any nonzero integer-valuedvector k = ( k , k , · · · , k m ), k τ + k τ + · · · + k m τ m = 0 . This topic has been subject to many important studies including Kolmogorov-Arnold-Moser(KAM) theory on Hamiltonian systems ([17],[19],[1]).Quasi-periodic motion is a common phenomenon in nature, e.g. arising in describing themovement of planets around the sun. The existence of a quasi-periodic motion for the nearlyintegrable regimes of the three-body problem with some transversality condition is given by theKAM theory. However many problems in nature are mixture of randomness and quasi-periodicmotions. For example the temperature process which is random has one year periodicity due to1he revolution of the earth around the sun and one day-night periodicity due to the rotation ofthe earth. Similarly, the energy demands should have similar nature. Thus to provide a rigorousmathematical theory is key in modelling random quasi-periodic phenomena in real world. Asfar as we know, such a concept still does not exist and the current paper is the first attempt inthis direction.The concepts of random periodic paths and periodic measures were introduced recently([22],[8],[9],[7],[10]). They are two different indispensable ways to describe random periodicity.The theory has led to progress in the study of bifurcations ([21]), random attractors ([2]),stochastic resonance ([6]), strange attractors ([12]) and modelling the El Nˆıno phenomenon([5]).In this paper, we study random quasi-periodicity of random dynamical systems or semi-flowsover a metric dynamical system (Ω , F , P, ( θ t ) t ∈ R ). First we define random quasi-periodic path ϕ of the stochastic-flows u ( t, s ) : Ω × R d → R d , t ≥ s as a random path satisfying u ( t, s, ϕ ( s, ω ) , ω ) = ϕ ( t, ω ) , t ≥ s, s ∈ R a.s. , and the pull-back random path t ϕ ( t, θ − t ω )is a quasi-periodic function for almost every sample path ω ∈ Ω.For a Markovian semi-flow, let p ( t, s, x, · ) , t ≥ s, be its transition probability. Then ameasure-valued function ρ : R → P ( R d ) is called a quasi-periodic measure if ρ is an entrancemeasure i.e. Z R d P ( t, s, x, Γ) ρ s ( dx ) = ρ t (Γ) for all Γ ∈ B ( R d ) , and the measure-valued map s ρ s is a quasi-periodic function.We will give a sufficient condition for the existence and uniqueness of random quasi-periodicpath for a stochastic differential equation on R d ( dX ( t ) = b ( t, X ( t )) dt + σ ( t, X ( t )) dW t , t ≥ s,X ( s ) = ξ, (1.2)where b, σ are quasi-periodic in the time variable t. As this is the first paper in this area, themain purpose here is to establish basic mathematical concepts and useful tools. We do not striketo technical details to try to provide best possible sufficient conditions in the current paper.We will prove the law of random quasi-periodic path is a quasi-periodic measure. We furthergive a sufficient condition for the density of the quasi-periodic measure to exist and to satisfythe Fokker-Planck equation.For simplicity, we only consider quasi-periodicity with two periods: τ and τ in the currentpaper. Our results also apply to general cases with any periods τ , τ , · · · , τ m without any extradifficulties. 2olving the reparameterised SDE is a key step in the analysis of finding random quasi-periodicpaths. Let ˜ b, ˜ σ be two functions such that˜ b ( t, t, x ) = b ( t, x ) , ˜ σ ( t, t, x ) = σ ( t, x )where ˜ b ( t , t , x ) , ˜ σ ( t , t , x ) are periodic in t , t with periods τ and τ respectively. Define˜ b r ,r ( t, x ) = ˜ b ( t + r , t + r , x )˜ σ r ,r ( t, x ) = ˜ σ ( t + r , t + r , x ) , then the solution K r ,r of SDE (1.2) when b, σ are replaced by ˜ b r ,r , ˜ σ r ,r , where r , r areregarded as parameters, satisfies K r,r ( t, s, x, ω ) = u ( t + r, s + r, x, θ − r ω )where u ( t, s, · , ω ) is the semi-flow generated by (1.2). Moreover we can prove under a dissipativecondition about the drifts b and ˜ b r ,r ,lim s →−∞ K r ,r ( t, s, x, ω ) = ϕ r ,r ( t, ω ) exists a.s.and ϕ ( r, ω ) = ϕ r,r (0 , θ − r ω )is a random quasi-periodic path of (1.2).Note the re-paramerterised SDE enjoys the following property: for all r , r , r ∈ R , t ≥ s , K r ,r ( t + r, s + r, x, θ − r ω ) = K r + r,r + r ( t, s, x, ω ) , P − a.s. on ω. (1.3)This is a very useful observation in our analysis, but the original time dependent SDE (1.2) doesnot have such a convenient relation.Lifting the semi-flow to ˜ X = [0 , τ ) × [0 , τ ) × R d is key to obtain an invariant measure fromthe quasi-periodic measure. Define˜Φ( t, ω )( s , s , x ) = ( t + s mod τ , t + s mod τ , K s ,s ( t, , x, ω ))and ˜ Y ( s, ω ) = ( s mod τ , s mod τ , ϕ ( s, ω )) . Then ˜ Y is a random quasi-periodic path of the cocycle ˜Φ. Moreover we will prove that˜ P ( t, ( s , s , x ) , ˜Γ) = P { ω : ˜Φ( t, ω )( s , s , x ) ∈ ˜Γ } , ˜Γ ∈ B ( ˜ X ) is Feller and˜ µ s (˜Γ) = P { ω : ˜ Y ( s, ω ) ∈ ˜Γ } = [ δ s mod τ × δ s mod τ × ρ s ](˜Γ)is a quasi-periodic measure with respect to ˜ P . We will show that { ¯˜ µ T = 1 T Z T ˜ µ s ds : T ∈ R + } is tight and a weak limit ¯˜ µ is an invariant measure with respect to ˜ P ∗ . Moreover, we will furthershow the invariant measure is unique and is given by the average1 τ τ Z τ Z τ δ s × δ s × ˜ ρ s ,s ds ds . Random path and entrance measure
In the stochastic differential equation (1.2), b : R × R d → R d , σ : R × R d → R d × d are continuousfunctions, W t is a two-sided R d -valued Brownian motion on probability space (Ω , F , P ) with W = 0 and W t − W s being a Gussian distribution N (0 , Σ( t − s )), Σ ∈ S d is a symmetricnondegenerate matrix, ξ is a R d -valued F s −∞ -measurable random variable, where F ba is thenatural filtration generated by ( W u − W v ) a ≤ u,v ≤ b . Now we consider the following assumptions. Condition 2.1.
There exists a constant α > such that for all t ∈ R , x, y ∈ R d (( x − y ) , ( b ( t, x ) − b ( t, y ))) ≤ − α ( x − y ) Condition 2.2.
The diffusion matrix σ ( t, x ) is continuous with respect to t and Lipschitz con-tinuous with Lipschitz constant β , i.e. for all t ∈ R , x, y ∈ R d k σ ( t, x ) − σ ( t, y ) k ≤ β | x − y | , and there exists M > such that sup t | b ( t, | + sup t k σ ( t, k ≤ M. Here k σ k = ( T r ( σ · σ T )) . Under Condition 2.1 and Condition 2.2, the solution of (1.2) exists, denoted by X ( t, s, ξ )satisfying for P − a.e. ω ∈ Ω X ( t, s, ξ ( ω ) , ω ) = X ( t, r, ω ) ◦ X ( r, s, ξ ( ω ) , ω ) , for all s ≤ r ≤ t. We call u : ∆ × R d × Ω → R d with u ( t, s, ω ) x = X ( t, s, x, ω ) a stochastic semi-flow, where∆ = { ( t, s ) : t ≥ s, t, s ∈ R } . Definition 2.3.
A random path of a semi-flow u : ∆ × R d × Ω → R d is a measurable map ϕ : R × Ω → R d such that for P-a.e. ω ∈ Ω u ( t + s, s, ϕ ( s, ω ) , ω ) = ϕ ( t + s, ω ) , for all t ≥ , s ∈ R . (2.1) In addition, if u is generated by an SDE, we say ϕ is a random path of this SDE. In the following, we will always use k · k to denote the norm in the L (Ω , dP ) space. Theorem 2.4.
Assume Conditions 2.1, 2.2 and α > β . Then there exists a unique uniformly L -bounded random path ϕ of SDE (1.2), i.e. sup t ∈ R k ϕ ( t ) k < ∞ . First we give some lemmas before we prove Theorem 2.4.
Lemma 2.5.
Assume Conditions 2.1, 2.2 and α > β . Let X s,ξt be the solution of SDE (1.2)with initial condition ( s, ξ ) , where ξ ∈ L (Ω) . Then there exists a constant C = C ( α, β, M ) suchthat for all t ≥ s , k X s,ξt k ≤ C (1 + k ξ k ) . roof. For any fixed λ , applying Itˆo’s formula to e λt | X s,ξt | , we have e λt | X s,ξt | = e λs | ξ | + Z ts (cid:16) λe λr | X s,ξr | + 2 e λr X s,ξr · b ( r, X s,ξr ) + e λr k σ ( r, X s,ξr ) k (cid:17) dr + Z ts e λr X s,ξr σ ( r, X s,ξr ) dW r . In Condition 2.1, let y = 0. Then for arbitrary ǫ >
0, by Young inequality and Condition 2.2 x · b ( t, x ) ≤ − α | x | + x · b ( t, ≤ − ( α − ǫ ) | x | + M ǫ , and k σ ( t, x ) k ≤ ( k σ ( t, x ) − σ ( t, k + k σ ( t, k ) ≤ ( β | x | + k σ ( t, k ) ≤ ( β + ǫ ) | x | + ( β ǫ + 1) M . We choose ǫ small enough such that α > β + 2 ǫ . Let λ = α − ǫ . Thus there exists a constant C ( α, β, M ) depending on α, β, M such that e α − ǫ ) t | X s,ξt | ≤ e α − ǫ ) s | ξ | + Z ts (cid:18) e α − ǫ ) r ( 12 ǫ + β ǫ + 1) M + e α − ǫ ) r ( β + ǫ ) | X s,ξr | (cid:19) dr + Z ts e α − ǫ ) r X s,ξr σ ( r, X s,ξr ) dW r ≤ e α − ǫ ) s | ξ | + C ( α, β, M ) e α − ǫ ) t + ( β + ǫ ) Z ts e α − ǫ ) r | X s,ξr | dr + Z ts e α − ǫ ) r X s,ξr σ ( r, X s,ξr ) dW r . Taking expectation of both sides, we have e α − ǫ ) t E | X s,ξt | ≤ e α − ǫ ) s E | ξ | + C ( α, β, M ) e α − ǫ ) t + ( β + ǫ ) Z ts e α − ǫ ) r E | X s,ξr | dr. By Gronwall inequality e α − ǫ ) t E | X s,ξt | ≤ e α − ǫ ) s E | ξ | + C ( α, β, M ) e α − ǫ ) t Z ts (cid:16) e α − ǫ ) s E | ξ | + C ( α, β, M ) e α − ǫ ) r (cid:17) ( β + ǫ ) exp (cid:18)Z tr ( β + ǫ ) dγ (cid:19) dr = e α − ǫ ) s E | ξ | + C ( α, β, M ) e α − ǫ ) t + Z ts e α − ǫ ) s E | ξ | ( β + ǫ ) e ( β + ǫ )( t − r ) dr + Z ts C ( α, β, M ) e α − ǫ ) r ( β + ǫ ) e ( β + ǫ )( t − r ) dr = e α − ǫ ) s E | ξ | + C ( α, β, M ) e α − ǫ ) t + e α − ǫ ) s E | ξ | (cid:16) e ( β + ǫ )( t − s ) − (cid:17) + C ( α, β, M ) e ( β + ǫ ) t β + ǫ α − β − ǫ (cid:16) e (2 α − β − ǫ ) t − e (2 α − β − ǫ ) s (cid:17) ≤ e α − ǫ ) s E | ξ | + C ( α, β, M ) e α − ǫ ) t + e α − ǫ ) s E | ξ | (cid:16) e ( β + ǫ )( t − s ) (cid:17) . C ( α, β, M ) is constant, which may be different from line to line. Then E | X s,ξt | ≤ e − α − ǫ )( t − s ) E | ξ | + C ( α, β, M ) + e − α − β − ǫ )( t − s ) E | ξ | ≤ C ( α, β, M )(1 + E | ξ | ) . Lemma 2.6.
Assume Conditions 2.1, 2.2 and α > β . Let X s,ξt and X s,ηt be two solutions ofSDE (1.2) with initial values ξ and η respectively, where ξ, η ∈ L (Ω) . Then k X s,ξt − X s,ηt k ≤ e − ( α − β / t − s ) k ξ − η k . Proof.
Note X s,ξt − X s,ηt = ξ − η + Z ts (cid:16) b ( r, X s,ξr ) − b ( r, X s,ηr ) (cid:17) dr + Z ts (cid:16) σ ( r, X s,ξr ) − σ ( r, X s,ηr ) (cid:17) dW r . Applying Itˆo’s formula to e αt | X s,ξt − X s,ηt | , by the dissipative Condition 2.1, we have e αt | X s,ξt − X s,ηt | = e αs | ξ − η | + Z ts h αe αr | X s,ξr − X s,ηr | + 2 e αr ( X s,ξr − X s,ηr ) · (cid:16) b ( r, X s,ξr ) − b ( r, X s,ηr ) (cid:17)i dr + Z ts e αr (cid:16) σ ( r, X s,ξr ) − σ ( r, X s,ηr ) (cid:17) dr + Z ts e αr ( X s,ξr − X s,ηr ) (cid:16) σ ( r, X s,ξr ) − σ ( r, X s,ηr ) (cid:17) dW r ≤ e αs | ξ − η | + Z ts β e αr | X s,ξr − X s,ηr | dr + Z ts e αr ( X s,ξr − X s,ηr ) (cid:16) σ ( r, X s,ξr ) − σ ( r, X s,ηr ) (cid:17) dW r . Taking expectation on both sides, we have e αt k X s,ξt − X s,ηt k ≤ e αs k ξ − η k + Z ts β e αr k X s,ξr − X s,ηr k dr. By Gronwall inequality, we have e αt k X s,ξt − X s,ηt k ≤ e αs k ξ − η k e β ( t − s ) , thus the lemma follows.Now we give the proof of Theorem 2.4 Proof of Theorem 2.4.
Existence: For any fixed ξ ∈ L (Ω). Let s < s < t , then X s ,ξt = X s ,X s ,ξs t . k X s ,ξt − X s ,ξt k . Applying Lemma 2.5 and Lemma 2.6, we have k X s ,ξt − X s ,ξt k = k X s ,X s ,ξs t − X s ,ξt k ≤ e − ( α − β / t − s ) k X s ,ξs − ξ k ≤ e − ( α − β / t − s ) (cid:16) k X s ,ξs k + k ξ k (cid:17) ≤ C ( α, β, ξ ) e − ( α − β / t − s ) . Thus there exists a L -limit of (cid:16) X s,ξt (cid:17) s ≤ t as s → −∞ . By Lemma 2.6, we know that this limitis independent of ξ . Define ϕ ( t ) := L − lim s →−∞ X s,ξt , then k ϕ ( t ) k ≤ lim sup s →−∞ k X s,ξt k ≤ C ( α, β, M, ξ ) ≤ C ( α, β, M ) . Next we will prove that ϕ is a random path of SDE (1.2). For any t ≥ s ≥ r , we have u ( t, s, X r,ξs ) = X r,ξt . By Lemma 2.6, we know that k u ( t, s, X r,ξs ) − u ( t, s, ϕ ( s )) k ≤ e − ( α − β / t − s ) k X r,ξs − ϕ ( s ) k . Itfollows that L − lim r →−∞ u ( t, s, X r,ξs ) = u ( t, s, ϕ ( s )) = ϕ ( t ) = L − lim r →−∞ X r,ξt , i.e. u ( t, s, ϕ ( s, ω ) , ω ) = ϕ ( t, ω ) , P − a.s. (2.2)Thus ϕ is a L -bounded random path of SDE (1.2).Uniqueness: If there are two uniformly L -bounded random path ϕ , ϕ of SDE (1.2), byLemma 2.6, we have k ϕ ( t ) − ϕ ( t ) k ≤ e − ( α − β / t − s ) k ϕ ( s ) − ϕ ( s ) k ≤ e − ( α − β / t − s ) (sup r ∈ R k ϕ ( r ) k + sup r ∈ R k ϕ ( r ) k ) → s → −∞ . Then ϕ ( t ) = ϕ ( t ) , for all t ∈ R , P − a.e. . Remark 2.7.
It is worth noticing that in the part of (2.2), the pathwise continuity of u ( t, s, · ) : R d → R d was not used. For a semi-flow u : △ × R d × Ω → R d with u ( t, s, x, ω ) = X s,xt ( ω ), we define the transition P : △ × R d × B ( R d ) → R + by P ( t, s, x, Γ) = P ( X s,xt ∈ Γ) for all t ≥ s , x ∈ R d and Γ ∈ B ( R d ).We further define P ∗ ( t, s ) : P ( R d ) → P ( R d ) by P ∗ ( t, s ) µ (Γ) = Z R d P ( t, s, x, Γ) µ ( dx ) , for all µ ∈ P ( R d ) , Γ ∈ B ( R d ) . (2.3)Here P ( R d ) := { all probability measures on ( R d , B ( R d )) } . efinition 2.8. We say a measure-valued map µ : R → P ( R d ) is an entrance measure ofSDE(1.2) if P ∗ ( t, s ) µ s = µ t for all t ≥ s, s ∈ R . Set M := { µ : R → P ( R d ) | sup t ∈ R Z R d x µ t ( dx ) < ∞} . Theorem 2.9.
Assume Conditions 2.1, 2.2 and α > β . Then there exists a unique entrancemeasure of SDE (1.2) in M . Before we prove Theorem 2.9, we need the following lemma.
Lemma 2.10.
Assume µ and µ are two probability measures on ( R d , B ( R d )) , and for anyopen set O we have µ ( O ) ≤ µ ( O ) . Then µ = µ .Proof. Let C := { all open sets on R d } . We know that µ ≤ µ on C .For any given O ∈ C , O c = R d \ O is a closed set. Define O cδ := { x : dist ( x, O c ) < δ } , where dist ( x, O c ) = inf y ∈O c | x − y | . Then we know that O cδ is open set and O cδ ↓ O c as δ ↓ µ ( O c ) = lim δ ↓ µ ( O cδ ) ≤ lim δ ↓ µ ( O cδ ) = µ ( O c ) . Since µ and µ are probability measures, we have1 − µ ( O ) ≤ − µ ( O ) , which implies µ ( O ) ≥ µ ( O ). Hence µ ≥ µ on C . This leads to µ = µ on C .Since C is a π -system and σ ( C ) = B ( R d ), thus µ = µ on B ( R d ).Now we give the proof of Theorem 2.9. Proof of Theorem 2.9.
Existence: Applying Theorem 2.4, we know that there exists a uniformly L -bounded random path ϕ of SDE (1.2). Let ρ t = L ( ϕ ( t )) be the law of ϕ ( t ). Since ϕ is therandom path of SDE (1.2), then for any Γ ∈ B ( R d ), we have P ∗ ( t, s ) ρ s (Γ) = Z R d P ( t, s, x, Γ) ρ s ( dx )= Z R d P ( X s,xt ∈ Γ) P ( ϕ ( s ) ∈ dx )= P ( X s,ϕ ( s ) t ∈ Γ)= P ( ϕ ( t ) ∈ Γ)= ρ t (Γ) . (2.4)Thus ρ is an entrance measure of SDE (1.2). And since ϕ is uniformly L -bounded, thensup t ∈ R Z R d x ρ t ( dx ) = sup t ∈ R E | ϕ ( t ) | < ∞ , ρ ∈ M .Uniqueness: We aim to prove that for any entrance measure µ of SDE (1.2) in M , µ t = ρ t for all t ∈ R . By Lemma 2.10, we just need to prove ρ t ( O ) ≤ µ t ( O ) for any open set O ⊂ R d .Since for any s < t , we have ρ t ( O ) − µ t ( O ) = ρ t ( O ) − Z R d P ( t, s, x, O ) µ s ( dx )= Z R d ( ρ t ( O ) − P ( X s,xt ∈ O )) µ s ( dx )= Z R d ( P ( ϕ ( t ) ∈ O ) − P ( X s,xt ∈ O )) µ s ( dx ) . Define O δ := { x : dist ( x, O c ) > δ } . Then O δ ↑ O as δ ↓ P ( X s,xt ∈ O ) = P ( X s,xt − ϕ ( t ) + ϕ ( t ) ∈ O ) ≥ P ( ϕ ( t ) ∈ O δ , | X s,xt − ϕ ( t ) | < δ ) ≥ P ( ϕ ( t ) ∈ O δ ) − P ( | X s,xt − ϕ ( t ) | ≥ δ ) . Thus it turns out from the above and the Chebyshev inequality that P ( ϕ ( t ) ∈ O ) − P ( X s,xt ∈ O ) ≤ P ( ϕ ( t ) ∈ O \ O δ ) + P ( | X s,xt − ϕ ( t ) | ≥ δ ) ≤ P ( ϕ ( t ) ∈ O \ O δ ) + 1 δ E | X s,xt − ϕ ( t ) | . Applying Lemma 2.5 and Lemma 2.6, we have E | X s,xt − ϕ ( t ) | = lim r →−∞ E | X s,xt − X r,xt | ≤ lim sup r →−∞ ,r s < t , we have ρ t ( O ) − µ t ( O ) = Z R d ( P ( ϕ ( t ) ∈ O ) − P ( X s,xt ∈ O )) µ s ( dx ) ≤ Z R d (cid:18) P ( ϕ ( t ) ∈ O \ O δ ) + 1 δ E | X s,xt − ϕ ( t ) | (cid:19) µ s ( dx ) ≤ P ( ϕ ( t ) ∈ O \ O δ ) + Cδ e − α − β / t − s ) Z R d (1 + x ) µ s ( dx ) . δ >
0, we have ρ t ( O ) − µ t ( O ) ≤ P ( ϕ ( t ) ∈ O \ O δ ) + lim sup s →−∞ Cδ (cid:18) r ∈ R Z R d x µ r ( dx ) (cid:19) e − α − β / t − s ) ≤ P ( ϕ ( t ) ∈ O \ O δ ) = ρ t ( O \ O δ ) . Since O δ ↑ O as δ ↓
0, we have ρ t ( O ) − µ t ( O ) ≤ lim δ ↓ ρ t ( O \ O δ ) = 0 , which implies ρ t ( O ) ≤ µ t ( O ).By the proof of Theorem 2.4, we know that ϕ ( t ) = L − lim s →−∞ X s,xt . Then we have thefollowing proposition. We use C b ( R d ) to be the linear space of all continuous and boundedfunctions on R d . Proposition 2.11.
The entrance measure ρ t is the limit of P ( t, s, x, · ) in P ( R d ) with weaktopology, i.e. for all f ∈ C b ( R d ) , we have lim s →−∞ Z R d f ( y ) P ( t, s, x, dy ) = Z R d f ( y ) ρ t ( dy ) . Proof.
Since R R d f ( y ) P ( t, s, x, dy ) = E f ( X s,xt ) and R R d f ( y ) ρ t ( dy ) = E f ( ϕ ( t )), we need to provethat for all f ∈ C b ( R d ), lim s →−∞ E f ( X s,xt ) = E f ( ϕ ( t )) . First we prove lim sup s →−∞ E f ( X s,xt ) ≤ E f ( ϕ ( t )). Otherwise there exists a sequence s n ↓ −∞ as n → ∞ and a constant λ = lim sup s →−∞ E f ( X s,xt ) > E f ( ϕ ( t )) such that lim n →∞ E f ( X s n ,xt ) = λ .Since lim n →∞ E [ | X s n ,xt − ϕ ( t ) | ] = 0, we know that there exists a subsequence { s n k } ⊆ { s n } suchthat X s nk ,xt a.s. −−→ ϕ ( t ) as k → ∞ . Thus f ( X s nk ,xt ) a.s. −−→ f ( ϕ ( t )). Then by Lebesgue’s dominatedconvergence theorem, we have lim k →∞ E f ( X s nk ,xt ) = E f ( ϕ ( t )) , which contradicts that lim k →∞ E f ( X s nk ,xt ) = λ > E f ( ϕ ( t )) . Hence lim sup s →−∞ E f ( X s,xt ) ≤ E f ( ϕ ( t )) . Similarly we can also prove that lim inf s →−∞ E f ( X s,xt ) ≥ E f ( ϕ ( t )) , which completes our proof. 10 Random quasi-periodic path, quasi-periodic measure and in-varant measure
In SDE (1.2), if we assume the coefficients b, σ are quasi-periodic functions in time t , can weobtain a kind of random quasi-periodic path? What should the “quasi-periodicity” of a randompath be defined? We give the following definition. Definition 3.1.
A measurable path ϕ : R × Ω → R d is called random quasi-periodic pathof periods τ , τ of a semi-flow u , where the reciprocals of τ and τ are rationally linearlyindependent, if it is a random path, i.e. u ( t, s, ϕ ( s, ω ) , ω ) = ϕ ( t, ω ) for t ≥ s , and there exists ˜ ϕ : R × R × Ω → R d such that ˜ ϕ ( t, t, ω ) = ϕ ( t, θ − t ω ) and ˜ ϕ ( t + τ , s, ω ) = ˜ ϕ ( t, s, ω ) , ˜ ϕ ( t, s + τ , ω ) = ˜ ϕ ( t, s, ω ) . (3.1) We also say ϕ is a random quasi-periodic path of an SDE if u is generated by this SDE. We give the quasi-periodic condition.
Condition 3.2.
Assume that b, σ in SDE (1.2) are quasi-periodic functions with periods τ , τ ,where the reciprocals of τ and τ are rationally linearly independent, which means there exists ˜ b : R × R × R d → R d and ˜ σ : R × R × R d → R d × n such that ˜ b ( t, t, x ) = b ( t, x ) , ˜ σ ( t, t, x ) = σ ( t, x ) for all t ∈ R , x ∈ R d satisfying ˜ b ( t + τ , s, x ) = ˜ b ( t, s, x ) , ˜ b ( t, s + τ , x ) = ˜ b ( t, s, x ) , (3.2) and ˜ σ ( t + τ , s, x ) = ˜ σ ( t, s, x ) , ˜ σ ( t, s + τ , x ) = ˜ σ ( t, s, x ) . (3.3) Condition 3.3.
Assume ˜ b, ˜ σ in Condition 3.2 satisfy the following conditions: (1) ( x − y ) (cid:16) ˜ b ( t, s, x ) − ˜ b ( t, s, y ) (cid:17) ≤ − α ( x − y ) for all x, y ∈ R d and t, s ∈ R ; (2) k ˜ σ ( t, s, x ) − ˜ σ ( t, s, y ) k ≤ β | x − y | for all x, y ∈ R d and t, s ∈ R ; (3) sup t,s ∈ R | ˜ b ( t, s, | + sup t,s ∈ R k ˜ σ ( t, s, k < ∞ . Conditions 3.2, 3.3 imply Conditions 2.1, 2.2. Now we give the following main theorem.
Theorem 3.4.
Assume Conditions 3.2, 3.3 and α > β . Then there exists a unique uniformly L -bounded random quasi-periodic path of SDE (1.2).Proof. Uniqueness: Applying Theorem 2.4, we know that if there exists a uniformly L -boundedrandom quasi-periodic path, it must be the random path ϕ defined in Theorem 2.4. So unique-ness holds.Existence: The solution of SDE (1.2) u ( t, s, x, ω ) can be written as u ( t, s, x, ω ) = x + Z ts b ( r, u ( r, s, x, ω )) dr + ( ω ) Z ts σ ( r, u ( r, s, x, · )) dW r . u ( t + r, s + r, x, ω ) = x + Z t + rs + r b ( v, u ( v, s + r, x, ω )) dv + ( ω ) Z t + rs + r σ ( v, u ( v, s + r, x, · )) dW v = x + Z ts b ( v + r, u ( v + r, s + r, x, ω )) dv + ( ω ) Z ts σ ( v + r, u ( v + r, s + r, x, · )) d ˜ W rv . (3.4)Here ˜ W rv = θ r W v . Replacing ω by θ − r ω in equation (3.4), we have u ( t + r, s + r, x, θ − r ω ) = x + Z ts b ( v + r, u ( v + r, s + r, x, θ − r ω )) dv + ( θ − r ω ) Z ts σ ( v + r, u ( v + r, s + r, x, · )) d ˜ W rv = x + Z ts b ( v + r, u ( v + r, s + r, x, θ − r ω )) dv + ( ω ) Z ts σ ( v + r, u ( v + r, s + r, x, θ − r · )) dW v , then u ( t + r, s + r, x, θ − r ω ) = x + Z ts ˜ b ( v + r, v + r, u ( v + r, s + r, x, θ − r ω )) dv + ( ω ) Z ts ˜ σ ( v + r, v + r, u ( v + r, s + r, x, θ − r · )) dW v . (3.5)Denote u r ( t, s, x, ω ) := u ( t + r, s + r, x, θ − r ω ), ˜ b r ,r ( t, x ) := ˜ b ( t + r , t + r , x ) and ˜ σ r ,r ( t, x ) :=˜ σ ( t + r , t + r , x ), then equation (3.5) can be written as u r ( t, s, x, ω ) = x + Z ts ˜ b r,r ( v, u r ( v, s, x, ω )) dv + ( ω ) Z ts ˜ σ r,r ( v, u r ( v, s, x, · )) dW v . (3.6)Since Condition 3.3 holds, then for all r , r ∈ R , ˜ b r ,r and ˜ σ r ,r satisfy( x − y ) (cid:16) ˜ b r ,r ( t, x ) − ˜ b r ,r ( t, y ) (cid:17) ≤ − α ( x − y ) , and k ˜ σ r ,r ( t, x ) − ˜ σ r ,r ( t, y ) k ≤ β | x − y | , for all t ∈ R , x, y ∈ R d . Thus the following equation K r ,r ( t, s, x, ω ) = x + Z ts ˜ b r ,r ( v, K r ,r ( v, s, x, ω )) dv + ( ω ) Z ts ˜ σ r ,r ( v, K r ,r ( v, s, x, · )) dW v , (3.7)12as a unique solution, denoted by K r ,r ( t, s, x, ω ). Since α > β , similar to the proof of Theorem2.4, we know there exist ϕ r ( t ) , ϕ r ,r ( t ) such that ( ϕ r ( t ) = L − lim s →−∞ u r ( t, s, x ) ϕ r ,r ( t ) = L − lim s →−∞ K r ,r ( t, s, x ) , (3.8)Compairing (3.6) and (3.7), obviously we know that for all r, t ∈ R K r,r = u r a.s. (3.9)and thus ϕ r,r ( t ) = ϕ r ( t ) a.s. for all r, t ∈ R . By quasi-periodicity of ˜ b and ˜ σ , we know that˜ b r + τ ,r = ˜ b r ,r = ˜ b r ,r + τ , ˜ σ r + τ ,r = ˜ σ r ,r = ˜ σ r ,r + τ . Thus it turns out that almost surely K r + τ ,r ( t, s, x ) = K r ,r ( t, s, x ) = K r ,r + τ ( t, s, x ) , for all t ≥ s, s ∈ R , then almost surely ϕ r + τ ,r ( t ) = ϕ r ,r ( t ) = ϕ r ,r + τ ( t ) , for all t ∈ R . Let ϕ be the unique random path of SDE (1.2), by Theorem 2.4 we have ϕ r ( t, ω ) = ϕ ( t + r, θ − r ω ) . Let ˜ ϕ ( r , r , ω ) := ϕ r ,r (0 , ω ), then˜ ϕ ( r + τ , r , ω ) = ˜ ϕ ( r , r , ω ) = ˜ ϕ ( r , r + τ , ω ) , and ˜ ϕ ( r, r, ω ) = ϕ r,r (0 , ω ) = ϕ r (0 , ω ) = ϕ ( r, θ − r ω ) . Therefore, the unique random path ϕ can be written as ϕ ( t, ω ) = ˜ ϕ ( t, t, θ t ω ) , for a.e. ω ∈ Ω , where ˜ ϕ ( t + τ , s, ω ) = ˜ ϕ ( t, s, ω ), ˜ ϕ ( t, s + τ , ω ) = ˜ ϕ ( t, s, ω ), for a.e. ω ∈ Ω. This means ϕ is arandom quasi-periodic path. Remark 3.5.
We can conduct similar operations as in (3.4) and (3.5) to re-parameterisedequation (3.7). Noticing ˜ b r ,r ( v + r, · ) = ˜ b r + r,r + r ( v, · ) , ˜ σ r ,r ( v + r, · ) = ˜ σ r + r,r + r ( v, · ) (3.10) and using the same argument as in the proof of (3.9), we can conclude important property (1.3).This property is similar to the shift property of the autonomous stochastic differential equationswhich leads to their cocycle property with a perfection argument. Though there is nothing similarto be said about the original SDEs due to the time dependency of the coefficients, this propertyholds due to ”time-invariance” of the re-parameterised coefficients in the sense of (3.10). .2 Existence and uniqueness of quasi-periodic measure First we give the definition of the quasi-periodic probability measure as follows.
Definition 3.6.
We say a map ρ : R → P ( R d ) is a quasi-periodic probability measure of periods τ , τ of SDE (1.2), where the reciprocals of τ and τ are rationally linearly independent, if P ∗ ( t, s ) ρ s = ρ t for all t ≥ s , and there exists ˜ ρ : R × R → P ( R d ) with ˜ ρ t,t = ρ t such that ˜ ρ t + τ ,s = ˜ ρ t,s , ˜ ρ t,s + τ = ˜ ρ t,s , (3.11) for all t, s ∈ R . Theorem 3.7.
Assume Conditions 3.2, 3.3 and α > β . Then there exists a unique quasi-periodic probability measure of periods τ , τ of SDE (1.2) in M .Proof. Uniqueness: Applying the proof of Theorem 2.9, we know that if there exists a quasi-periodic probability measure with periods τ , τ of SDE (1.2) in M , it must be the uniqueentrance measure of SDE (1.2) defined by the law of the random path.Existence: by Theorem 3.4, we know that SDE (1.2) has a uniformly L -bounded randomquasi-periodic path ϕ : R × Ω → R d with periods τ and τ , i.e. there exists a ˜ ϕ : R × R × Ω → R d such that ϕ ( t, θ − t · ) = ˜ ϕ ( t, t, · ) and ˜ ϕ ( t + τ , s ) = ˜ ϕ ( t, s ) = ˜ ϕ ( t, s + τ ) for all t, s ∈ R . Let ρ t = L ( ϕ ( t )) , ˜ ρ t,s = L ( ˜ ϕ ( t, s )) (3.12)be the laws of ϕ ( t ) and ˜ ϕ ( t, s ) respectively. Since ϕ is the random path of SDE (1.2), then byequation (2.4) we have P ∗ ( t, s ) ρ s = ρ t for all t ≥ s . Since θ − t preserves probability measure P ,then ρ t = L ( ϕ ( t )) = L ( ˜ ϕ ( t, t )) = ˜ ρ t,t . By the construction of ˜ ρ , we have˜ ρ t + τ ,s = L ( ˜ ϕ ( t + τ , s )) = L ( ˜ ϕ ( t, s )) = ˜ ρ t,s and ˜ ρ t,s + τ = L ( ˜ ϕ ( t, s + τ )) = L ( ˜ ϕ ( t, s )) = ˜ ρ t,s . Also since ϕ is uniformly L -bounded, thensup t ∈ R Z R d x ρ t ( dx ) = sup t ∈ R E | ϕ ( t ) | < ∞ , which means ρ ∈ M . Example 3.8 (Ornstein-Uhlenbeck equation) . We include the following example with a numberof reasons. First, O-U process is one of the simplest stochastic process that one would analysefor new concepts. Second, it is instructive and does illustrate clearly the idea of random quasi-periodicity and two kinds of formulations as well as their relation. Third, the formulae for itsrandom quasi-periodic path and quasi-periodic measure can be written down explicitly. Last,but not least, this equation is relevant in various different applications e.g. modelling energyconsumptions or temperature variants with two obvious daily and seasonal periodicities.The Ornstein-Uhlenbeck process with mean reversion of single-period was used in modellingelectricity prices ([4],[18]), daily temperature ([3]), biological neurons ([14]) etc. The quasi-periodic O-U process we introduce here allows a feature of multiple periods which is natural in any real world situations e.g energy consumptions, temperature, business cycles, economicscycles. While it is not the purpose of this paper to study these interesting applied problems intheir specific contexts, our work in this paper provides a mathematical theory of random quasi-periodicity for this purpose.Here we consider the following mean reversion multidimensional Ornstein-Uhlenbeck equa-tion on R d dX t = ( S ( t ) − AX t ) dt + σ ( t ) dW t (3.13) where S ( t ) , σ ( t ) are deterministic quasi-periodic functions with periods τ , τ and A ∈ S d with A > , which means that A is a symmetrical matrix with positive eigenvalues { λ n } dn =1 . Theanalysis is given as follows.Applying It ˆ o ’s formula to e tA X t , we have X t = e − ( t − s ) A X s + Z ts e − ( t − r ) A S ( r ) dr + Z ts e − ( t − r ) A σ ( r ) dW r t ≥ s. (3.14) Let ϕ ( t ) := Z t −∞ e − ( t − r ) A S ( r ) dr + Z t −∞ e − ( t − r ) A σ ( r ) dW r . (3.15) Then we have ϕ ( t ) = e − ( t − s ) A ϕ ( s ) + Z ts e − ( t − r ) A S ( r ) dr + Z ts e − ( t − r ) A σ ( r ) dW r , (3.16) which means that ϕ is a random path of SDE (3.13). Next we will show that ϕ is also a randomquasi-periodic path. We first rewrite ϕ ( t ) by ϕ ( t, ω ) = Z t −∞ e − ( t − r ) A S ( r ) dr + (cid:20)Z t −∞ e − ( t − r ) A σ ( r ) dW r (cid:21) ( ω )= Z −∞ e rA S ( r + t ) dr + (cid:20)Z −∞ e rA σ ( r + t ) dW r (cid:21) ( θ t ω ) . (3.17) Since
S, σ are quasi-periodic functions with periods τ , τ , then there exist ˜ S, ˜ σ such that S ( t ) =˜ S ( t, t ) and σ ( t ) = ˜ σ ( t, t ) and ( ˜ S ( t + τ , s ) = ˜ S ( t, s ) = ˜ S ( t, s + τ )˜ σ ( t + τ , s ) = ˜ σ ( t, s ) = ˜ σ ( t, s + τ ) . (3.18) Then we have ϕ ( t, ω ) = Z −∞ e rA ˜ S ( r + t, r + t ) dr + (cid:20)Z −∞ e rA ˜ σ ( r + t, r + t ) dW r (cid:21) ( θ t ω ) . Let ˜ ϕ ( t, s, ω ) = Z −∞ e rA ˜ S ( r + t, r + s ) dr + (cid:20)Z −∞ e rA ˜ σ ( r + t, r + s ) dW r (cid:21) ( ω ) . (3.19)15 hen we have ϕ ( t, θ − t ω ) = ˜ ϕ ( t, t, ω ) and ˜ ϕ ( t + τ , s, ω ) = ˜ ϕ ( t, s, ω ) , ˜ ϕ ( t, s + τ , ω ) = ˜ ϕ ( t, s, ω ) , (3.20) which shows that ϕ is a random quasi-periodic path of SDE (3.13) with periods τ , τ .Let ρ t = L ( ϕ ( t )) . By Theorem 3.7, we know that ρ t is the unique quasi-periodic probabilitymeasure with periods τ , τ of SDE (3.13). Moreover, from (3.15), we know that ρ t ( · ) = N (cid:18)Z t −∞ e − ( t − r ) A S ( r ) dr, Z t −∞ e − ( t − r ) A σ ( r ) σ ( r ) T e − ( t − r ) A dr (cid:19) ( · ) , where N is the multivariate normal distribution. Let ˜ ρ t,s = L ( ˜ ϕ ( t, s )) . Then from (3.19), weknow that ˜ ρ t,s ( · ) = N (cid:18)Z −∞ e rA ˜ S ( r + t, r + s ) dr, Z −∞ e rA (˜ σ ˜ σ T )( r + t, r + s ) e rA dr (cid:19) ( · ) . It is obvious that ρ t = ˜ ρ t,t .In Subsection 3.3, we will develop a way to lift a quasi-periodic stochastic flow to the cylinder [0 , τ ) × [0 , τ ) × R d and prove ˜ µ t,s = δ t × δ s × ˜ ρ t,s is a quasi-periodic measure. This setup willenable us to prove that the average τ τ R τ R τ ˜ µ t,s dtds is an invariant measure on the cylinder.Our result also implies that for this particular case, it is the unique invariant measure for thelifted quasi-periodic Ornstein-Uhlenbeck process. In Section 3.1, we have the existence and uniqueness of random quasi-periodic path, and inthis case, we will lift the semi-flow u and obtain an invariant measure. Consider the cylinder˜ X = [0 , τ ) × [0 , τ ) × R d with the following metric d (˜ x, ˜ y ) = d ( t , s ) + d ( t , s ) + | x − y | , for all ˜ x = ( t , t , x ) , ˜ y = ( s , s , y ) ∈ ˜ X , where d , d are the metrics on [0 , τ ) , [0 , τ ) defined by d i ( t i , s i ) = min( | t i − s i | , τ i − | t i − s i | ) , for all t i , s i ∈ [0 , τ i ) , i = 1 , . Denote by B ( ˜ X ) the Borel measurable set on ˜ X deduced by metric d . Then we have the followinglemma. Lemma 3.9.
Assume Conditions 3.2, 3.3 and α > β . We lift the semi-flow u : △× R d × Ω → R d to a random dynamical system on a cylinder ˜ X = [0 , τ ) × [0 , τ ) × R d by the following: ˜Φ( t, ω )( s , s , x ) = ( t + s mod τ , t + s mod τ , K s ,s ( t, , x, ω )) , where K r ,r is the solution of (3.7). Then ˜Φ : R + × ˜ X × Ω → ˜ X is a cocycle on ˜ X over themetric dynamical system (Ω , F , P, ( θ t ) t ∈ R ) .Moreover, assume ϕ : R × Ω → R d is a random path of the semi-flow u. Then ˜ Y : R × Ω → ˜ X defined by ˜ Y ( s, ω ) = ( s mod τ , s mod τ , ϕ ( s, ω )) is a random path of the cocycle ˜Φ on ˜ X . roof. We first prove that ˜Φ is a cocycle on ˜ X . Note K r ,r is periodic in r , r with periods τ , τ . It follows that for any ( s , s , x ) ∈ ˜ X , t, s ∈ R + , we have˜Φ( t, θ s ω ) ◦ ˜Φ( s, ω )( s , s , x )= ˜Φ( t, θ s ω )( s + s mod τ , s + s mod τ , K s ,s ( s, , x, ω ))= ( t + s + s mod τ , t + s + s mod τ , K s + s ,s + s ( t, , K s ,s ( s, , x, ω ) , θ s ω )) . Now we compute the K s + s ,s + s ( t, , K s ,s ( s, , x, ω ) , θ s ω ) term. By equation (3.7), we knowthat K s ,s ( s, , x, ω ) (3.21)= x + Z s ˜ b s ,s ( v, K s ,s ( v, , x, ω )) dv + ( ω ) Z s ˜ σ s ,s ( v, K s ,s ( v, , x, · )) dW v = x + Z s b ( v + s , v + s , K s ,s ( v, , x, ω )) dv +( ω ) Z s σ ( v + s , v + s , K s ,s ( v, , x, · )) dW v , and K s + s ,s + s ( t, , K s ,s ( s, , x, ω ) , θ s ω ) (3.22)= K s ,s ( s, , x, ω ) + Z t b ( v + s + s , v + s + s , K s + s ,s + s ( v, , K s ,s ( s, , x, ω ) , θ s ω )) dv +( θ s ω ) Z t σ ( v + s + s , v + s + s , K s + s ,s + s ( v, , K s ,s ( s, , x, θ − s · ) , · )) dW v = x + Z s b ( v + s , v + s , K s ,s ( v, , x, ω )) dv +( ω ) Z s σ ( v + s , v + s , K s ,s ( v, , x, · )) dW v + Z t + ss b ( v + s , v + s , K s + s ,s + s ( v − s, , K s ,s ( s, , x, ω ) , θ s ω )) dv +( ω ) Z t + ss σ ( v + s , v + s , K s + s ,s + s ( v − s, , K s ,s ( s, , x, · ) , θ s · )) dW v P − a.e. on ω. Let Q ( r, , x, ω ) = ( K s ,s ( r, , x, ω ) , ≤ r ≤ s,K s + s ,s + s ( r − s, , K s ,s ( s, , x, ω ) , θ s ω ) , s < r ≤ t + s. (3.23)From equation (3.21) and (3.22), we know that for all 0 ≤ r ≤ t + s , Q ( r, , x, ω ) solves thefollowing equation Q ( r, , x, ω ) = x + Z r ˜ b s ,s ( v, Q ( v, , x, ω )) dv + ( ω ) Z r ˜ σ s ,s ( v, Q ( v, , x, · )) dW v . which implies Q ( r, , x, ω ) = K s ,s ( r, , x, ω ) P − a.s. on ω for all 0 ≤ r ≤ t + s . In particular, K s + s ,s + s ( t, , K s ,s ( s, , x, ω ) , θ s ω ) = Q ( t + s, , x, ω ) = K s ,s ( t + s, , x, ω ) . (3.24)17ence ˜Φ( t, θ s ω ) ◦ ˜Φ( s, ω )( s , s , x )=( t + s + s mod τ , t + s + s mod τ , K s + s ,s + s ( t, , K s ,s ( s, , x, ω ) , θ s ω ))=( t + s + s mod τ , t + s + s mod τ , K s ,s ( t + s, , x, ω ))= ˜Φ( t + s, ω )( s , s , x ) , which implies the cocycle property of ˜Φ.Next, since ϕ is a random path of the semi-flow u and ˜ Y ( s, ω ) = ( s mod τ , s mod τ , ϕ ( s, ω )),then ˜Φ( t, θ s ω ) ˜ Y ( s, ω ) =( t + s mod τ , t + s mod τ , K s,s ( t, , ϕ ( s, ω ) , θ s ω ))=( t + s mod τ , t + s mod τ , u s ( t, , ϕ ( s, ω ) , θ s ω ))=( t + s mod τ , t + s mod τ , u ( t + s, s, ϕ ( s, ω ) , θ − s θ s ω ))=( t + s mod τ , t + s mod τ , ϕ ( t + s, ω ))= ˜ Y ( t + s, ω ) , which means ˜ Y is a random path of the cocycle ˜Φ on ˜ X .Consider the Markovian transition ˜ P : R + × ˜ X × B ( ˜ X ) → [0 ,
1] generated by the cocycle ˜Φ,i.e., ˜ P ( t, ( s , s , x ) , ˜Γ) = P ( ω : ˜Φ( t, ω )( s , s , x ) ∈ ˜Γ) , for all t ∈ R + , ( s , s , x ) ∈ ˜ X , ˜Γ ∈ B ( ˜ X ). Similarly, for any ˜ µ ∈ P ( ˜ X ), we define˜ P ∗ t ˜ µ (˜Γ) = Z ˜ X ˜ P ( t, ( s , s , x ) , ˜Γ)˜ µ ( ds × ds × dx ) . Then we have the following theorem.
Theorem 3.10. If ρ : R → P ( R d ) is the entrance measure of semi-group P ∗ , i.e. P ∗ ( t, s ) ρ s = ρ t , then ˜ µ : R → P ( ˜ X ) defined by ˜ µ t = δ t mod τ × δ t mod τ × ρ t is an entrance measure of semi-group ˜ P ∗ , i.e., ˜ P ∗ t ˜ µ s = ˜ µ t + s . Moreover, ˜ µ is also a quasi-periodic measure. roof. For any ˜Γ ∈ B ( ˜ X ), let ˜Γ s := { x ∈ R d | ( s mod τ , s mod τ , x ) ∈ ˜Γ } . Then we have˜ P ∗ t ˜ µ s (˜Γ) = Z ˜ X ˜ P ( t, ( s , s , x ) , ˜Γ)˜ µ s ( ds × ds × dx )= Z R d ˜ P ( t, ( s mod τ , s mod τ , x ) , ˜Γ) ρ s ( dx )= Z R d P ( ω : ˜Φ( t, ω )( s mod τ , s mod τ , x ) ∈ ˜Γ) ρ s ( dx )= Z R d P ( ω : ( t + s mod τ , t + s mod τ , u s ( t, , x, ω )) ∈ ˜Γ) ρ s ( dx )= Z R d P ( ω : u ( t + s, s, x, θ − s ω ) ∈ ˜Γ t + s ) ρ s ( dx )= Z R d P ( ω : u ( t + s, s, x, ω ) ∈ ˜Γ t + s ) ρ s ( dx )= Z R d P ( t + s, s, x, ˜Γ t + s ) ρ s ( dx )= P ∗ ( t + s, s ) ρ s (˜Γ t + s )= ρ t + s (˜Γ t + s ) = ˜ µ t + s (˜Γ) . Moreover, let ˆ µ s ,s = δ s mod τ × δ s mod τ × ˜ ρ s ,s , (3.25)we know that ˜ µ s = ˆ µ s ,s and ˆ µ s + τ ,s = ˆ µ s ,s , ˆ µ s ,s + τ = ˆ µ s ,s , (3.26)which completes our proof.For the above entrance measure ˜ µ , set¯˜ µ T := 1 T Z T ˜ µ s ds and M := { ¯˜ µ T : T ∈ R + } . (3.27)We have the following lemma. Lemma 3.11.
Assume Conditions 3.2, 3.3 and α > β . Then M is tight, and hence is weaklycompact in P ( ˜ X ) .Proof. We just need to prove that for any ǫ >
0, there exists a compact set ˜Γ ǫ ∈ B ( ˜ X ) such thatfor all T ∈ R + , we have ¯˜ µ T (˜Γ ǫ ) > − ǫ. ρ t is the law of the L -bounded random path ϕ ( t ), then { ρ t : t ∈ R } is tight because ρ t ( B N (0)) = P ( | ϕ ( t ) | < N )= 1 − P ( | ϕ ( t ) | ≥ N ) ≥ − k ϕ ( t ) k N ≥ − sup t ∈ R k ϕ ( t ) k N . (3.28)Then for the given ǫ >
0, there exists a compact set Γ ǫ ⊂ R d such that for all t ∈ R , ρ t (Γ ǫ ) > − ǫ. It is well-known that [0 , τ ) , [0 , τ ) are both homeomorphic to the circle S under metrics d , d respectively. Hence they are compact and ˜Γ ǫ = [0 , τ ) × [0 , τ ) × Γ ǫ is compact on ˜ X . Moreover¯˜ µ T (˜Γ ǫ ) = 1 T Z T ˜ µ s (˜Γ ǫ ) ds = 1 T Z T ρ s (Γ ǫ ) ds > − ǫ, (3.29)which completes our proof.For any f ∈ C ( ˜ X ), which is defined as the collection of B ( ˜ X ) measurable functions, wedefine ˜ P t f (˜ x ) = Z ˜ X ˜ P ( t, ˜ x, d ˜ y ) f (˜ y ) , for any ˜ x ∈ ˜ X . (3.30)We have the following Feller property of the semi-group ˜ P t , t ≥ Proposition 3.12.
Assume Conditions 3.2, 3.3 and α > β . In addition, we assume that ˜ b ( t, s, x ) , ˜ σ ( t, s, x ) are continuous with respect to ( t, s ) uniformly in x . Then the semi-group ˜ P t , t ≥ , defined by (3.30), is Feller, i.e. for all f ∈ C b ( ˜ X ) , ˜ P t f ∈ C b ( ˜ X ) .Proof. Obviously k ˜ P t f k ∞ ≤ k f k ∞ , then we just need to prove that ˜ P t f is continuous. It issufficient to prove that for any sequence ˜ x n = ( r n , r n , x n ) , ˜ x = ( r , r , x ) ∈ ˜ X with ˜ x n n →∞ −−−→ ˜ x ,we have ˜ P t f (˜ x n ) n →∞ −−−→ ˜ P t f (˜ x ). Since˜ P t f (˜ x ) = Z [0 ,τ ) × [0 ,τ ) × R d ˜ P ( t, ( r , r , x ) , ds × ds × dy ) f ( s , s , y )= Z [0 ,τ ) × [0 ,τ ) × R d P ( ˜Φ( t, · )( r , r , x ) ∈ ds × ds × dy ) f ( s , s , y )= Z R d P ( K r ,r ( t, , x ) ∈ dy ) f ( t + r mod τ , t + r mod τ , y )= E f ( t + r mod τ , t + r mod τ , K r ,r ( t, , x )) . (3.31)Let f t ( r , r , x ) := f ( t + r mod τ , t + r mod τ , x ). Then we have | ˜ P t f (˜ x n ) − ˜ P t f (˜ x ) | = | E f t ( r n , r n , K r n ,r n ( t, , x n )) − E f t ( r , r , K r ,r ( t, , x )) |≤| E f t ( r n , r n , K r n ,r n ( t, , x n )) − E f t ( r n , r n , K r ,r ( t, , x )) | + | E f t ( r n , r n , K r ,r ( t, , x )) − E f t ( r , r , K r ,r ( t, , x )) | =: A n + A n . (3.32)20ince f ∈ C b ( ˜ X ), then f t ∈ C b ( ˜ X ) and f t ( r n , r n , K r ,r ( t, , x )) a.s. −−→ f t ( r , r , K r ,r ( t, , x )) as n → ∞ . By Lebesgue’s dominated convergence theorem, we have A n = | E f t ( r n , r n , K r ,r ( t, , x )) − E f t ( r , r , K r ,r ( t, , x )) | n →∞ −−−→ . (3.33)Furthermore, by the uniformly continuous of ˜ b, ˜ σ , we know that ˜ b r n ,r n uniformly −−−−−−→ n →∞ ˜ b r ,r and˜ σ r n ,r n uniformly −−−−−−→ n →∞ ˜ σ r ,r . Let b n = sup t ∈ R ,x ∈ R d | ˜ b r n ,r n ( t, x ) − ˜ b r ,r ( t, x ) | and σ n = sup t ∈ R ,x ∈ R d | ˜ σ r n ,r n ( t, x ) − ˜ σ r ,r ( t, x ) | . Then lim n →∞ b n = lim n →∞ σ n = 0. Set K n ( t ) = K r n ,r n ( t, , x n ) and K ( t ) = K r ,r ( t, , x ).Applying Itˆo formula to | K n ( t ) − K ( t ) | , we have | K n ( t ) − K ( t ) | = | x n − x | + Z t K n ( s ) − K ( s ))(˜ b r n ,r n ( s, K n ( s )) − ˜ b r ,r ( s, K ( s ))) ds + Z t k ˜ σ r n ,r n ( s, K n ( s )) − ˜ σ r ,r ( s, K ( s )) k ds + Z t K n ( s ) − K ( s ))(˜ σ r n ,r n ( s, K n ( s )) − ˜ σ r ,r ( s, K ( s ))) dW s . (3.34)Note 2( K n ( s ) − K ( s ))(˜ b r n ,r n ( s, K n ( s )) − ˜ b r ,r ( s, K ( s )))= 2( K n ( s ) − K ( s ))(˜ b r n ,r n ( s, K n ( s )) − ˜ b r ,r ( s, K n ( s )))+2( K n ( s ) − K ( s ))(˜ b r ,r ( s, K n ( s )) − ˜ b r ,r ( s, K ( s ))) ≤ b n | K n ( s ) − K ( s ) | − α | K n ( s ) − K ( s ) | ≤ b n λ + λ | K n ( s ) − K ( s ) | − α | K n ( s ) − K ( s ) | , (3.35)and k ˜ σ r n ,r n ( s, K n ( s )) − ˜ σ r ,r ( s, K ( s )) k = k ˜ σ r n ,r n ( s, K n ( s )) − ˜ σ r ,r ( s, K n ( s )) + ˜ σ r ,r ( s, K n ( s )) − ˜ σ r ,r ( s, K ( s )) k ≤ ( k ˜ σ r n ,r n ( s, K n ( s )) − ˜ σ r ,r ( s, K n ( s )) k + k ˜ σ r ,r ( s, K n ( s )) − ˜ σ r ,r ( s, K ( s )) k ) ≤ ( σ n + β | K n ( s ) − K ( s ) | ) ≤ σ n (1 + β λ ) + ( λ + β ) | K n ( s ) − K ( s ) | , (3.36)where λ > α − β > λ . Comparing with (3.35) and (3.36), we takeexpectation both side on (3.34) to have k K n ( t ) − K ( t ) k ≤ | x n − x | + b n λ + σ n (1 + β λ ) n →∞ −−−→ . K r n ,r n ( t, , x n ) L −−−→ n →∞ K r ,r ( t, , x ). Let R N = { ω : | K r ,r ( t, , x, ω ) | ≤ N } and R nN = { ω : | K r n ,r n ( t, , x n , ω ) | ≤ N } . Then by Chebyshev inequality we have lim N →∞ (inf n ∈ N P ( R nN ∩ R N )) = 1. Since f is continuous,then it is uniformly continuous on all compact subset of ˜ X . Then for arbitrary ǫ >
0, thereexists δ ǫN > t , t , x ) , ( s , s , y ) ∈ [0 , τ ) × [0 , τ ) × B N (0), where B N (0) isa ball centered at 0 with radius N in R d , and d ( t , s ) + d ( t , s ) + | x − y | < δ ǫN , we have | f (( t , t , x )) − f (( s , s , y )) | < ǫ . Set C nδ ǫN = { ω : | K r n ,r n ( t, , x n ) − K r ,r ( t, , x ) | < δ ǫN } . Then also by Chebyshev inequality lim n →∞ P ( C nδ ǫN ) = 1. Hence for all ω ∈ C nδ ǫN ∩ R nN ∩ R N , | f t ( r n , r n , K r n ,r n ( t, , x n )) − f t ( r n , r n , K r ,r ( t, , x )) | < ǫ. Thereforelim sup n →∞ A n = lim sup n →∞ | E f t ( r n , r n , K r n ,r n ( t, , x n )) − E f t ( r n , r n , K r ,r ( t, , x )) |≤ ǫ + 2 k f k ∞ lim sup n →∞ [(1 − P ( C nδ ǫN )) + (1 − P ( R nN ∩ R N ))]= ǫ. (3.37)Since ǫ > A n n →∞ −−−→
0. We complete the proof of ˜ P t f (˜ x n ) n →∞ −−−→ ˜ P t f (˜ x ).From Lemma 3.11 and Proposition 3.12, we have the existence of invariant measure under˜ P ∗ . Theorem 3.13.
Assume Conditions 3.2, 3.3 and α > β . In addition, we assume that ˜ b ( t, s, x ) , ˜ σ ( t, s, x ) are continuous with respect to ( t, s ) uniformly in x . Then there exists auniqueness invariant probability measure with respect to the semi-group ˜ P ∗ which is given by τ τ Z τ Z τ δ s × δ s × ˜ ρ s ,s ds ds . Proof.
Existence: From Lemma 3.11, we know that M defined by (3.27) is tight and henceweakly compact. This means that there exists a sequence { T n } n ≥ with T n ↑ ∞ as n → ∞ anda probability measure ¯˜ µ ∈ P ( ˜ X ) such that ¯˜ µ T n w −→ ¯˜ µ . Moreover, for any fixed t >
0, since˜ P ∗ t ¯˜ µ T n − ¯˜ µ T n = 1 T n Z T n ˜ P ∗ t ˜ µ s ds − T n Z T n ˜ µ s ds = 1 T n Z T n ˜ µ t + s ds − T n Z T n ˜ µ s ds = 1 T n Z t + T n t ˜ µ s ds − T n Z T n ˜ µ s ds = 1 T n Z t + T n T n ˜ µ s ds − T n Z t ˜ µ s ds, (3.38)22o lim sup n →∞ k ˜ P ∗ t ¯˜ µ T n − ¯˜ µ T n k BV ≤ lim sup n →∞ T n ( Z t k ˜ µ s k BV ds + Z T n + tT n k ˜ µ s k BV ds ) ≤ lim sup n →∞ tT n = 0 . Hence ˜ P ∗ t ¯˜ µ T n w −→ ¯˜ µ . On the other hand, for any f ∈ C b ( ˜ X ), by Proposition 3.12, we have˜ P t f ∈ C b ( ˜ X ), and thereforelim n →∞ Z ˜ X f (˜ y ) ˜ P ∗ t ¯˜ µ T n ( d ˜ y ) = lim n →∞ Z ˜ X Z ˜ X f (˜ y ) ˜ P ( t, ˜ x, d ˜ y )¯˜ µ T n ( d ˜ x )= lim n →∞ Z ˜ X ˜ P t f (˜ x )¯˜ µ T n ( d ˜ x )= Z ˜ X ˜ P t f (˜ x )¯˜ µ ( d ˜ x )= Z ˜ X f (˜ y ) ˜ P ∗ t ¯˜ µ ( d ˜ y ) . (3.39)This means ˜ P ∗ t ¯˜ µ T n w −→ ˜ P ∗ t ¯˜ µ . Summarizing above we have that ˜ P ∗ t ¯˜ µ = ¯˜ µ .Moreover, using the same method as in the proof of Proposition 3.12, we know that thequasi-periodic path ˜ ϕ of SDE (1.2) is continuous under L norm, i.e.lim ( t,s ) → ( t ,s ) k ˜ ϕ ( t, s ) − ˜ ϕ ( t , s ) k = 0 . Then similar to the proof of Proposition 2.11, we know that ˜ ρ is continuous under the weaktopology in P ( R d ), i.e. for all f ∈ C b ( R d ),lim ( t,s ) → ( t ,s ) Z R d f ( x )˜ ρ t,s ( dx ) = Z R d f ( x )˜ ρ t ,s ( dx ) . Let ˆ µ defined by (3.25). It is easy to check that ˆ µ is also continuous under the weak topologyin P ( ˜ X ). Since τ and τ are rationally linearly independent, by definition 5.1 in [20], T t :[0 , τ ) × [0 , τ ) → [0 , τ ) × [0 , τ ) defined by T t ( s , s ) = ( t + s mod τ , t + s mod τ ) , for all s , s ∈ [0 , τ ) × [0 , τ )is a minimal ratation. Then applying Theorem 6.20 in [20], we know that τ τ L is a uniqueergodic probability measure on [0 , τ ) × [0 , τ ), where L present the Lebesgue measures. Henceby Birkhoff’s ergodic theory,¯˜ µ T = 1 T Z T ˜ µ t dt = 1 T Z T ˆ µ T t (0 , dt T →∞ −−−−→ Z [0 ,τ ) × [0 ,τ ) ˆ µ s ,s τ τ ds ds .
23o ¯˜ µ = Z [0 ,τ ) × [0 ,τ ) ˆ µ s ,s τ τ ds ds = 1 τ τ Z τ Z τ δ s × δ s × ˜ ρ s ,s ds ds is an invariant measure with respect to ˜ P ∗ . Uniqueness: We need to prove that for any invariant probability measure υ , we have υ = ¯˜ µ .By Lemma 2.10, we only need to prove that for any open set ˜ O ∈ B ( ˜ X ), we have υ ( ˜ O ) ≥ ¯˜ µ ( ˜ O ).Define ˜ O r ,r = { x ∈ R d : ( r mod τ , r mod τ , x ) ∈ ˜ O} , ˜ O r ,r δ = { x : dist ( x, ( ˜ O r ,r ) c ) > δ } , and ˜ O δ = [ ( s ,s ) ∈ [0 ,τ ) × [0 ,τ ) ( s , s ) × ˜ O s ,s δ . We know that ˜ O r ,r , ˜ O r ,r δ and ˜ O δ are open sets, ˜ O r ,r δ ↑ ˜ O r ,r and ˜ O δ ↑ ˜ O as δ ↓
0. Then υ (cid:16) ˜ O (cid:17) = lim T →∞ T Z T ˜ P ∗ t υ (cid:16) ˜ O (cid:17) dt = lim T →∞ T Z T Z ˜ X ˜ P (cid:16) t, ( s , s , x ) , ˜ O (cid:17) υ ( d ˜ x ) dt = lim T →∞ Z ˜ X T Z T P (cid:16) K s ,s ( t, , x, · ) ∈ ˜ O t + s ,t + s (cid:17) dtυ ( d ˜ x ) . (3.40)Applying Remark 3.5 and measure preserving transformation θ t , it follows that υ (cid:16) ˜ O (cid:17) = lim T →∞ Z ˜ X T Z T P (cid:16) K t + s ,t + s (0 , − t, x, · ) ∈ ˜ O t + s ,t + s (cid:17) dtυ ( d ˜ x ) . Similar to the proof of Theorem 2.4, Lemma 2.5 and Lemma 2.6, it can be shown that thesolution K r ,r of (3.7) has the following estimate k K r ,r ( t, s, x ) − ˜ ϕ r ,r ( t ) k ≤ Ce − ( α − β / t − s ) , for all r , r ∈ R , t ≥ s , where C = C ( α, β, ˜ M ) only depends on α, β, ˜ M with ˜ M = sup t,s ∈ R ( | ˜ b ( t, s, | + k ˜ σ ( t, s, k ) . Then for all δ >
0, by Chebyshev’s inequality, we have P (cid:16) K t + s ,t + s (0 , − t, x, · ) ∈ ˜ O t + s ,t + s (cid:17) ≥ P (cid:16) ˜ ϕ t + s ,t + s (0) ∈ ˜ O t + s ,t + s δ , | K t + s ,t + s (0 , − t, x ) − ˜ ϕ t + s ,t + s (0) | < δ (cid:17) ≥ P (cid:16) ˜ ϕ t + s ,t + s (0) ∈ ˜ O t + s ,t + s δ (cid:17) − P (cid:0) | K t + s ,t + s (0 , − t, x ) − ˜ ϕ t + s ,t + s (0) | ≥ δ (cid:1) ≥ ˜ ρ t + s ,t + s (cid:16) ˜ O t + s ,t + s δ (cid:17) − C δ e − α − β / t =ˆ µ t + s ,t + s (cid:16) ˜ O δ (cid:17) − C δ e − α − β / t . (3.41)24hus it turns out from (3.40), (3.41) and Fatou’s Lemma that υ (cid:16) ˜ O (cid:17) ≥ lim inf T →∞ Z ˜ X T Z T (cid:18) ˆ µ t + s ,t + s (cid:16) ˜ O δ (cid:17) − C δ e − α − β / t (cid:19) dtυ ( d ˜ x ) ≥ Z ˜ X (cid:18) lim inf T →∞ T Z T ˆ µ t + s ,t + s (cid:16) ˜ O δ (cid:17) dt − lim T →∞ C δ ( α − β / T (cid:19) υ ( d ˜ x ) ≥ Z ˜ X (cid:18) lim inf T →∞ T Z T ˆ µ t + s ,t + s (cid:16) ˜ O δ (cid:17) dt (cid:19) υ ( d ˜ x ) . (3.42)Again by Birkhoff’s ergodic theory, we know that for all ( s , s ) ∈ R T Z T ˆ µ t + s ,t + s dt T →∞ −−−−→ ¯˜ µ. Then since O δ is open, and by Proposition 2.4 in [13], we have υ (cid:16) ˜ O (cid:17) ≥ ¯˜ µ (cid:16) O δ (cid:17) . Since O δ ↑ O as δ ↓
0, the desired result follows from the continuity of measures with respect toan increasing sequence of sets.
Remark 3.14.
It is not obvious how to check directly that τ τ R τ R τ δ s × δ s × ˜ ρ s ,s ds ds is an invariant measure with respect to ˜ P ∗ without appealing to the tightness argurement. By a similar proof of Lemma 3.11, Proposition 3.12 and Theorem 3.13, it is not difficult toderive a general theorem. Here we denote by X a metric space, B ( X ) the Borel σ -algebra on X , B b ( X ) the linear space of all B ( X )-bounded measurable functions and P ( X ) the collectionof all probability measures on ( X , B ( X )). Assume that P ( t, x, Γ) , t ≥ , x ∈ X , Γ ∈ B ( X ), is aMarkovian transition function on X . Denote by P t , t ≥ B b ( X ) → B b ( X ) and P ∗ t , t ≥ P ( X ) → P ( X ), the Markovian semi-groups associated with P ( t, x, · ). We say ρ : R → P ( X )is an entrance measure with respect to P ∗ if P ∗ t ρ s = ρ t + s for all t ∈ R + , s ∈ R . We say ρ isquasi-periodic if exists a measure-valued function ˜ ρ s ,s satisfying the same relation with ρ s asin Definition 3.6. However we do not have the uniqueness of invariant measure in the generalcase. Theorem 3.15.
Assume the entrance measure ρ with respect to P ∗ t , t ≥ , is a quasi-periodicmeasure with periods τ and τ , where the reciprocals of τ and τ are rationally linearly inde-pendent. If { ¯ ρ T = T R T ρ s ds : T ∈ R + } is tight and the Markovian semi-group P t , t ≥ , isFeller, then there exists one invariant measure given by τ τ Z τ Z τ ˜ ρ s ,s ds ds . In this section, we will give a sufficient condition to guarantee the existence of the density of theentrance measure. We need an extra condition.25 ondition 4.1.
Assume that b, σ in SDE (1.2) satisfy the following conditions: (1) σ is invertible and sup t ∈ R k σ − ( t, x ) k < ∞ ; (2) b ( t, x ) is continuous with respect to t, x . We now give the definition of the well-known BMO space and some lemmas which will usedin this section.
Definition 4.2.
Denote by BMO(s,t) the space of all ( F rs ) s ≤ r ≤ t -adapted R d -valued process M with k M k BMO ( s,t ) := sup T ∈T ts (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) E (cid:20)Z tT | M r | dr |F Ts (cid:21)(cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ < ∞ , where s < t and T ts is the set of stopping times taking their values in [ s, t ] . Then we have the following lemma.
Lemma 4.3.
Let M ∈ BM O ( s, t ) . Then there exists p > such that E (cid:20)(cid:18) E (cid:18)Z ts M r dW r (cid:19)(cid:19) p (cid:21) < ∞ , where E (cid:16)R ts M r dW r (cid:17) := exp { R ts M r dW r − R ts | M r | dr } .Proof. By Theorem 3.1 in [16], we know that if k M k BMO ( s,t ) ≤ Φ( p ) for some p >
1, where Φis a continuous monotone function from (1 , ∞ ) to R + with Φ(1+) = ∞ and Φ( ∞ ) = 0, then E (cid:16)R ts M r dW r (cid:17) is in L p .We also need the following lemma which is almost the same as Lemma 4.1 in [11]. Lemma 4.4.
Assume Conditions 2.1, 2.2 and 4.1 hold. Let X s,xt be the solution of SDE (1.2)and Z s,xt be the solution of the following SDE ( dZ t = σ ( t, Z t ) dW t , t ≥ s,Z s = x ∈ R d . (4.1) Then the laws of X s,xt and Z s,xt are equivalent, i.e. P X s,xt ( B ) = ˜ P Z s,xt ( B ) , for all B ∈ B ( R d ) , where d ˜ PdP = E (cid:16)R ts σ − ( r, Z s,xr ) b ( r, Z s,xr ) dW r (cid:17) Proof.
This lemma can be proved by almost the same proof as them of Lemma 4.1 in [11].Now we have the following theorem. 26 heorem 4.5.
Assume Conditions 2.1, 2.2 and 4.1 hold. If α > β , then P ( t, s, x, · ) andthe entrance measure ρ t are absolutely continuous with respect to the Lebesgue measure L on ( R d , B ( R d )) , and hence have the density p ( t, s, x, y ) and q ( t, y ) respectively.Proof. First we prove that P ( t, s, x, · ) is absolutely continuous with respect to L , i.e. for anyΓ ∈ B ( R d ), L (Γ) = 0 implies P ( t, s, x, Γ) = P ( X s,xt ∈ Γ) = 0. By Lemma 4.4, we know that P ( X s,xt ∈ Γ) = ˜ P ( Z s,xt ∈ Γ) = E ˜ P [1 Γ ( Z s,xt )]= E (cid:20) E (cid:18)Z ts σ − ( r, Z s,xr ) b ( r, Z s,xr ) dW r (cid:19) Γ ( Z s,xt ) (cid:21) , (4.2)where Z s,xt is the solution of SDE (4.1). Set T n := inf t ≥ s {| Z s,xt | ≥ n } . Since E ˜ P [sup r ∈ [ s,t ] | Z s,xr | ] < ∞ , then we have ˜ P ( T n > t ) = ˜ P ( sup r ∈ [ s,t ] | Z s,xr | ≤ n ) → n → ∞ . Thus P ( X s,xt ∈ Γ) = E ˜ P [1 Γ ( Z s,xt )]= E ˜ P [1 Γ ( Z s,xt )1 [ s,T n ] ( t )] + E ˜ P [1 Γ ( Z s,xt )1 ( T n , ∞ ) ( t )] ≤ lim n →∞ [ E ˜ P [1 Γ ( Z s,xt )1 [ s,T n ] ( t )] + ˜ P ( T n < t )]= lim n →∞ E ˜ P [1 Γ ( Z s,xt )1 [ s,T n ] ( t )]= lim n →∞ E (cid:20) [ s,T n ] ( t ) E (cid:18)Z ts σ − ( r, Z s,xr ) b ( r, Z s,xr ) dW r (cid:19) Γ ( Z s,xt ) (cid:21) . (4.3)Since1 [ s,T n ] ( t ) E (cid:18)Z ts σ − ( r, Z s,xr ) b ( r, Z s,xr ) dW r (cid:19) ≤ E (cid:18)Z ts [ s,T n ] ( r ) σ − ( r, Z s,xr ) b ( r, Z s,xr ) dW r (cid:19) , we have P ( X s,xt ∈ Γ) ≤ lim inf n →∞ E (cid:20) E (cid:18)Z ts [ s,T n ] ( r ) σ − ( r, Z s,xr ) b ( r, Z s,xr ) dW r (cid:19) Γ ( Z s,xt ) (cid:21) . (4.4)We only need to prove that if L (Γ) = 0, then for all n E (cid:20) E (cid:18)Z ts [ s,T n ] ( r ) σ − ( r, Z s,xr ) b ( r, Z s,xr ) dW r (cid:19) Γ ( Z s,xt ) (cid:21) = 0 . Let a n ( r ) = 1 [ s,T n ] ( r ) σ − ( r, Z s,xr ) b ( r, Z s,xr ). By Condition 4.1, we know that there exists C > r ∈ R | a n ( r ) | ≤ C . Thensup T ∈T ts (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) E (cid:20)Z tT | a n ( r ) | dr |F Ts (cid:21)(cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ≤ C √ t − s, which means a n ∈ BM O ( s, t ). By Lemma 4.3, there exists p > γ n := (cid:18) E (cid:20)(cid:18) E (cid:18)Z ts a n ( r ) dW r (cid:19)(cid:19) p (cid:21)(cid:19) p < ∞ . Z s,xt = x + R ts σ ( r, Z s,xr ) dW r , note that R ts σ ( r, Z s,xr ) dW r is in law a Brownian motionwith time ˆ σ t = R ts k σ ( r, Z s,xr ) k dr , i.e. there exists a standard Brownian motion ˜ W such that R ts σ ( r, Z s,xr ) dW r d = ˜ W ˆ σ t . Also notice √ d = k σ ( t, x ) σ − ( t, x ) k ≤ k σ ( t, x ) kk σ − ( t, x ) k , thus k σ ( t, x ) k ≥ √ d k σ − ( t, x ) k ≥ √ d sup t ∈ R ,x ∈ R d k σ − ( t, x ) k =: σ, which suggests that ˆ σ t ≥ σ ( t − s ). Using Proposition 6.17 in Chapter 2 in [15], we have E [1 Γ ( Z s,xt )] = E h Γ ( x + ˜ W ˆ σ t ) i = E h E h Γ ( x + ˜ W ˆ σ t ) |F ˆ σ t − σ ( t − s ) ii = E h E h Γ ( x + y + ˜ W σ ( t − s ) ) i (cid:12)(cid:12)(cid:12) y = ˜ W ˆ σt − σ ( t − s ) i . (4.5)Note E h Γ ( x + y + ˜ W σ ( t − s ) ) i = 1(2 πσ ( t − s )) d/ | det Σ | / Z R d Γ ( x + y + z ) e − (1 / σ ( t − s )) | Σ − / z | dz ≤ πσ ( t − s )) d/ | det Σ | / L (Γ) , where W ∼ N (0 , Σ). Then E [1 Γ ( Z s,xt )] ≤ πσ ( t − s )) d/ | det Σ | / L (Γ) . Let q be the dual number of p . Then by Cauchy-Schwarz inequality, E (cid:20) E (cid:18)Z ts [ s,T n ] ( r ) σ − ( r, Z s,xr ) b ( r, Z s,xr ) dW r (cid:19) Γ ( Z s,xt ) (cid:21) ≤ γ n { E [1 Γ ( Z s,xt )] } q ≤ C n · L (Γ) q , (4.6)where C n = γ n · (cid:16) πσ ( t − s )) d/ | det Σ | / (cid:17) q .So if L (Γ) = 0, then E h E (cid:16)R ts [ s,T n ] ( r ) σ − ( r, Z s,xr ) b ( r, Z s,xr ) dW r (cid:17) Γ ( Z s,xt ) i = 0, and hence P ( t, s, x, Γ) = P ( X s,xt ∈ Γ) = 0. Thus P ( t, s, x, · ) is absolutely continuous with respect to theLebesgue measure and by Radon-Nikodym theorem, the density of P ( t, s, x, · ) with respect tothe Lebesgue measure exists.For the entrance measure ρ t , since ρ t (Γ) = P ∗ ( t, s ) ρ s (Γ) = Z R d P ( t, s, x, Γ) ρ s ( dx ) , (4.7)then if L (Γ) = 0, we have ρ t (Γ) = 0. This also suggests that ρ t is absolutely continuous withrespect to L and thus its density exists. 28e already know the conditions to guarantee the existence of the density p ( t, s, x, y ) and q ( t, y ) of the two- parameter Markov transition kernel P ( t, s, x, · ) and entrance measure ρ t re-spectively. By Fubini theorem, we know that ρ t (Γ) = Z R d P ( t, s, x, Γ) ρ s ( dx ) = Z Γ Z R d p ( t, s, x, y ) ρ s ( dx ) dy = Z Γ Z R d p ( t, s, x, y ) q ( s, x )( dx ) dy. Then it is obvious that q ( t, y ) = Z R d p ( t, s, x, y ) q ( s, x )( dx ) . (4.8)Moreover it is well-known that p ( · , s, x, · ) satisfies the following Fokker-Planck equation ∂ t p ( t, s, x, y ) = L ∗ ( t ) p ( t, s, x, y ) , (4.9)where L ∗ ( t ) p is the Fokker Planck operator given by L ∗ ( t ) p = − d X i =1 ∂ x i ( b i ( t, y ) p ) + 12 d X i,j =1 ∂ x i x j (cid:0) σσ Tij ( t, y ) p (cid:1) . (4.10)Now we have the following theorem. Theorem 4.6.
Assume the same assumptions as in Theorem 4.5. Let q ∈ C , ( R × R d ) T L ( R d ) with k q ( t, · ) k L ( R d ) = 1 for all t , and define ρ : R → P ( R d ) by ρ t (Γ) = Z Γ q ( t, y ) dy, for all t ∈ R . Then ρ is an entrance measure if and only if ∂ t q = L ∗ ( t ) q. (4.11) Hence the solution of (4.11) and the entrance measure have one to one correspondence.Proof.
Assume first that ρ is an entrance measure. We already know that p, q satisfy (4.8) and p ( t, s, x, y ) satisfies Fokker-Planck equation (4.9). We take the derivative with respect to t onboth side of (4.8) to have ∂ t q ( t, x ) = Z R d ∂ t p ( t, s, y, x ) q ( s, y ) dy = Z R d L ∗ ( t ) p ( t, s, y, x ) q ( s, y ) dy = Z R d − d X i =1 ∂ x i ( b i ( t, x ) p ( t, s, y, x )) q ( s, y ) dy + Z R d d X i,j =1 ∂ x i x j (cid:0) σσ Tij ( t, x ) p ( t, s, y, x ) (cid:1) q ( s, y ) dy =: I + II. (4.12)29or the first part, we have I = − d X i =1 Z R d [ ∂ x i ( b i ( t, x )) p ( t, s, y, x ) + b i ( t, x ) ∂ x i ( p ( t, s, y, x ))] q ( s, y ) dy = − d X i =1 ∂ x i ( b i ( t, x )) Z R d p ( t, s, y, x ) q ( s, y ) dy − d X i =1 b i ( t, x ) ∂ x i Z R d p ( t, s, y, x ) q ( s, y ) dy = − d X i =1 ∂ x i ( b i ( t, x )) q ( t, x ) − d X i =1 b i ( t, x ) ∂ x i q ( t, x )= − d X i =1 ∂ x i ( b i ( t, x ) q ( t, x )) . (4.13)Similarly, for the second part, we have II = 12 d X i,j =1 ∂ x i x j (cid:0) σσ Tij ( t, x ) q ( t, x ) (cid:1) . Hence the density function q ( t, x ) of entrance measure ρ t satisfies ∂ t q = L ∗ ( t ) q. Conversely, if q is the solution of (4.9), then by Fubini’s theorem, we have for all Γ ∈ B ( R d ) P ∗ ( t, s ) ρ s (Γ) = Z R d P ( t, s, y, Γ) ρ s ( dy )= Z R d Z Γ p ( t, s, y, x ) dxq ( s, y ) dy = Z Γ Z R d p ( t, s, y, x ) q ( s, y ) dydx = Z Γ q ( t, x ) dx = ρ t (Γ)which means ρ is a entrance measure. Remark 4.7.
Since the entrance measure is unique under the assumption in Theorem 4.6, weknow that the solution of the following Fokker-Planck equation ( ∂ t q = L ∗ ( t ) qq (0 , · ) ∈ C ( R d ) T L ( R d ) , k q (0 , · ) k L ( R d ) = 1 . (4.14) is unique. Now assume that u r ( t, s, x ) and K r ,r ( t, s, x ) are the solutions of equation (3.6) and (3.7)respectively, and the corresponding semi-groups P r , P r ,r defined as ( P r ( t, s, x, Γ) := P ( u r ( t, s, x ) ∈ Γ) P r .r ( t, s, x, Γ) := P ( K r ,r ( t, s, x ) ∈ Γ) . (4.15)30e can also define P t, ∗ ( t, s ) ( resp. P r ,r , ∗ ( t, s )) as in (2.3) when we replace { P ∗ ( t, s ) , P ( t, s, x, Γ) } by { P r, ∗ ( t, s ) , P r ( t, s, x, Γ) } ( resp. { P r ,r , ∗ ( t, s ) , P r .r ( t, s, x, Γ) } ). Let ϕ r ( t ) , ϕ r ,r ( t ) be de-fined as in (3.8), and ρ rt , ρ r ,r t be the laws of ϕ r ( t ) , ϕ r ,r ( t ) respectively. Then we have P r, ∗ ( t, s ) ρ rs = ρ rt , P r ,r , ∗ ( t, s ) ρ r ,r s = ρ r ,r t . Similar to Condition 4.1, we give the following condition.
Condition 4.8.
Assume that ˜ b, ˜ σ in Condition 3.2 satisfy the following conditions: (1) ˜ σ is invertible and sup t,s ∈ R k ˜ σ − ( t, s, x ) k < ∞ ; (2) ˜ b ( t, s, x ) is continuous with respect to t, s, x . Then by Theorem 2.9 and Theorem 4.5, we can directly deduce the following theorem
Theorem 4.9.
Assume Conditions 3.2, 3.3 and 4.8 hold. If α > β , then ρ r , ρ r ,r are the en-trance measures of equation (3.6) and (3.7) respectively. Moreover P r ( t, s, x, · ) , P r ,r ( t, s, x, · ) and the entrance measures ρ rt , ρ r ,r t are absolutely continuous with respect to the Lebesguemeasure L on ( R d , B ( R d )) , and hence have the density p r ( t, s, x, y ) , p r ,r ( t, s, x, y ) , q r ( t, y ) , q r ,r ( t, y ) respectively. Similarly, we know that q r ( t, x ) = Z R d p r ( t, s, y, x ) q r ( s, y )( dy )and q r ,r ( t, x ) = Z R d p r ,r ( t, s, y, x ) q r ,r ( s, y )( dy ) . Moreover, q r , q r ,r satisfy the following Fokker-Planck equations ∂ t q r = L r, ∗ ( t ) q r , ∂ t q r ,r = L r ,r , ∗ ( t ) q r ,r , where L r, ∗ and L r ,r , ∗ are given by (4.10) where b, σ are replaced by ˜ b r , ˜ σ r and ˜ b r ,r , ˜ σ r ,r respectively.By the proof of Theorem 3.4, we know that u r ( t, s, x, · ) = u ( t + r, s + r, x, θ − r · ) and ϕ r ( t, · ) = ϕ ( t + r, θ − r · ). Since θ − r preserves the probability measure P , then P r ( t, s, x, · ) = P ( t + r, s + r, x, · )and ρ rt = ρ t + r . Hence their densities have the following relations p r ( t, s, x, y ) = p ( t + r, s + r, x, y ) , q r ( t, x ) = q ( t + r, x ) . Acknowledgements
We would like to thank Kening Lu and Hans Crauel for raising our interests to consider randomquasi-periodicity in various occasions. We acknowledge the financial support of a Royal Soci-ety Newton Fund (Ref NA150344) and an EPSRC Established Career Fellowship to HZ (RefEP/S005293/1). 31 eferences [1] V. I. Arnold, Proof of a Theorem by A. N. Kolmogorov on the invariance of quasi-periodic motions undersmall perturbations of the Hamiltonian,
Russian Math (1963a) Survey
18 : 13-40.[2] P. W. Bates, K.N. Lu and B.X. Wang, Attractors of non-autonomous stochastic lattice systems in weightedspaces,
Physica D , Vol. 289 (2014), 32-50.[3] F. E. Benth and J. Saltyte-Benth, The volatility of temperature and pricing of weather derivatives,
Quan-titative Finance , Vol. 7 (2007), 553-561.[4] F. E. Benth, J. Kallsen and T. Meyer-Brandis, A Non-Gaussian Ornstein-Uhlenbeck Process for ElectricitySpot Price Modelling and Derivatives Pricing,
Applied Mathematical Finance , Vol 14 (2007), 153-169.[5] M. Chekroun, E. Simonnet and M. Ghil, Stochastic climate dynamics: random attractors and time-dependent invariant measures,
Physica D , 240 (2011), 1685-1700.[6] A. M. Cherubini, J. S W Lamb, M. Rasmussen and Y. Sato, A random dynamical systems perspective onstochastic resonance,
Nonlinearity , Vol 30 (2017), 2835-2853.[7] C. R. Feng, Y. Wu and H. Z. Zhao, Anticipating Random Periodic Solutions I. SDEs with MultiplicativeLinear Noise,
J. Funct. Anal. , Vol 271 (2016), 365-417.[8] C.R. Feng, H.Z. Zhao and B. Zhou, Pathwise random periodic solutions of stochastic differential equations,
J. Differential Equations , Vol. 251 (2011), 119-149.[9] C.R. Feng and H.Z. Zhao, Random periodic solutions of SPDEs via integral equations and Wiener-Sobolevcompact embedding
J. Funct. Anal. , Vol. 262 (2012), 4377-4422.[10] C.R. Feng and H.Z. Zhao, Random periodic processes, periodic measures and ergodicity, 2018, arXiv:1408.1897 .[11] C. Feng, H. Zhao and J. Zhong, Existence of Geometric Ergodic Periodic Measures of Stochastic DifferentialEquations, 2019, arXiv:1904.08091 .[12] W. Huang and Z. Lian, Horseshoe and periodic orbits for quasi-periodic forced systems, arXiv:1612.08394 .[13] N. Ikeda and S. Watanabe,
Stochastic Differential Equations and Diffusion Processes, 2nd Edition , North-Holland, Amsterdam 1989.[14] A. Iolov, S.Ditlevsen and A. Longtin, Fokker-Planck and Fortet Equation-Based Parameter Estimation fora Leaky Integrate-and-Fire Model with Sinusoidal and Stochastic Forcing,
The Journal of MathematicalNeuroscience , Vol 4 (2014), article no. 4.[15] I. Karatzas and S. E. Shreve,
Brownian motions and stochastic calculus, 2nd Edition , Springer-Verlay NewYork, 1991.[16] N. Kazamaki,
Continuous Exponential Martingales and BMO , Springer, 1994.[17] A. N. Kolmogorov, On the conservation of conditionally periodic motions under small perturbation of theHamiltonian,
Dokl. Akad. Nauk. , (1954) SSR 98: 527-530.[18] J. J. Lucia and E. Schwartz, Electricity prices and power derivatives: Evidence from the Nordic PowerExchange,
E.S. Review of Derivatives Research , Vol 5 (2002), 5-50.[19] J. K. Moser, On invariant curves of area-preserving mappings of an annulus,
Nach. Akad. Wiss. Gttingen,Math. Phys. (1962) Kl. II 1 : 1-20.[20] P. Walters,
An Introduction to Ergodic Theory, Graduate Tests in Mathematics , 79, Springer-Verlag NewYork (1982).[21] B.X. Wang, Existence, stability and bifurcation of random complete and periodic solutions of stochasticparabolic equations,
Nonlinear Analysis , Vol. 103 (2014), 9-25.[22] H.Z. Zhao and Z.H. Zheng, Random periodic solutions of random dynamical systems,
J. Differential Equa-tions , Vol. 246 (2009), 2020-2038., Vol. 246 (2009), 2020-2038.