Một số bài toán định tính của phương trình vi phân ngẫu nhiên không ôtônôm với nhiễu brown phân thứ TT

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS Phan Thanh Hong SOME QUALITATIVE PROBLEMS OF NONAUTONOMOUS STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS Speciality: Probability and Statistics Theory Speciality code: 46 01 06 SUMMARY DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI - 2021 download by : skknchat@gmail.com The dissertation was written on the basis of the author’s research works carried at Institute of Mathematics, Vietnam Academy of Science and Technology Supervisor: Dr Luu Hoang Duc First referee: Second referee: Third referee: To be defended at the Jury of Institute of Mathematics, Vietnam Academy of Science and Technology on The dissertation is publicly available at: • The National Library of Vietnam • The Library of Institute of Mathematics download by : skknchat@gmail.com Introduction A fractional Brownian motion (in short fBm) is a family of centered Gaussian processes B H = BtH t ∈ R or R+ , indexed by the Hurst parameter H ∈ (0, 1) with continuous sample paths and the covariance function R H (s, t) = 2H t + s2H − |t − s|2H In the last decade, stochastic differential equations driven by fractional Brownian motions (in short fSDE) have attracted a lot of research interest Since B H is not a semimartingale if H = 12 , one cannot apply the classical Ito theory to construct a stochastic integral w.r.t the fBm Among a numerate attempts to define a (stochastic) integral with respect to fractional Brownian motion, the deterministic approach consists of two directions of development: rough path theory and fractional calculus, in which the integrals can be defined in the pathwise sense For the case H > 1/2, the integral defined by rough path theory is understood in the Young ă sense and coincides with that defined by fractional derivative on the space of Holder continuous functions This dissertation focuses on studying the nonautonomous stochastic differential equations driven by m−dimensional fractional Brownian motions with Husrt index H > 1/2 of the form dxt = f (t, xt )dt + g(t, xt )dBtH , (1) which, like Ito differential equations, is understood as an integral equation of form t x t = x0 + t f (s, xs )ds + t0 g(s, xs )dBsH , t ∈ [0, T ] (2) where f : [0, T ] × Rd → Rd , g : [0, T ] × Rd → Rd×m are time dependent coefficient functions, the first integral is of Riemann type and the second one is understood pathwise in the Young sense At first we prove that, under similar assumptions to those in D Nualart (2002), the existence and uniqueness theorem for system (1) still holds in the space of continuous functions with bounded p-variation norm When applying to stochastic differential equations driven by fractional Brownian motions, by considering an appropriate probability space, it is proved that the system generates a random dynamical system (in short RDS, see B Schmalfuß et al (2010, 2014) and M Scheutzow (2017) However in the nonautonomous situation, one only expects the system to generate a stochastic two-parameter flow on the phase space These results allow us to study the long term behaviour of the systems under the framework of RDS theory One main part of the thesis is devoted to the results on the asymptotic behaviour of the solution Note that this problem is still open even download by : skknchat@gmail.com for the case H > 1/2, see recent results in M Garrido-Atienza et al (2010), L H Duc, et al (2018) We develop the semi-group technique to study the existence of the random pullback attractor provided that the linear part has negative eigenvalue and the nonlinear pertubations are small These techniques show the capability to deal with the rough equation in the work by L H Duc (2019), or in the paper for infinite dimensional case by N D Cong et al (2020), L Galeati (2021) We are also interested in studying Lyapunov spectrum of nonautonomous linear systems Notice that Lyapunov spectrums and its splitting are the main content of the celebrated multiplicative ergodic theorem (MET) by V I Oseledets (1968) It was also investigated by V M Millionshchikov (1968, 1986, 1987) for linear nonautonomous differential equations In the stochastic setting, the MET is also formulated for random dynamical systems in L Arnold (1998) Further investigations can be found in N D Cong (2001, 2004) for stochastic flows generated by nonautonomous linear stochastic differential equations driven by standard Brownian motions To our knowledge there has not been any works on this topic for the stochastic system driven by fBm The thesis consists of chapters Chapter recalls the definition and properties of fBm as well as the construction of the canonical space for fBm, some basic results on Young integrals, greedy sequences of times, the concepts of stochastic flows and random dynamical systems In Chapter we present the existence and uniqueness theorem for system (1) as well as its backward equation The main tools in use are greedy sequences of times, Shauder-Tychonoff fixed point theorem and a Gronwall-type lemma We establish the estimate for the norm of the solution then prove finiteness of its moments The continuity and differentiability of the solution are proved The final part of this chapter presents the generation of stochastic flows which become random dynamical systems for autonomous systems In Chapter we discuss Lyapunov spectrum of linear systems We show that Lyapunov exponents can be computed based on the discretization scheme Moreover, the spectrum can be computed independently of ω for triangular systems (i.e both A, C are upper triangular matrices) which are Lyapunov regular Finally, under some further conditions on coefficient matrices we construct a probability measure together with a Bebutov flow generated by the system The regularity almost all of the spectrum is proved as a consequence of the MET In Chapter we extend the results we obtain in L H Duc et al (2019, 2020) for nonautonomous equations We establish criteria for the existence of a nonautonomous pullback attractor of the system Moreover we show that the attractor is a singleton and is also a forward attractor in case g is linear or bounded At the end of this chapter, we show the generation of the random dynamical system w.r.t the metric dynamical system generated by Bebutov flows in the product space The flow then possesses a singleton random attractor in both pullback and forward directions as a consequence of the results presented at the beginning of this chapter The dissertation is written based on refereed papers [1 - 4], [6] and also a preprint [5] download by : skknchat@gmail.com Chapter Preliminaries In this chapter we recall some notations and summarize some standard results which we will need in the sequel 1.1 Fractional Brownian motions Definition 1.1 (Y Mishura (2008)) A two-sided one-dimensional fractional Brownian motion of Hurst parameter H ∈ (0, 1) is a centered continuous Gaussian process B H = ( BtH ), t ∈ R with covariance function R H (s, t) = (|t|2H + |s|2H − |t − s|2H ), s, t ∈ R Due to Kolmogorov Theorem, for each < ν < H is fixed, there exists a modifiă cation of B H with paths are of ν−Holder continuity on every compact sets of R For H = 12 , B H with H = 1.1.1 is not a semimartingale Canonical spaces Due to B Mandelbrot and J Van Ness (1968), the process B H = ( BtH )t∈R defined as Ito integral BtH := k H (t, u)dWu (1.1) cH R where W is a standard Bm and 1 k H (t, u) := [(t − u) ∨ 0] H − − [(−u) ∨ 0] H − , t, u ∈ R with c H is a normalized constant has a continuous modification which is two-sided fBm of Hurst parameter H For W is the coordinate process defined on the canonical space, and consider the continuous version which we keep the same notation B H one obtains a measurable mapping B H : (C0 (R, R), B) → (C0 (R, R), B) Then there exists (C0 (R, R), B , P H , θ ) the canonical space for fBm, in which P H := B H P Theorem 1.1 (M Garrido-Atienza et al (2011) )(C0 (R, R), B , P H , θ ) is an ergodic metric dynamical system download by : skknchat@gmail.com Definition 1.2 Let m be a positive integer An m−dimensional fractional Brownian moH ) where B H are independent onetion B H of Hurst index H ∈ (0, 1) is ( B1H , B2H , · · · , Bm i dimensional fractional Brownian motions of Hurst index H Similar to the one-dimensional case, one can construct the canonical space (Ω, F , P, θ ) for m−dimensional B H Here Ω is C0 (R, Rm ), F is the Borel σ−algebra, P is the distribution of B H and θ is the Wiener shift operator Moreover, the space is ergodic For our purpose, we need to work with the space of bounded p - variation paths 0,p−var (R, Rm ) and reon any compact intervals Hence we take the trace of B on C0 strict P H on this new σ−algebra to build a new metric dynamical system which we keep the old notation (Ω, F , P, θ ) Moreover, this new one is ergodic (T Caraballo et al (2011)) 1.2 Pathwise stochastic integrals with respect to fractional Brownian motions 1.2.1 Young integrals In this subsection we recall some facts about Young integral Definition 1.3 (P Friz et al (2010)) Given f ∈ C q−var ([ a, b], Rd×m ) and g ∈ C p−var ([ a, b], Rm ) we say that z ∈ C([ a, b], Rd ) is an indefinite Young integral of f against g if there exist a sequence f n ∈ C 1−var ([ a, b], Rd×m ), gn ∈ C 1−var ([ a, b], Rm ) satisfying supn ||| f n ||| p−var,[ a,b] , supn ||| gn |||q−var,[ a,b] < ∞ such that lim f n − f ∞,[ a,b] = lim gn − g ∞,[ a,b] = n→∞ n→∞ · a and f (t)dg(t) tends to z uniformly on [ a, b] as n → ∞ If z is unique we write z = · t t s a f ( t ) dg ( t ) and set s f ( u ) dg ( u ) : = a f ( u ) dg ( u ) − a f ( u ) dg ( u ) Theorem 1.2 (P Friz et al (2010)) Given f ∈ C q−var ([ a, b], Rd×m ) and g ∈ C p−var ([ a, b], Rm ) with · a unique indefinite Young integral of f against g, estimate t s + 1q > 1, there exists f (t)dg(t) Moreover, the Young-Loeve f (u)dg(u) − f (s)[ g(t) − g(s)] ≤ 1−2 1− 1p − 1q p ||| g||| p−var,[s,t] ||| f |||q−var,[s,t] holds for all [s, t] ⊂ [ a, b] From the Young-Loeve estimate (1.2), one can see that t s f (u)dg(u) = lim ∑ |Π|→0 t ∈Π[s,t] i f (ξ i )[ g(ti+1 ) − g(ti )] where the limit is taken over the class of finite partitions Π of [s, t] download by : skknchat@gmail.com (1.2) 1.2.2 Stochastic integrals w.r.t fractional Brownian motions Given a stochastic process Xt and a fractional Brownian motion BtH with Hurst index H > 1/2, since B H is not a semimartingale one can not define Xt dBtH using Ito stochastic calculus Instead, we use Young integral to construct the stochastic integral of X w.r.t fBm in the pathwise sense To that we work with locally bounded p−variation version of B H Assume that sample paths of Xt is of locally bounded q− variation, 1p + 1q > 1, a.s Then for each realization xt , ωt of X and B H the integral b a Xt dBtH is defined pathwise as the Young integral b xu dωu a 1.2.3 Young integrals on infinite domains Given f ∈ C q−var ([ a, b], Rd×m ) and g ∈ C p−var ([ a, b], Rm ) with [ a, b] ⊂ R+ Then ∞ t0 define b a We define Gt := q > for all t t → ∞ t0 t0 + f (s)dg(s) exists for all ≤ a < b For fixed t0 ∈ R+ , we f (sd) g(s) as the limit lim this case, p f (s)dg(s) = ∞ t f ( s ) dg ( s ) ∞ f (s)dg(s) if the limit exists and is finite In f (s)dg(s) − ∞ t0 f (s)dg(s) and ∆k := [k, k + 1] Proposition 1.1 If ∑k≥0 f q−var,∆k < ∞ and supk≥0 ||| g||| p−var,∆k < ∞ then Gt is well defined for all t ∈ R+ Moreover, G is of bounded p- variation and ||| G ||| p−var,[a,b] ≤ ||| g||| p-var,[a,b] | f ( a)| + (K + 1) ||| f |||q−var,[a,b] 1.3 Greedy sequences of times Let us recall here the concept of greedy sequences first introduced by T Lyons (2013) Let ω be an element in C p−var (R, Rm ) For any given µ > we construct a nondecreasing sequence of times {τn = τn (ω )} such that τ0 ≡ a and τ (ω ) := inf{t ∈ [ a, b] : |||ω ||| p−var,[a,t] ≥ µ} ∧ b (1.3) N = N[ a,b],µ (ω ) := sup{n : τn ≤ b} (1.4) Assign Note that for τi (ω ) < b and |||ω ||| p−var,[τi (ω ),b] ≥ µ, τi+1 (ω ) is intuitively the first time |||ω ||| p−var,[τi (ω ),·] reaches µ Since the function κ (t) := |||ω ||| p-var,[t0 ,t] is continuous and nondecreasing w.r.t t with κ (t0 ) = we obtain |||ω ||| p−var,[τi (ω ),τi+1 (ω )] = µ, i = 0, · · · , N − download by : skknchat@gmail.com (1.5) Lemma 1.1 The following estimate holds N[ a,b],µ (ω ) ≤ + 1.4 p |||ω ||| p-var,[a,b] p µ (1.6) Stochastic flows Stochastic two-parameter flows Definition 1.4 (H Kunita (1990)) A collection Ψt,s ( x, ω ), s, t ∈ T, x ∈ Rd of Rd − valued random variables defined on (Ω, F , P) is called a stochastic flow of homeomorphisms if there exists N ∈ F with P( N ) = such that for ω ∈ Ω\ N the following conditions hold: (i) (s, t, x ) → Ψt,s ( x, ω ) is a continuous map, (ii) Ψs,s (·, ω ) = id for any s ∈ T, (iii) Ψt,s (·, ω ) = Ψt,u (Ψu,s (·, ω ), ω ) for any s, t, u ∈ T, (iv) Ψ− t,s (·, ω ) = Ψs,t (·, ω ) for any s, t ∈ T, (v) For any s, t ∈ [0, T ] the mapping Ψt,s (·, ω ) is a homeomorphism of Rd It is called stochastic (two-parameter) flow of C k −diffeomorphisms of Rd on T if satisfies (i)-(v) and: (vi) Ψt,s ( x, ω ) is k −times differentiable w.r.t x for all s, t ∈ T and the derivative is continuous w.r.t (s, t, x ) If, in addition, Ψt,s (·, ω ) is a linear operator of Rd for any s, t ∈ T then the family Ψt,s (·, ω ) is called a two-parameter stochastic flow of linear operators of Rd Random dynamical systems Assume T = R and there exists a semigroup (θt ) which forms a metric dynamical system (Ω, F , P, (θt )t∈T ) If on a full measure set which is (θt )− invariant we have for all s < t and x ∈ Rd Ψt,s ( x, ω ) = Ψt−s,0 ( x, θs ω ), then the flow can be considered as a random dynamical system Definition 1.5 A measurable random dynamical system on Rd over a metric dynamical system (Ω, F , P, (θt )t∈T ) is a measurable mapping ϕ : R × Rd × Ω → Rd , (t, x, ω ) → ϕ(t, ω ) x with the properties (i ) ϕ(0, ω ) = Id for all ω ∈ Ω, (ii ) ϕ(t + s, ω ) = ϕ(t, θs ω ) ◦ ϕ(s, ω ) for all s, t ∈ R, ω ∈ Ω If, in addition, x → ϕ(t, ω ) x is continuous for all t, ω then ϕ is called a continuous random dynamical system download by : skknchat@gmail.com Chapter Stochastic differential equations driven by fBms In this chapter we consider the equation dxt = f (t, xt )dt + g(t, xt )dBtH , (2.1) in which B H is m - dimensional fractional Brownian motion with Hurst index H is greater than 12 System (2.1) is understood in the pathwise sense Namely, for each fixed sample path ω of the noise, we consider the deterministic equation dxt = f (t, xt )dt + g(t, xt )dωt , (2.2) which can be written in the integral form t t x t = x0 + f (s, xs )ds + g(s, xs )dωs , (2.3) where the first integral is of Riemannian type, meanwhile the second one is understood in the Young sense The content of this chapter are mainly from the papers [1] and [2] This chapter establishes the theorem on the existence and uniqueness of solution to (2.1) and studies properties of the solution 2.1 Assumptions (H1 ) f (t, x ) is continuous and there exists C f > 0, b ∈ L1 ([0, T ], Rd ) and for every N ≥ there exists L N > such that the following properties hold:  (i ) Local Lipschitz continuity    | f (t, x ) − f (t, y)| ≤ L | x − y|, ∀ x, y ∈ Rd , | x |, |y| ≤ N, ∀t ∈ [0, T ], N  (ii ) Boundedness    | f (t, x )| ≤ C f | x | + b(t), ∀ x ∈ Rd , ∀t ∈ [0, T ] (H2 ) g(t, x ) is differentiable in x and there exist some constants Cg > 0, < β, δ ≤ 1, a control function h(s, t) defined on ∆[0, T ] and for every N ≥ there exists M N > such that the following properties hold: download by : skknchat@gmail.com   (i ) Lipschitz continuity      | g(t, x ) − g(t, y)| ≤ Cg | x − y|, ∀ x, y ∈ Rd , ∀t [0, T ], ă continuity (ii ) Local Holder Dx g(t, x ) − Dy g(t, y) ≤ M N | x − y|δ , ∀ x, y ∈ Rd , | x |, |y| ≤ N, ∀t ∈ [0, T ],   ă (iii ) Generalized Holder continuity in time     | g(t, x ) − g(s, x )| + Dx g(t, x ) − Dx g(s, x ) ≤ h(s, t) β ,     ∀ x ∈ Rd , ∀s, t ∈ [0, T ], s < t (H3 ) The parameters in (H1 ) and (H2 ) statisfy the inequalities δ > p − 1, β > − p By condition (H3 ), one can choose consecutively constants q0 , q such that 1 + > 1, q0 β > 1, q0 ≥ q0 δ ≥ q > p p q0 (2.4) Introduce the notation K := 1−2 2.2 1− 1p − q1 (2.5) Existence and uniqueness theorem for deterministic equations In this section, we work with the deterministic Young equation t x t = x0 + t f (s, xs )ds + g(s, xs )dωs , t ∈ [0, T ] (2.6) where ω belongs to C p−var ([0, T ], Rm ) We now consider x ∈ C q−var ([t0 , t1 ], Rd ) with some [t0 , t1 ] ⊂ [0, T ] and define the mapping F ( x ) t = x t0 + I ( x ) t + J ( x ) t t : = x t0 + t0 t f (s, xs )ds + t0 g(s, xs )dωs , ∀ t ∈ [ t0 , t1 ] Lemma 2.1 (Gronwall-type Lemma) Let ≤ p ≤ q be arbitrary and satisfy (2.7) p + 1q > Assume that ω ∈ C p−var ([0, T ], R) and y ∈ C q−var ([0, T ], Rd ) satisfy 1/q |yt − ys | ≤ Aˆ s,t + a1 t s t yu du + a2 s yu dωu , ∀s, t ∈ [0, T ], download by : skknchat@gmail.com s < t, (2.8) 2.2.1 Special case: linear equations In this subsection, we consider the linear YDE dxt = A(t) xt dt + C (t) xt dωt , x0 ∈ Rd , t ∈ [0, T ], (2.13) It can be seen that the condition (H2 )(iii ) in Assumption 2.1 does not hold for this system Therefore we need to reprove the theorem on the existence and uniqueness of solution for (2.13) The proof is in the similar manner with some modified estimates Theorem 2.4 Assume that A ∈ C([0, T ], Rd×d ), C ∈ C q−var ([0, T ], Rd×d ) with q > p and 1q + 1p > Then equation (2.13) has a unique solution in the space C p−var ([0, T ], Rd ) which satisfies (i ) x (ii ) ∞,[ a,b] x ≤ | x a |e p-var,[a,b] A ≤ | x a |e ∞,[0,T ] ( b − a )+1+(8K A 1−2 1− 1p − 1q p p C q−var,[0,T ] |||ω ||| p−var,[ a,b] , ∞,[0,T ] ( b − a )+2[1+(8K for all [ a, b] ⊂ [0, T ], where K ∗ = 2.3 ∗)p ∗)p) C p p |||ω ||| p−var,[a,b] ] q−var,[0,T ] (2.14) , (2.15) Continuity and differentiability of the solution We denote X (·, t0 , x0 , ω ) the solution of (2.6) on [0, T ] which starts at x0 ∈ Rd at time t0 Theorem 2.5 Suppose that the assumptions of Theorem 2.1 are satisfied then the solution mapping X : [0, T ] × [0, T ] × Rd × C p−var ([0, T ], Rm ) → Rd , (t, s, x0 , ω ) → X (t, s, x0 , ω ), is continuous Now we assume additionally (H4 ) f is of C0,1 on [0, T ] × Rd , i.e f (t, x ) and Dx f (t, x ) are continuous w.r.t (t, x ) ∈ [0, T ] × Rd Theorem 2.6 Suppose that the assumptions (H1 )- (H4 ) are satisfied then the solution mapping X : Rd → C p−var ([0, T ], Rd ), x0 → X (·, x0 ), is differentiable 10 download by : skknchat@gmail.com 2.4 The stochastic differential equations driven by fBm Consider system (2.1) dxt = f (t, xt )dt + g(t, xt )dBtH , xt0 ∈ Rd , t ∈ [0, T ] As a consequence of Theorem 2.1, (2.1) is solved pathwise for each ω ∈ (Ω, F , P) Theorem 2.7 Under the assumptions (H1 ) − (H3 ), system (2.1) possesses a unique solution X (·, t0 , x0 , ω ) ∈ C p−var ([0, T ], Rd ) such that for each t ∈ [0, T ] X (t, t0 , x0 , ·) : (Ω, F , P) → Rd is measurable Proposition 2.2 For each T > there exists constant D = D ( T ) and random variable ξ = ξ (ω ) such that almost all B·H (ω ) p−var,[0,T ] ≤ Dξ (ω ) where Eeκξ < ∞ with some κ > Proposition 2.3 For each t0 , t ∈ [0, T ], x0 ∈ Rd the mapping solution X (t, t0 , x0 , ·) is of finite moments of any order n ∈ R+ Moreover the tail probability of X is estimated as P( X (·, t0 , x0 , ω ) ≤( ∞,[0,T ] TD )p log λ − log D π ≥ λ) ∞ ∑ 2nH exp −22n(1− H )−1 ( n =1 log λ − log D 2p ) TD where D is a constant 2.5 The generation of stochastic two parameter flows In this section we show that equation (2.1) generates a stochastic two-parameter flow on Rd In the autonomous situation, we show that the solution then satisfies the cocycle property, thus generates a random dynamical system Theorem 2.8 Assume that the conditions (H1 ) − (H3 ) hold on any compact interval of R The family of Cauchy operators of (2.2) generates a stochastic two parameter flow of homeomorphisms of Rd Theorem 2.9 Suppose that assumptions (H1 ) − (H4 ) are satisfied on every compact interval of R Then equation (2.1) generates a stochastic two-parameter flow of C1 − diffeomophisms of Rd on R At the end of this chapter we restrict the discussion to the autonomous system dxt = f ( xt )dt + g( xt )dBtH , (2.16) where f , g are time independent and sastisfy the assumptions (H1 ) − (H3 ) on every compact set in R Theorem 2.10 System (2.16) generates a continuous random dynamical system 11 download by : skknchat@gmail.com Chapter Lyapunov spectrum of nonautonomous linear fSDEs This chapter is devoted to study the Lyapunov spectrum of the linear system dxt = A(t) xt dt + C (t) xt dBtH , x0 ∈ Rd , t ≥ 0, (3.1) where B H is one dimensional fBm defined on the canonical space and A, C are continuous matrix valued functions The chapter is written on the basis of [3] Recall that the classical definition of Lyapunov exponent for function h : R+ → R given by χ(ht ) := lim log |ht | (3.2) t→∞ t 3.1 Some useful properties of sample paths of fBm To begin with, we introduce some properties of the paths of B H and the path wise Young integral driven by B H By Proposition 2.2 Γk := E 1/k k B·H (ω ) p−var,[0,1] then lim t cs dωs t→∞ t = 0, a.s Indefinitely Young integrals w.r.t B H To study the Lyapunov exponent of system (3.1) we consider here some estimates on the Lyapunov exponent of functions as indefinitely Young integrals w.r.t to fixed ω ∈ Ω Note that the sequence |||ω ||| p−var,[k,k+1] , k k ≥ is bounded t Lemma 3.2 Consider Gt = gs dωs , where g is of bounded q−variation function on every compact interval, 1p + 1q > If χ( gt ), χ(||| g|||q−var,[t,t+1] ) ≤ λ ∈ [0, +∞) then χ( Gt ), χ(||| G |||q−var,[t,t+1] ) ≤ λ Lemma 3.3 Let g be a bounded q−variation function on every compact interval, satisfying ∞ χ( gt ), χ(||| g|||q−var,[t,t+1] ) ≤ −λ ∈ (−∞, 0) then Gt = t g(s)dω (s) exists for all t ∈ R+ and χ( Gt ), χ(||| G |||q−var,[t,t+1] ) ≤ −λ 3.3 The generation of stochastic flow of linear operators In analogy with Theorem 2.8, we obtain the result on the generation of the stochastic two - parameter flow of the linear system dxt = A(t) xt dt + C (t) xt dBtH , xt0 = x0 ∈ Rd , t ∈ [0, T ] (3.5) under the assumption of Theorem 2.4 Theorem 3.1 Suppose that the assumptions of Theorem 2.4 are satisfied on every compact interval of R Then equation (3.5) generates a stochastic two-parameter flow of linear operators of Rd on R 3.4 Lyapunov spectrum for nonautonomous linear fSDEs From now on, we study the system dxt = A(t) xt dt + C (t) xt dBtH , x0 ∈ Rd , t ≥ 0, (3.6) under the following condition on the coefficient matrices A, C Assumptions (H1 ) Aˆ := A ∞,R+ < ∞ (H2 ) For some δ > 0, Cˆ := C q−var,δ,R+ := sup C q−var,[s,t] < ∞ 0≤ t − s ≤ δ One can assume, without loss of generality that δ = Put ˆ [8K Cˆ ] p } M0 := max{2 A, 13 download by : skknchat@gmail.com (3.7) 3.4.1 Exponents and spectrum For a linear differential equation, the Lyapunov spectrum of the system is the collection of Lyapunov exponents of its solutions and also can be defined based on the two-parameter flow of linear operator generated by the system Definition 3.1 (N D Cong (2001)) (i ) Given a stochastic two-parameter flow Φt,s (ω ) of linear operators of Rd on the time interval [t0 , ∞), the extended-real numbers (real numbers or symbol ∞ or −∞) (3.8) λk (ω ) := inf sup lim log |Φt,t0 (ω )y|, k = 1, , d, V ∈Gd−k+1 y∈V t→∞ t are called Lyapunov exponents of the flow Φt,s (ω ) The collection {λ1 (ω ), , λd (ω )} is called Lyapunov spectrum of the flow Φt,s (ω ) (ii ) For any u ∈ [t0 , ∞) the linear subspaces of Rd Eku (ω ) := y ∈ Rd lim log |Φt,u (ω )y| ≤ λk (ω ) , k = 1, , d, (3.9) t→∞ t are called Lyapunov subspaces at time u of the flow Φt,s (ω ) The flag of nonincreasing linear subspaces of Rd Rd = E1u (ω ) ⊃ E2u (ω ) ⊃ · · · ⊃ Edu (ω ) ⊃ {0} is called Lyapunov flag at time u of the flow Φt,s (ω ) Proposition 3.1 (i) The Lyapunov exponents λk (ω ), k = 1, , d, of Φt,s (ω ) are measurable functions of ω ∈ Ω (ii) For any u ∈ [t0 , ∞), the Lyapunov subspaces Eku (ω ), k = 1, , d, of Φt,s (ω ) are measurable and invariant with respect to the flow in the following sense Φt,s (ω ) Eks (ω ) = Ekt (ω ), for all s, t ∈ [t0 , ∞), ω ∈ Ω, k = 1, , d Theorem 3.2 Let Φt,s (ω ) be the flow generated by (3.6) and {λ1 (ω ), , λd (ω )} be its Lyapunov spectrum hence of equation (3.6) Then under assumption (H1 ), (H2 ), the Lyapunov exponents λk (ω ), k = 1, , d, can be computed via a discrete-time interpolation of the flow, i.e λk (ω ) := inf sup lim log |Φt,t0 (ω )y|, k = 1, , d (3.10) V ∈Gd−k+1 y∈V N t→∞ t Morever, the spectrum is bounded by a constant, namely p |λk (ω )| ≤ + M0 (1 + Γ p ), k = 1, , d, (3.11) where M0 is determined by (3.7) Corollary 3.1 (Integrability condition) Under the assumptions (H1 ) and (H2 ), Φt,s (ω ) satisfies the following integrability condition E p sup t0 ≤ s ≤ t ≤ t0 +1 log+ Φt,s (ω )±1 ≤ + M0 + Γ p , for any t0 ≥ 0, where M0 is determined by (3.7) and we use the notation log+ a := max{log a, 0}, a > 14 download by : skknchat@gmail.com (3.12) 3.4.2 Lyapunov spectrum of triangular systems Let us consider the deterministic system dXt = A(t) Xt dt + C (t) Xt dωt (3.13) in which, X = ( x1 , x2 , , xd ), A(t) = ( aij (t)), C (t) = (cij (t)) are d dimensional upper triangular matrices of coefficient functions satisfy the condition (H1 ), (H2 ) Theorem 3.3 Under assumptions (H1 ) – (H2 ), if there exist the exact limits t→∞ t akk := lim t akk (s)ds, k = 1, d (3.14) then the spectrum of system (3.13) hence (3.6) is given by { a11 , a22 , , add } 3.4.3 Lyapunov regularity The concept regularity has been introduced by Lyapunov for linear ODEs, and since then has attracted lots of interests (see e.g V M Millionshchikov (1968), L Arnold (1998), N D Cong (2004), or L Barreira (2017)) For a linear YDE, we define the concept of Lyapunov regularity via the generated two-parameter flow of linear operators in Rd Theorem 3.4 (Lyapunov theorem on regularity of triangular system) Suppose that the matrices A(t), C (t) are upper triangular and satisfy (H1 ) – (H2 ) Then system (3.13) is Lyat punov regular if and only if there exists lim 1t t akk (s)ds, k = 1, d t→∞ 3.5 Almost sure Lyapunov regularity In case A(·) ≡ A, C (·) ≡ C, the stochastic two-parameter flow Φt,s (ω ) of (3.6) generates a linear random dynamical system Φ By applying the multiplicative ergodic Theorem (3.6) possesses a Lyapunov spectrum consisting of exact and nonrandom Lyapunov exponents For general cases, we consider stronger conditions that (H1 ) Aˆ := A ∞,R < ∞ and lim sup | A(t) − A(s)| = δ→0 |t−s| such that the following properties hold: 17 download by : skknchat@gmail.com   (i ) Lipschitz continuity      | g(t, x ) − g(t, y)| ≤ Cg | x − y|, ∀ x, y ∈ Rd , t R, ă (ii ) Local Holder continuity     Dx g(t, x ) − Dy g(t, y) ≤ M | x − y|δ , ∀ x, y ∈ Rd , | x |, |y| N, t R, N ă (iii ) Holder continuity in time      | g(t, x ) − g(s, x )| + Dx g(t, x ) − Dx g(s, x ) ≤ k(|t − s|) β =: h∗ (|t − s|)      ∀ x ∈ Rd , ∀s, t ∈ R    ∗  (iv) limt→∞ log h (t) = |t| (H4 ) The parameters in (H1 ), (H2 ) and (H3 ) satisfy the inequalities: δ > p − 1, β > − , and p λ A − C A C f > Condition (H2 )(iii ) is equivalent to supt∈R the notation t +1 f := sup t ∈R t t +1 |b(s)|ds t < ∞, then we introduce |b(s)|ds < ∞ (4.3) Under these conditions, system (4.1) possesses a unique solution on whole R which starts at an arbitrary time t0 and inherits the properties presented in Chapter We end this introduction with a remark that throughout this chapter we always denote by D a generic constant 4.1 Nonautonomous attractors In what follows we recall the notion of the (global) pullback attractor of a stochastic two-parameter flow Definition 4.1 (H Crauel et al (2015)) For a given stochastic two-parameter flow Ψt,s (ω ), a family of sets At (ω ) of Rd , t ∈ R is called the random pullback (forward) attractor of Ψ if there exist a full probability measurable set Ω0 ⊂ Ω such that for each ω ∈ Ω0 , At (ω ) (i ) is compact set for t ∈ R, (ii ) is invariant, i.e Ψt,s (ω )As (ω ) = At (ω ) for all s ≤ t in R, (iii ) globally pullback (forward) attracting, i.e for every t ∈ R and every D bounded lim d(Ψt,s (ω ) D |At (ω )) = 0, s→−∞ ( lim d(Ψt,s (ω ) D |At (ω )) = 0) t→+∞ In general, one may consider the attracting on a family of nonempty sets ( Dt ) instead of a single set as in Definition 4.1 Below, we consider the family D of Dt which is a subset of the closed ball B¯ (0, rt ) where the radius rt satisfies lim log+ rt = 0, t→±∞ t 18 download by : skknchat@gmail.com in which log+ a = max{log a, 0}, a ∈ R+ The pullback attracting property now can be written as lim d(Ψt,s (ω ) Ds |At (ω )) = 0, s→−∞ In the case the flow induced a random dynamical system ϕ, one has notation of random pullback attractor A(ω ) which independent of t In random dynamical setting, the variance property writes ϕ(t, ω )A(ω ) = A(θt ω ) The notion of random attractor for RDS is also defined together with a universal family of sets The uniˆ ∈ Dˆ verse Dˆ is a family of random sets which is closed w.r.t inclusions, i.e if D ˆ2 ⊂ D ˆ then D ˆ ∈ Dˆ and D Definition 4.2 (L Arnold (1998)) Given a random dynamical system ϕ on Rd and a universal D (i) A random subset A is called invariant, if ϕ(t, ω )A(ω ) = A(θt ω ) ∀t ∈ R, a.s ω ∈ Ω (ii) An invariant random compact set A ∈ Dˆ is called a pullback random attractor in Dˆ , if A attracts any closed random set Dˆ ∈ Dˆ in the pullback sense, i.e ˆ (θ−t ω )|A(ω )) = 0, ∀ D ˆ ∈ Dˆ , a.s ω ∈ Ω lim d( ϕ(t, θ−t ω ) D t→∞ (4.4) (ii) An invariant random compact set A ∈ Dˆ is called a forward random attractor in Dˆ , if A is invariant and attracts any closed random set Dˆ ∈ Dˆ in the forward sense, i.e ˆ (ω )|A(θt ω )) = 0, ∀ D ˆ ∈ D , a.s ω ∈ Ω lim d( ϕ(t, ω ) D t→∞ (4.5) Here we take into account the universe Dˆ to be a family of tempered random sets ˆ ( ω ) D 4.2 Existence of random attractors Consider equation (4.1) in which A, f , g satisfy conditions (H1 ) − (H4 ) It is proved in Chapter that (4.1) generates a stochastic two-parameter flow of homeomorphism Ψt,s (ω ), t ≥ s in which Ψt,s (ω ) x0 is the solution to (4.1) at time t with initial value x0 at time s defined for each ω ∈ (Ω, F , P, θ )-the canonical space of B H Recall that the following equality holds on a subset Ω of probability lim n→∞ n p |||θ−k x ||| p−var,[0,1] ∑ n k =1 p = E B·H p p−var,[0,1] =: Γ p < ∞ (4.6) In this section we fix ω in the space Ω and consider the deterministic equation dxt = [ A(t) xt + f (t, xt )]dt + g(t, xt )dωt , t ∈ R, xt0 ∈ Rd In the following we are going to establish some prior estimates We are now in position to state the first main result of this chapter 19 download by : skknchat@gmail.com (4.7) Theorem 4.1 Assume that (H1 ) − (H4 ) are satisfied Then under the condition λ > Gˆ := C A eλ A +2L (1 + C A A ) where L := A t ( ω ) 2( K + 1) Cg Γ p p + 2( K + 1) Cg Γ p , (4.8) A + C f , the flow generated by system (4.1) possesses a pullback attractor In the following, we consider two particular cases: g is linear and g is bounded We will see that under weaken condition the existence of nonautonomous random pullback attractor still holds in these cases 4.3 Special case: g linear The case g is of linear, i.e g has the form g(t, x ) = C (t) x, we study the asymptotic dynamic of the equation dxt = [ A(t) xt + f (t, xt )]dt + C (t) xt dBtH , t ∈ R, x (t0 ) = xt0 ∈ Rd where C satisfies (HC ) C is continuous and Cˆ := sup C k q−var,∆k (4.9) < ∞ Theorem 4.2 Assume that (H1), (H2), (HC ) are satisfied Then there exists ε > such that if Cˆ < the flow generated by the system (4.9) possesses a pullback attractor At (ω ) In what follow we impose a stronger condition on f to study the difference between two solutions of the system which facilitates the proof of singleton attractor (H5 ): f is global Lipchitz continuous with Lipchitz constant C f (here we use an abuse notation for simplicity) We use the linearity of g to obtain one more result in the following Theorem Theorem 4.3 Under the assumption in Theorem 4.2 and (H5 ), the pullback attractor At (ω ) is singleton for each t and moreover is forward attractor 4.4 Special case: g bounded From now on we consider the case h∗ (|t − s|) = Cg |t − s| in (H3 )(iii ) The following theorem claims the existence of the pullback attractor relaxing the condition on the smallness of Cg Theorem 4.4 In addition to (H1 ) − (H4 ) assume further that g is bounded by g Then system (4.1) possesses a random pullback attractor almost sure ∞ < ∞ Theorem 4.5 Under (H1) − (H4) and additionally g is bounded and Dg is of global Lipchitz continuity w.r.t x, i.e Dx g(t, x ) − Dy g(t, y) ≤ Cg | x − y| for all t ∈ R, x, y ∈ Rd , there exists > such that if Cg , Cg < the pullback attractor is singleton set almost sure Moreover the attractor is forward one 20 download by : skknchat@gmail.com 4.5 Bebutov flow and its generation Recall that for each h defined on R × Rd denote by ht -the translate of h given by ht (s, x ) = h(t + s, x ), (s, x ) ∈ R × Rd for each t ∈ R We also denote by S the shift mapping on (C(R × Rd , Rd ), d1 )-the space of continuous functions on R × Rd equipped with the compact open topology St f = S(t, f ) = f t , ∀ f ∈ C(R × Rd , Rd ) Then define the hull of f , which denoted by H f - the closure of the sets {Sτ f |τ ∈ R} in C(R × Rd , Rd ) Similarly, as introduced in Chapter 3, for A ∈ (C b (R, Rd×d ), · ∞,R ) - the space of continuous bounded functions from R to Rd×d with supermum norm we recall the notation S A -shift mapping and consider H A - the hull of A, is the the closure of the sets {SτA A|τ ∈ R} Due to G Sell (1967), S A , S define a dynamical system on C(R, Rd×d ), C(R × Rd , Rd ) respectively Moreover, The hull H A is compact in C(R, Rd ) if (and only if) A is bounded and uniformly continuous on R, H f is compact in C(R × Rd , Rd ) if (and only if) f is bounded and uniformly continuous on every set R × K, where K is a compact set in Rd These results and Theorems 4.4, 4.5 promote the following additional condition to (H1 ) − (H5 ) (H6 ) A is uniformly continuous, f is uniformly continuous on R × K for each K compact in Rd ; f , g are bounded by f ∞ , g ∞ respectively Moreover, Dg is of Lipchitz continuity w.r.t x with Lipchitz constant Cg In the similar manner, we consider the hull of g Firstly, we fix − p < β < β Denote by H g the closure of {Sτ g|τ ∈ R} in the space (C β0 ;1,0 (R × Rd , Rd×m ), d2 ) in which ∞ f − g α,1,0;Kn d2 ( f , g ) : = ∑ n , (4.10) + f − g α,1,0;K n n =1 where f −g α,1,0;K1 ×K2 ||| f − g|||α,K1 ×K2 + ||| f − g|||α,K1 ×K2 := sup ||| f (·, x ) − g(·, x )|||α−Hol,K1 := f −g 1,0;K1 ×K2 x ∈K2 with K1 , K2 are compact sets in R, Rd respectively Proposition 4.1 If g satisfies (H3 ) and (H6 ), then so does each g∗ ∈ H g Moreover, H g is a compact set in (C β0 ;1,0 (R × Rd , Rd×m ), d2 ) Theorem 4.6 S defines a dynamical system on hull of g We deduce from Krylov-Bogoliubov theorem that there are probability measures P A , P f , Pg on measurable space (H A , B A ), (H f , B f ), (H g , B g ) with Borel σ−algebras B A , B f , B g , that are invariant under the shifts mapping introduced above respec¯ the Catersian product H A × H f × H g × Ω with the product tively Denote by Ω 21 download by : skknchat@gmail.com ¯ = P A × P f × Pg × P H and Borel σ−field denoted by B¯ and the product measure P ¯ →Ω ¯ given by consider the product dynamical system θ¯ : R × Ω ˜ St f˜, St g, ˜ f˜, g, ¯ ˜ f˜, g, ˜ θt ω ), ( A, ˜ ω ) ∈ Ω ˜ ω ) = (StA A, θ¯(t, A, ¯ B¯ , P, ¯ θ¯) is a metric dynamical system Lemma 4.1 (Ω, ˜ f˜, g˜ in H A , H f , H g satisfy (H1 ) − (H6 ) Proposition 4.2 Each A, ˜ f˜, g, ¯ the equation ˜ ω ) ∈ Ω, Theorem 4.7 For each ω¯ = ( A, dxt = [ A˜ (t) xt + f˜(t, xt )]dt + g˜ (t, xt )dBtH (ω ), t ∈ R, x (0) = x0 ∈ Rd , (4.11) possesses a unique solution x (t, x0 , ω¯ ) := X (t, 0, x0 , ω¯ ) that inherits all the properties as introduced in Chapter Moreover, x (t, x0 , ·) is measurable ˜ f˜, g, ¯ denote by Φ∗ (t, ω¯ ) x0 the value of the of the ˜ ω) ∈ Ω Now for each ω¯ = ( A, solution of (4.11) at the time t ∈ R with the initial time s = 0, i.e X (t, 0, x0 , ω ) We have t xt+s = xs + [SsA A˜ (u) xs+u + Ss f˜(u, xs+u )]du + t Ss g˜ (u, xs+u )dθs ωu It means that Φ∗ satisfies the cocycle property Φ∗ (t + s, ω¯ ) x0 = Φ∗ (t, θ¯s ω¯ ) ◦ Φ∗ (s, ω¯ ) x0 Therefore, we have proved the following theorem Theorem 4.8 The system (4.11) generates a random dynamical system over the metric dy¯ B¯ , P, ¯ θ¯) namical system (Ω, Theorem 4.9 The system (4.11) possesses a random pullback attractor Moreover, if Cg , Cg is small enough the attractor is a singleton almost surely, thus the path wise convergence is in both the pullback and forward directions 22 download by : skknchat@gmail.com Conclusions The main results of this dissertation include: 1) The existence and uniqueness of the solution of the nonautonomous stochastic differential equations driven by fractional Brownian motions and the properties of the solution 2) The generation of the stochastic two-parameter flow by the equation, and particularly the random dynamical system in the case of autonomy 3) Three theorems on the Lyapunov spectrum of the linear systems: the discretization scheme to compute the Lyapunov spectrum, the formula for the spectrum for regular triangular equation, the regularity almost sure of the nonautonomous equation in the sense of an probability measure 4) The criterion for the existence of global pullback attractor for the generated flow If the diffusion part is linear or bounded then system possesses a singleton attractor provided that the noise intensity is small 5) The construction of the Bebutov flow for nonautonomous fSDE which is a random dynamical system with an appropriate metric dynamical system 23 download by : skknchat@gmail.com List of Author’s Related Papers [1 ] N D Cong, L H Duc, P T Hong, Nonautonomous Young differential equations revisited, Journal of Dynamics and Differential Equations 30, (2018), 19211943 ă [2 ] L H Duc, P T Hong, Young differential delay equations driven by Holder continuous paths, book chapter Modern Mathematics and Mechanics, Springer International Publishing AG., (2019), 313-333 [3 ] N D Cong, L H Duc, P T Hong, Lyapunov spectrum of nonautonomous linear Young differential equations Journal of Dynamics and Differential Equations 32, (2020), 1749–1777 [4 ] L H Duc, P T Hong, N D Cong, Asymptotic stability for stochastic dissipative systems with a Holder noise SIAM Journal on Control and Optimization 57 (4), (2019), 3046–3071 [5 ] L H Duc, P T Hong, Asymptotic dynamics of Young differential equations Accepted by Journal of Dynamics and Differential Equations https://link.springer.com/article/10.1007%2Fs10884-021-10095-1 (2021) [6 ] N D Cong, L H Duc, P T Hong, Pullback attractors for stochastic Young differential delay equations, Journal of Dynamics and Differential Equations 34, (2022), 605–636 The results of this dissertation were presented at • The weekly seminar of the Department of Probability and Statistics, Institute of Mathematics, Vietnam Academy of Science and Technology • SEAM school ”Dynamical Systems and Birfucations Analysis”, August 2018, Penang, Malaysia • Workshop ”International Workshop on Probability Theory and Related Fields”, February 2019, Hanoi, Vietnam • Workshop ”Piecewise Deterministic Markov Processes and Applications”, July, 2019, Hanoi, Vietnam • Workshop ”Probability theory and Its applications”, August, 2019, Hanoi, Viet- nam • The 6th National conference ”Probability and Statistics: Research, applications and teaching”, November, 2020, Cantho, Vietnam 24 download by : skknchat@gmail.com ...The dissertation was written on the basis of the author’s research works carried at Institute of Mathematics, Vietnam Academy of Science and Technology Supervisor: Dr Luu Hoang Duc First... for the existence of a nonautonomous pullback attractor of the system Moreover we show that the attractor is a singleton and is also a forward attractor in case g is linear or bounded At the... pullback attracting property now can be written as lim d(Ψt,s (ω ) Ds |At (ω )) = 0, s→−∞ In the case the flow induced a random dynamical system ϕ, one has notation of random pullback attractor