Mục tiêu của luận án: Mục đích của luận án là chứng minh sự tồn tại duy nhất nghiệm của hệ phương trình vi phân ngẫu nhiên nhiễu bởi chuyển động Brown phân thứ, nghiên cứu tính chất của nghiệm, nghiên cứu số mũ Lyapunov của hệ tuyến tính và nghiên cứu tiêu chuẩn cho sự tồn tại tập hút pullback không autonome của dòng hai tham số ngẫu nhiên sinh bởi hệ. Nội dung của luận án gồm 3 phần chính. Phần 1: Phương trình vi phân ngẫu nhiên nhiễu bởi chuyển động Brown phân thứ. Phần 2: Phổ Lyapunov của hệ phương trình tuyến tính không autonome. Phần 3: Tập hút ngẫu nhiên của hệ phương trình không autonome. Các kết quả chính của luận án: Luận án đã đạt được các kết quả chính sau đây: 1. Chứng minh định lý về sự tồn tại duy nhất nghiệm của hệ phương trình không autonome nhiễu bởi chuyển động Brown phân thứ và một số tính chất của nghiệm. Chứng minh sự sinh dòng hai tham số ngẫu nhiên từ hệ và đặc biệt là sự sinh hệ động lực ngẫu nhiên khi các hàm hệ số không phụ thuộc thời gian. 2. Một số kết quả về phổ Lyapunov của hệ tuyến tính: lược đồ rời rạc hoá để tính toán phổ, công thức hiển cho phổ của hệ tam giác chính quy, tính chính quy hầu chắc chắn (theo nghĩa một độ đo xác suất) của hệ. 3. Tiêu chuẩn cho sự tồn tại tập hút pullback không autonome của dòng hai tham số ngẫu nhiên sinh bởi hệ: trường hợp tổng quát và hai trường hợp đặc biệt với hệ số khuếch tán tuyến tính, bị chặn. Xây dựng dòng Bebutov từ hệ và chỉ ra sự tồn tại tập hút ngẫu nhiên một điểm của dòng theo nghĩa pullback và forward.
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 DISSERTATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI - 2021 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 DISSERTATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS Supervisor: Dr Luu Hoang Duc HANOI - 2021 Confirmation This dissertation was written based on my research works at the Institute of Mathematics, Vietnam Academy of Science and Technology under the supervision of Dr Luu Hoang Duc I declare hereby that all the presented results have never been published by others April, 2021 The author Phan Thanh Hong i Acknowledgment First and foremost I am extremely grateful to my advisor Dr Luu Hoang Duc for continuous support of my academic research, for his invaluable advice, patience, motivation, and immense knowledge His guidance helped me in all the time of research and writing of this thesis I thank him for his encouragement and recommendation to the IMU Breakout Graduate Fellowship I would also like to express my special appreciation to Prof Dr.Sc Nguyen Dinh Cong for his enormous support I benefited a lot from his advices in the past few years Despite numerous other interests and busy academic life, Prof Cong has taken the time to read the draft and made precious suggestions for the contents of my thesis My sincere thanks also goes to all the members in the Probability and Statistics Department of the Institute of Mathematics I received many suggestions and experience through the seminars of the Department Furthermore, I thank my colleages at Thang Long University, for their support throughout my PhD study I specially thank Prof Dr.Sc Ha Huy Khoai for his support and encouragement I gratefully acknowledge the IMU Breakout Graduate Fellowship Program and the International Center for Research and Postgraduate Training in Mathematics - Institute of Mathematics for their financial support It is my honor to receive the grants And last but not least, I could not have finished this work without the unconditional support from my parents, my husband and my little children I would like to express my sincere gratitude to all of them ii Contents Table of Notation v Introduction vi Chapter Preliminaries 1.1 Fractional Brownian motions 1.1.1 Nonsemimartingale properties 1.1.2 Canonical spaces 1.2 Pathwise stochastic integrals with respect to fractional Brownian motions 1.2.1 Young integrals 1.2.2 Fractional integrals and fractional derivatives 1.2.3 Stochastic integrals w.r.t fractional Brownian motions 1.2.4 Young integrals on infinite domains 1.3 Greedy sequences of times 1.4 Stochastic flows 10 10 11 12 Chapter Stochastic differential equations driven by fractional Brownian motions 2.1 Assumptions 2.2 Existence and uniqueness theorem for deterministic equations 2.2.1 Existence and uniqueness of a global solution 2.2.2 Estimate of the solution growth 2.2.3 Special case: linear equations 2.3 Continuity and differentiability of the solution 2.3.1 The continuity of the solution 2.3.2 The differentiability of the solution 2.4 The stochastic differential equations driven by fBm 2.5 The generation of stochastic two parameter flows 2.6 Conclusions and discussions 14 15 16 17 27 29 31 32 33 38 40 43 iii 1 Chapter Lyapunov spectrum of nonautonomous linear fSDEs 3.1 The generation of stochastic flow of linear operators 3.2 Lyapunov exponent of Young integrals w.r.t B H 3.3 Lyapunov spectrum for nonautonomous linear fSDEs 3.3.1 Exponents and spectrum 3.3.2 Lyapunov spectrum of triangular systems 3.3.3 Lyapunov regularity 3.4 Almost sure Lyapunov regularity 3.5 Conclusions and discussions Chapter Random attractors for nonautonomous fSDEs 4.1 Nonautonomous attractors 4.2 Existence of random attractors 4.3 Special case: g linear 4.4 Special case: g bounded 4.5 Bebutov flow and its generation 4.6 Conclusions and discussions 44 44 46 50 51 55 60 62 66 67 69 73 84 86 94 102 General Conclusions 103 List of Author’s Related Papers 104 References 114 iv Table of Notations a∨b ∆n ∆[ a, b] |·| x ∞,[a,b] x p−var,[ a,b] x β−Hol,[ a,b] x L p (a,b) C([ a, b], Rd ) C ∞ ([ a, b], Rd ) C p−var ([ a, b], Rd ) C α-Hol ([ a, b], Rd ) C 0,p−var ([ a, b], Rd ) C 0,α−Hol ([ a, b], Rd ) 0,p−var C0 ([ a, b], Rd ) C00,α−Hol ([ a, b], Rd ) G( z ) a.s fBm SDE fSDE RDS w.r.t the maximum of a and b the closed interval [n, n + 1], n ∈ Z the simplex {(s, t) ∈ [ a, b]|s ≤ t} norm on the Euclide space Rd the supermum norm of function x on [ a, b] the p variation norm of function x on [ a, b] ă the Holder norm of function x on [ a, b] L p −norm of function x the space of Rd -valued continuous function on [ a, b] the subspace of smooth functions in C([ a, b], Rd ) the subspace of bounded p-variation functions in C([ a, b], Rd ) ă the subspace of Holder functions in C([ a, b], Rd ) the closure of C ∞ ([ a, b], Rd ) in C p−var ([ a, b], Rd ) the closure of C ∞ ([ a, b], Rd ) in C α−Hol ([ a, b], Rd ) the subspace of functions which vanish at in C 0,p−var ([ a, b], Rd ) the subspace of functions which vanish at in C 0,α−Hol ([ a, b], Rd ) Gamma function almost sure fractional Brownian motions stochastic differential equation SDE driven by fractional Brownian motions random dynamical system with respect to v 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 2H R H (s, t) = t + s2H − |t − s|2H It was originally defined and studied by Kolmogorov ( [57]) and then was developed by Mandelbrot and Van Ness in [65] It is a self-similar process ă with stationary increments and has Holder continuous sample paths with index β ∈ (0, H ) a.s For H > 1/2, the increments are positive correlated and for H < 1/2 they are negative correlated Moreover, it is a long memory process when H > 21 ( [71]) These significant properties make fractional Brownian motions a natural candidate to model the noise in applications to mathematical finance ( [18], [50], [37]), in hydrology, communication networks and in other fields (see for instance [48], [84]) When modelling real data which often include noises, stochastic differential equations is a powerful tool If noises are assumed to be fractional Brownian motions the problem of modelling becomes a stochastic differential equation driven by fBms which is understood in the integral form This leads to the need of definition of integral w.r.t fractional Brownian motions However, B H is not a semimartingale if H = 21 , one cannot apply the classical Ito theory to construct a stochastic integral w.r.t the fBm by taking the limit in the sense of probability convergence of a sequence of Darboux sums A modern development in the field of Stochastic Analysis deals with stochastic integrators which are more general than semimartingales 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 A comprehensive presentation of these theories can be found in Friz and Victoir [42] and in Samko et al [88] Both theory relies on properties of the sample paths For the case H > 1/2, the integral defined by rough path theory vi is understood in the Young sense and coincides with that defined by fractional ă derivative on the space of Holder continuous functions In the last decades, after the successful construction of integral w.r.t fBm, stochastic differential equations driven by fractional Brownian motions (in short fSDE) have attracted a lot of research interest In this thesis we study 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) to take advantage of the simplicity of Young integral System (1), like Ito differential equations, is understood as an integral equation of form t x t = x0 + t f (s, xs )ds + 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 The first important question is on the existence and uniqueness of solution to (1) The first study on the differential equations driven by rough signals dates back to [61] which is then generalized to introduce rough path theory ( [62], [63]) Using this approach, the existence of the solution of equations in a certain space of continuous functions with bounded p-variation is proved in [61] and [30], [78] The results are then generalized for the case < p < by [63] and [42], see also recent work by [77], [40] According to their settings, f , g are time - independent and/or g is often assumed to be differentiable and bounded in itself and its derivatives All can be applied to the stochastic differential equations driven by fBm (fSDE) Another approach follows Zăahle [86] by using fractional derivatives where the non autonomous systems are treated (see [74]) Similar results are established for system in infinite dimensional case, see for instance in [66], [7] Since our target is the equations driven by fBm with H > 1/2 which can be studied by these two approaches, we aim to close the gap between the two methods and develop techniques to study more on the infinite dimensional cases ( [34]) and on the dynamic of these systems ( [23], [22]) At first we prove that, under similar assumptions to those in [74], 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 vii an appropriate probability space, it is proved that the system generates a random dynamical system (in short RDS, see [16], [44], and [5]) 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 some qualitative problems of the systems under the framework of RDS theory with typical topics: random attractor, stability, invariant manifolds and so on (see for instane [72], [3], [4]) In the scope of this thesis we focus on studying the Lyapunov spectrum of linear systems and the random attractor of semilinear equations Note that these problems are still open even for the case H > 1/2 (see recent results in [42], [32]) Random attractor is one of the most important notation of random dynamical system Its generalization, nonautonomous random attractor is introduced to stochastic flow where the state of the system depends on both the initial and present time ( [24]) 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 In the case g is linear, the attractor is singleton and also forward attractor For the nonlinear case, under some additional conditions we point out that the attractor is one point in the sense of the Bebutov flow generated by the equation which is a RDS on the appropriate space of noise These techniques show the capability to deal with the rough equation in the work by [39], or in the paper for infinite dimensional case by [22], [43] 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 Oseledets [75] It was also investigated by Millionshchikov in [67–70] for linear nonautonomous differential equations In the stochastic setting, the MET is also formulated for random dynamical systems in [4, Chapter 3] Further investigations can be found in [19, 20] 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 We use the approach developed in [19] to study the Lyapunov spectrum of the system We show that Lyapunov exponents can be computed based on the discretization scheme And moreover, the spectrum is bounded by a nonrandom constant We are also interested in the question on the non-randomness of Lyapunov exponents In case the system is driven by standard Brownian viii = Sn + α n u n + β n ≤ Sn + α n Sn + β n ≤ Tn (1 + αn ) + β n n −1 ≤ max{ a, u0 } ∏ (1 + αk ) + k =0 n ≤ max{ a, u0 } ∏ (1 + αk ) + k =0 n −1 ∑ k =0 n n −1 βk ∏ j = k +1 n −1 ∑ βk ∏ k =0 (1 + α j ) (1 + α n ) + β n (1 + α j ) = Tn+1 j = k +1 Since un ≤ Sn , (A2) holds Spaces of functions Variation and Holder ă spaces The content in this part is from the book [42] Let C([ a, b], Rd ) denote the space of all continuous paths x : [ a, b] → Rd , t → xt equipped with the supremum norm · ∞,[ a,b] given by x ∞,[ a,b] = supt∈[ a,b] | xt |, where | · | is the Euclidean norm in Rd For p ≥ and [ a, b] ⊂ R, a continuous path x : [ a, b] → Rd is called of finite p-variation if n ||| x ||| p-var,[a,b] := sup ∑ | x t i +1 − x t i | 1/p p Π[ a,b] i =1 < ∞, where the supremum is taken over the whole class of finite partitions Π[ a, b] = { a = t0 < t1 < · · · < tn = b} of [ a, b] The subspace C p−var ([ a, b], Rd ) ⊂ C([ a, b], Rd ) consists of all paths x with finite p-variation and equipped with the p-var norm x p-var,[ a,b] := | x a | + ||| x ||| p-var,[a,b] , is a nonseparable Banach space [42, Theorem 5.25, p 92] In the following definition, the notion of control or control function is defined on the simplex ∆[ a, b] := {(s, t) : a ≤ s ≤ t ≤ b} Definition A.4 ( [42, Definition 1.7]) A continuous map ω : ∆[ a, b] −→ R+ is called a control (on [ a, b]) if it is zero on the diagonal and superadditive, i.e (i ) For all t ∈ [ a, b], ω (t, t) = 0, (ii ) For all s ≤ t ≤ u in [ a, b], ω (s, t) + ω (t, u) ≤ ω (s, u) Example A.4 The following functions are controls on [ a, b] 106 ω (s, t) = (t − s)θ , θ ≥ 1, q ω (s, t) = ||| x ||| p−var,[s,t] where x ∈ C p−var ([ a, b], Rd ), p ≥ is given and q ≥ p, t s bu du ω (s, t) = where b is a nonnegative integrable function on [ a, b], (φ ◦ ω )(s, t) where ω (s, t) is a control and φ : [0, ∞) → [0, ∞) is increasing, convex and vanish at The following lemmas, which are more general results of Propostion 5.10 and Exercise 5.11 in [42], give us useful properties of controls in relation with the variations of a path Lemma A3 Let ω j be a finite sequence of control functions on [ a, b], Cj > 0, j = 1, k, p ≥ and x : [ a, b] → Rd be a continuous path satisfying k | xt − xs | ≤ ∑ Cj ω j 1/p (s, t), ∀s < t ∈ [ a, b], i= j then k ||| x ||| p−var,[a,b] ≤ ∑ Cj ω1/p (a, b) j =1 Proof Consider an arbitrary finite partition Π = (si ), i = 0, , n + of [ a, b] By assumption and Minskowski inequality we have 1/p n ∑ | x s i +1 − x s i | p k i =0 j =1 k n j =1 i =0 ∑ ∑ ≤ i =0 ≤ 1/p C j ω j ( s i , s i +1 ) 1/p ∑ ∑ k ≤ p 1/p n ∑ Cj ω j p C j ω j ( s i , s i +1 ) 1/p ( a, b) j =1 ✷ This implies the conclusion of the lemma Lemma A4 Let x ∈ C p−var ([ a, b], Rd ), p ≥ If a = a1 < a2 < · · · < ak = b, then k −1 ∑ i =1 p ||| x ||| p-var,[a ,a i i+1 ] p ≤ ||| x ||| p-var,[a,b] ≤ (k − 1) p−1 k −1 ∑ |||x||| p-var,[ai,ai+1 ] p i =1 Proof The proof is similar to that in [42, p 84], by using triangle inequality and power means inequality n zi ≤ n i∑ =1 n r zi n i∑ =1 1/r 107 , zi 0, r ă For < α ≤ denote by C α-Hol ([ a, b], Rd ) the space of all Holder continuous paths x : [ a, b] → Rd with exponential α, equipped with the norm x α-Hol,[ a,b] := | x a | + ||| x |||α-Hol,[ a,b] = | x a | + | xt − xs | < ∞ α ( t − s ) a≤s p, xn converges along a subsequence to some x ∈ C p−var ([ a, b], Rd ) (ii ) If ( xn ) is bounded and supn ||| x |||α−Hol,[a,b] < ∞ then for each α < α, xn converges along a subsequence to some x ∈ C α−Hol ([ a, b], Rd ) Closure of smooth paths in variation norm, Holder ă norm Define C 0,pvar ([ a, b], Rd ), C 0,α−Hol ([ a, b], Rd ) as the closure of C ∞ ([ a, b], Rd )-the space of smooth functions on [ a, b], in C p−var ([ a, b], Rd ) and C α−Hol ([ a, b], Rd ) respectively Thus they are Banach space and so are the spaces 0,p−var C0 ([ a, b], Rd ) := { x ∈ C 0,p−var ([ a, b], Rd )| x0 = 0}, and C00,α−Hol ([ a, b], Rd ) := { x ∈ C 0,α−Hol ([ a, b], Rd )| x0 = 0} Moreover, these are separable spaces which can be defined as C 0,p−var ([ a, b], Rd ) = x ∈ C p−var ([ a, b], Rd ) | lim sup δ →0 ∑ Π( a,b), t ∈ Π i |Π|≤δ | x t i +1 − x t i | p = , and C 0,α−Hol ([ a, b], Rd ) = x ∈ C α−Hol ([ a, b], Rd ) | lim sup δ→0 a≤s and put Kδ := {c ∈ A¯ | m[a,b] (c, δ) ≥ ε} The Kδ are closed for all δ Due to the fact that lim m[ a,b] (c, δ) = for all c ∈ C˜ δ →0 Kδ = ∅ Then there exists δ = δ(ε) > such that Kδ = ∅, which we have δ >0 proves (3.33) 111 For the ”only if” part, assume (3.32) and (3.33) and prove the compactness of H¯ Since C˜ is a complete metric space, it suffices to prove that every sequence {cn }∞ n=1 ⊂ H has a convergent subsequence Now following the arguments of [55, Theorem 4.9, p 63] line by line, we can construct a convergent subsequence {c˜n }∞ n=1 by the ”diagonal sequence” such that c˜n (r ) → c (r ) as n → ∞ for any rational number r ∈ Q With (3.32) and (3.33), H satisfies the condition in [55, Theorem 4.9, p 63], hence c˜n converge uniformly to a continuous function c in every [ a, b] ⊂ R Fix [ a, b], by (3.33) for each ε > there exist δ0 > such that if δ ≤ δ0 , |c˜ (t)−c˜ (s)| sup s,t∈[a,b] n |t−s|nα ≤ ε for all n Hence |s−t|≤δ sup s,t∈[ a,b] |s−t|≤δ |c(t) − c(s)| ≤ε |t − s|α ˜ Finally, we prove that c˜n converge to c in the Holder ă and then c C seminorm on every compact interval [ a, b] Namely, with ε, δ0 given, there exist n0 such that for all n ≥ n0 , c˜n − c ∞,[ a,b] ≤ δ0α ε We then have for n ≥ n0 |(c˜n − c)(t) − (c˜n − c)(s)| ≤ |t − s|α s,t∈[ a,b] sup sup s,t∈[ a,b] |t−s|≤δ0 |(c˜n − c)(t) − (c˜n − c)(s)| |t − s|α + sup s,t∈[ a,b] |t−s|≥δ0 |(c˜n − c)(t) − (c˜n − c)(s)| |t − s|α ≤ m[a,b] (c˜n , δ0 ) + m[a,b] (c, δ0 ) + c˜n − c ∞,[ a,b] δ0α ≤ 4ε This implies |||c˜n − c|||α−Hol,[ a,b] converge to as n → ∞ This complete the proof ✷ Tempered variables Let (Ω, F , P) be a probability space equipped with an ergodic metric dynamical system θ, which is a P measurable mapping θ : T × Ω → Ω, T is either R or Z, and θt+s = θt ◦ θs for all t, s ∈ T Recall that a random variable ρ : Ω → [0, ∞) is called tempered if log+ ρ(θt ω ) = 0, t→±∞ t lim 112 a.s which, as shown in [52, p 220], [44], is equivalent to the sub-exponential growth lim e−c|t| ρ(θt ω ) = t→±∞ a.s ∀c > Note that our definition of temperedness corresponds to the notion of temperedness from above given in [4, Definition 4.1.1(ii)] Lemma A5 (i) If h1 , h2 ≥ are tempered random variables then h1 + h2 and h1 h2 are tempered random variables (ii) If h1 ≥ is a tempered random variable, h2 ≥ is a measurable random variable and h2 ≤ h1 almost surely, then h2 is a tempered random variable (iii) Let h1 be a nonnegative measurable function If log+ h1 ∈ L1 then h1 is tempered Proof (i ) See [4, Lemma 4.1.2, p 164] (ii ) Immediate from the definition of tempered random variable (iii ) See [4, Proposition 4.1.3, p 165] ✷ Lemma A6 Let c : Ω → [0, ∞) be a tempered random variable, and δ > be an arbitrary fixed positive number Put ∞ d(ω ) := ∑ e−δk c(θ−k ω ) k =1 Then d(·) is a nonnegative almost everywhere finite and tempered random variable Proof Put dn (ω ) := ∑nk=1 e−δk c(θ−k ω ) Then dn (·), n ∈ N, is an increasing sequence of nonnegative random variable, hence converges to the nonnegative random ˜ ⊂ Ω of variable d(·) By temperedness of c(·) we can find a measurable set Ω ˜ there exists n0 (ω ) > such that for all full measure such that for all ω ∈ Ω n ≥ n0 (ω ) we have c(θ−n ω ) ≤ enδ/2 Hence dn (ω ), n ∈ N, is an increasing sequence of positive numbers tending to finite value d(ω ) Thus d(·) is finite ˜ we have almost everywhere Furthermore, for m ∈ N and x ∈ Ω ∞ d (θ−m ω ) = ∑e −δk ∞ c (θ−k θ−m ω ) ≤ e k =1 This implies that lim supm→∞ 4.1.3], d(·) is tempered δm ∑ e−δl c(θ−l ω ) = eδm d(ω ) l =1 m log+ d(θ−m ω ) ≤ δ Following [4, Proposition ✷ 113 References [1] E Alos, O Mazet, D Nualart Stochastic calculus with respect to Gaussian processes The Annals of Probability, 29, No 2, (2001), 766–801 [2] H Amann Ordinary Differential Equations: An Introduction to Nonlinear Analysis Walter de 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