The asymptotic behaviour of solutions for stochastic evolution equations with pantograph delay

March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo Stochastics and Dynamics c World Scientific Publishing Company THE ASYMPTOTIC BEHAVIOUR OF SOLUTIONS FOR STOCHASTIC EVOLUTION EQUATIONS WITH PANTOGRAPH DELAY YARONG LIU School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China [email protected]du.cn YEJUAN WANG∗ School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China [email protected]du.cn TOMAS CARABALLO Depto. de Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, c/ Tarfia s/n, 41012 Seville, Spain carabal[email protected] Received (Day Month Year) Revised (Day Month Year) The polynomial stability problem of stochastic delay differential equations has been studied in recent years. In contrast, there are relatively few works on stochastic partial differential equations with pantograph delay. The present paper is devoted to investigating large-time asymptotic properties of solutions for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. We first show that the mild solutions of stochastic pantograph delay evolution equations with nonlinear multiplicative noise tend to zero with general decay rate (including both polynomial and logarithmic rates) in the pth moment and almost sure senses. The analysis is based on the Banach fixed point theorem and various estimates involving the gamma function. Moreover, by using a generalized version of the factorization formula and exploiting an approximation technique and a convergence analysis, we construct the nontrivial equilibrium solution, defined for t∈R, for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. In particular, the uniqueness, H¨older regularity in time and general stability, in the pth moment and almost sure senses, of the nontrivial equilibrium solution are established. Keywords: Pantograph delay; Moment general stability; Almost sure general stability; Nonlinear multiplicative noise; Nontrivial equilibrium solution. AMS Subject Classification: 60H15, 60G15, 35B35 ∗Corresponding author. 1 March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo 2Liu, Wang & Caraballo 1. Introduction The stochastic delay differential equations (SDDEs) take into account the perturbations and delays often present in the real world. The stochastic pantographdelay differential equations are a kind of SDDEs with unbounded delays. Pantograph equations arise in a wide range of applications such as small vertical displacements of a stretched string under gravity [6, 20]. In recent years, stochastic evolution equations have received a great deal of attention. However, there is little work on stochastic evolution equations with pantograph delay. In this work, Our purpose is to investigate the long time behavior of the following stochastic pantograph delay evolution equations with nonlinear multiplicative noise du(t) = −Au(t)dt +f(t, u(ηt))dt +g(t, u(ηt))dBQ(t), t ≥0, η ∈(0,1),(1.1) satisfying the initial condition u(0) = u0.(1.2) Here −Ais a closed, densely defined linear operator generating an analytic semigroup S(t), t≥0 on the space Hand BQis a K-valued Brownian motion. In what follows, we assume that the mappings [0,∞)3t7→ f(t, µ)∈Hand [0,∞)3t7→ g(t, µ)∈L0 Q(K,H) are measurable for any µ∈Lp(Ω; Hλ), where H,K,L0 Q(K,H) and Lp(Ω; Hλ) will be introduced later. Firstly, we are interested in the existence, uniqueness, pth moment general stability and almost sure general stability of mild solutions for problem (1.1)-(1.2). The analysis is based on the Banach fixed point theorem and various estimates involving the gamma function. Note that many existing works are concerned with the polynomial and exponential stability of SDEs with delay or without delay by using the Razumikhin technique and Lyapunov functions; see for example, [7,11,17,22,24,25] and the references therein. For some related works on stability of stochastic differential equations, we mention the interesting papers [2,10,12,16,18] and references therein. However, there are some systems which are not exponentially stable or polynomially stable, but the solutions do tend to zero asymptotically. Therefore, it is necessary to study general stability. Authors in [8] have considered almost sure stability with general decay rate of the exact solutions for stochastic pantograph differential equations. The moment general stability of exact solutions of the stochastic pantograph differential equation has been investigated in [9]. Moreover, we construct a unique solution u∗, defined for all t∈R, for problem (1.1)-(1.2). In particular, the mean-pH¨older regularity, pth moment general stability and mean-palmost sure general stability of u∗are also established. The existence and uniqueness of u∗follow from constructing a Cauchy convergent sequence of linear versions and using the generalized version of the factorization formula and convergence analysis. Then the Banach fixed theorem allows us to show that the limit u∗has pth moment and almost sure stability with general decay rate. To the best of our knowledge, there are no results on the construction and stability of the March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo Stochastic pantograph delay evolution equations 3 nontrivial equilibrium solution for stochastic differential equations with delay. The mean-square exponential stability of the nontrivial equilibrium solution on t∈R has been established for stochastic reaction-diffusion equations in [19]. It is worth mentioning that, because of the difficulties caused by pantograph delays, we cannot prove the stability with the exponential decay as in [19, Theorem 3]. The presence of pantograph delays also makes the analysis more complicated. Here by using the Banach fixed theorem, we obtain the pth moment stability of the nontrivial equilibrium solution on t∈Rwith general decay rate (including the polynomial rate and the logarithmic rate). Furthermore, the almost sure general stability of the nontrivial equilibrium solution is also addressed. In addition, H¨older regularity in time of the nontrivial equilibrium solution is given. The paper is organized as follows. In Section 2, we present some notations and technical lemmas. The results of pth moment and almost sure α-type stability are considered in Section 3 for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. In Section 4, we shall show the existence and uniqueness of the nontrivial equilibrium solution by constructing the stochastic process u∗. Furthermore, pth moment and almost sure stability with a general decay function α(t) are established for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. We also establish H¨older regularity in time of the nontrivial equilibrium solution. A summary of this work is provided in Section 5. In the end the proof of Theorem 2.1 and a technical proposition are given in the appendix. Throughout this paper, we denote by Cand Creal positive constants which can vary from a line to another and even in the same line. Moreover, let constants Cs and C(s) denote Cand Cdepend on some variable s, respectively. 2. Preliminaries We define the Banach space Hλ=D(Aλ), where D(Aλ) denotes the domain of the fractional power operator Aλ:H→H. The norm is given by kgkλ:= kAλgkfor g∈Hλ. Denote by Lp(Ω; Hλ) = Lp(Ω,F,P;Hλ) the set of all strongly-measurable, Lpintegrable Hλ-valued random variable. For any g∈Lp(Ω; Hλ) define its norm by kg(·)kLp(Ω;Hλ)=Ekg(·)kp λ1 p. Let Cc, d;Lp(Ω; Hλ)denote the Banach space of all continuous functions from (c, d) into Lp(Ω; Hλ) equipped with the sup norm supt∈[c,d]Ekg(t)kp λ1 p. Let Kbe a separable Hilbert space endowed with a complete orthonormal basis {ei}i∈N. We denote by Hanother Hilbert space with norm k·kand inner product (·,·). Denote by L(K,H) the space of all bounded linear operators from Kinto H. We use the same notation k·kfor the norms of Kand L(K,H), and use (·,·) to denote the inner product of Kfor convenience. Let Q∈L(K,K) be an operator March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo 4Liu, Wang & Caraballo defined by Qei=λieiwith finite trace trQ =P∞ i=1 λi<∞. Let φ∈L(K,H) and define kφk2 Q:= Tr(φQφ∗) = ∞ X i=1  pλiφei 2,(2.1) where φ∗is the adjoint of the operator φ. If kφk2 Q<∞, then φis called a Q-HilbertSchmidt operator. Here L0 Q(K,H) denotes the space of all Q-Hilbert-Schmidt operators from Kinto H. Let (Ω,F,(Ft)t≥0,P) be a complete probability space where Fis the σ-algebra of measurable subsets of Ω, Pis the probability measure and Ftis a right-continuous filtration. Here {Ft}t≥0denotes the filtration generated by Bi(t), that is, Ft:= σ{Bi(s) : 0 ≤s≤t;i≥1}.(2.2) Denote by BQ(t) the Brownian motion adapted to the filtration (Ft)t≥0. We assume that BQ(t) = ∞ X i=1 pλiBi(t)ei, t ≥0, where {Bi(t); t≥0}i≥1is a sequence of one-dimensional standard Brownian motions mutually independent over (Ω,F,P). The following lemma is needed in this paper. Lemma 2.1. (See [3, Theorem 4.36])If φ: [0, T ]×Ω→L0 Q(K,H)is a progressively measurable function satisfying ERT 0kφ(s)k2 Qdsp 2<∞, then for any t∈[0, T], E  Zt 0 φ(s)dBQ(s)   p≤CpEZt 0 kφ(s)k2 Qdsp 2,(2.3) where Cp>0is a positive constant depending on pand p≥2. We will also need the following theorem which is a corollary of the stochastic Fubini theorem (see, e.g. [4, Theorem 5.2.5]). For the convenience of the reader the proof is given in Appendix A. This result is often referred to as the factorization formula. Theorem 2.1. Assume that for some α∗∈(0,1) and all t∈[t0, T], Zt t0 (t−s)α∗−1hEZs t0 (s−r)−2α∗ S(t−r)φ(r) 2 Qdrp 2i1 pds < +∞.(2.4) Then BA(t) = sin α∗π πZt t0 (t−s)α∗−1S(t−s)Yα∗(s)ds, t ∈[t0, T],(2.5) where t0∈R,p≥2and BA(t) = Zt t0 S(t−s)φ(s)dBQ(s), Yα∗(s) = Zs t0 (s−r)−α∗S(s−r)φ(r)dBQ(r). March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo Stochastic pantograph delay evolution equations 5 3. General stability of mild solutions for PDEs with nonlinear multiplicative noise This section mainly focuses on pth moment and almost sure stability with general decay rate α(t). First we introduce the following α-type function: (S0) (1) α∈C(R+,R+) is increasing; (2) α(0) >0 and limt→∞ α(t) = ∞; (3) α(t) satisfies that lim sup t→∞ e−δpt 4Zt 2 0 e−δ(t 2−τ)(α(ητ))−1dτ →0 and lim sup t→∞ e−δpt 4α(t)→0, where ηand δare given in (1.1) and the assumption (S1) below, respectively. (4) There exists a positive constant c∗such that lim sup t→∞ α(t) α(ηt/2) =c∗, where η∈(0,1) is given in (1.1). It is clear that α(t) = 1 + tξ∗(0 < ξ∗<1) and α(t) = log(2 + t) satisfy the above requirements. To study the stability of mild solutions with general decay rate α(t), we need the following assumptions: (S1) There exist a real number δ > 0 and positive constants C0, Cλ,0≥1 such that for any x∈H,  AλS(t)x ≤Cλ,0e−δtt−λkxk, t > 0,  S(t)x ≤C0e−δtkxk, t ≥0. (S2) There exist nonnegative functions L1, L2∈L∞(R+) such that for any u, v ∈ Lp(Ω; Hλ) and t≥0, E f(t, u)−f(t, v) p≤L1(t)Eku−vkp λ, E g(t, u)−g(t, v) p Q≤L2(t)Eku−vkp λ. (S3) There exist nonnegative functions l1, l2∈L∞(R+) such that for any t≥0, kf(t, 0)kp≤l1(t),kg(t, 0)kp Q≤l2(t), and Z∞ 0α(τ)l1(τ)qdτ1 q:= Ξ1<∞, Z∞ 0α(τ)l2(τ)q1dτ1 q1:= Ξ2<∞, where 1/p + 1/q = 1 and 1/q1= 1 −1/p1with p1∈(1,1 2λ) and λ∈(0,1 p). March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo 6Liu, Wang & Caraballo Let us state the following definition of mild solution to problem (1.1)-(1.2). Definition 3.1. Let T > 0 and u0be an F0-measurable initial process satisfying Eku0kp λ<∞. An Ft-measurable stochastic process u(t) is called a mild solution of problem (1.1)-(1.2) on [0, T] if u∈C(0, T ;Lp(Ω; Hλ)) and for t∈[0, T], u(t) = S(t)u0+Zt 0 S(t−τ)f(τ, u(ητ))dτ+Zt 0 S(t−τ)g(τ, u(ητ))dBQ(τ),P-a.s.(3.1) Remark 3.1. In fact, by similar arguments in Section 4, the solution u(t) defined by Eq. (3.1) has continuous trajectories with probability 1. The following theorem shows that mild solutions to Eq. (1.1)-(1.2) are pth moment α-type stable. Theorem 3.1. Let p≥2,λ∈(0,1 p)and u0∈Lp(Ω; Hλ). Suppose that assumptions (S0)-(S3)hold. Let kL1kL∞(R+)and kL2kL∞(R+)be sufficiently small such that 2pCp λ,0δλ−1Γ(1 −λ)pkL1kL∞(R+)<1, 2pCpCp λ,0(2δ)2λ−1Γ(1 −2λ)p 2kL2kL∞(R+)<1, 22pc∗Cp λ,0δλ−1Γ(1 −λ)pkL1kL∞(R+)<1, 22pc∗CpCp λ,0(2δ)2λ−1Γ(1 −2λ)p 2kL2kL∞(R+)<1, (3.2) where c∗,δ,Cλ,0and Cpare given in the assumptions (S0)-(S1)and Lemma 2.1, respectively. Then problem (1.1)-(1.2) has a unique global mild solution usatisfying sup r∈[0,∞) α(r)Eku(r)kp λ<∞.(3.3) Proof. We consider the abstract phase space Cp,λ ϑ=Cϑ0,∞;Lp(Ω; Hλ)equipped with the norm kukϑ= sup t∈[0,∞) ϑ(t)Eku(t)kp λ, u ∈C0,∞;Lp(Ω; Hλ), where ϑ(t) = (α(T), t ∈[0, T], α(t), t ≥T, (3.4) with T > 0 given later. Then Cp,λ ϑ,k·kϑis a Banach space. In order to apply the Banach fixed point theorem, we shall prove that the mapping e Tdefined by (e Tu)(t) = S(t)u0+Zt 0 S(t−τ)f(τ, u(ητ))dτ +Zt 0 S(t−τ)g(τ, u(ητ))dBQ(τ),(3.5) is contractive and bounded on Cp,λ ϑ. Step 1. It follows immediately from (3.5) that ϑ(t)E (e Tu)(t)−(e Tv)(t) p λ March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo Stochastic pantograph delay evolution equations 7 ≤2p−1ϑ(t)EZt 0 S(t−τ)f(τ, u(ητ)) −f(τ, v(ητ)) λdτp + 2p−1ϑ(t)E  Zt 0 AλS(t−τ)g(τ, u(ητ)) −g(τ, v(ητ))dBQ(τ)   p := R1+R2.(3.6) In view of assumptions (S1)-(S2) and H¨older’s inequality, we deduce that for t∈ [0, T] and any u, v ∈Cp,λ ϑ, R1≤2p−1α(T)Cp λ,0EZt 0 e−δ(t−τ)(t−τ)−λ f(τ, u(ητ)) −f(τ, v(ητ)) dτp ≤2p−1α(T)Cp λ,0Zt 0 e−δ(t−τ)(t−τ)−λdτp−1 ×Zt 0 e−δ(t−τ)(t−τ)−λE f(τ, u(ητ)) −f(τ, v(ητ)) pdτ ≤2p−1Cp λ,0δλ−1Γ(1 −λ)pkL1kL∞(R+)ku−vkϑ.(3.7) On the other hand, for t≥Tand any u, v ∈Cp,λ ϑ, R1≤22p−2α(t)EZt 2 0 S(t−τ)f(τ, u(ητ)) −f(τ, v(ητ)) λdτp + 22p−2α(t)EZt t 2 S(t−τ)f(τ, u(ητ)) −f(τ, v(ητ)) λdτp := R1 1+R2 1.(3.8) Applying H¨older’s inequality and assumptions (S1)-(S2) results in R1 1≤22p−2α(t)Cp λ,0EZt 2 0 e−δ(t−τ)(t−τ)−λ f(τ, u(ητ)) −f(τ, v(ητ)) dτp ≤22p−2α(t)Cp λ,0t 2−pλZt 2 0 e−δ(t−τ)dτp−1 ×Zt 2 0 e−δ(t−τ)E f(τ, u(ητ)) −f(τ, v(ητ)) pdτ (3.9) ≤22p−2α(t)Cp λ,0ku−vkϑkL1kL∞(R+)t 2−pλ e−δpt/2 δp−1Zt 2 0 e−δ(t/2−τ)(α(ητ))−1dτ and R2 1≤22p−2Cp λ,0α(t)EZt t 2 e−δ(t−τ)(t−τ)−λ f(τ, u(ητ)) −f(τ, v(ητ)) dτp ≤22p−2Cp λ,0α(t)Zt t 2 e−δ(t−τ)(t−τ)−λdτp−1 ×Zt t 2 e−δ(t−τ)(t−τ)−λE f(τ, u(ητ)) −f(τ, v(ητ)) pdτ March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo 8Liu, Wang & Caraballo ≤22p−2Cp λ,0δλ−1Γ(1 −λ)pku−vkϑkL1kL∞(R+) α(t) α(ηt/2).(3.10) Combining (3.7) and (3.8)-(3.10), we can find Tlarge enough such that for all t≥0, R1<1 2ku−vkϑ,(3.11) thanks to assumptions (3.2) and (S0). Now it remains to estimate the stochastic term. In view of Lemma 2.1, H¨older’s inequality and assumptions (S1)-(S2), we find that for t∈[0, T], R2≤2p−1α(T)CpEZt 0 AλS(t−τ)g(τ, u(ητ)) −g(τ, v(ητ)) 2 Qdτp 2 ≤2p−1α(T)CpCp λ,0EZt 0 e−2δ(t−τ)(t−τ)−2λ g(τ, u(ητ)) −g(τ, v(ητ)) 2 Qdτp 2 ≤2p−1α(T)CpCp λ,0Zt 0 e−2δ(t−τ)(t−τ)−2λdτp−2 2 ×Zt 0 e−2δ(t−τ)(t−τ)−2λE g(τ, u(ητ)) −g(τ, v(ητ)) p Qdτ ≤2p−1CpCp λ,0(2δ)2λ−1Γ(1 −2λ)p 2kL2kL∞(R+)ku−vkϑ,(3.12) and for t≥T, R2≤22p−2α(t)E  Zt 2 0 S(t−τ)g(τ, u(ητ)) −g(τ, v(ητ))dBQ(τ)   p λ + 22p−2α(t)E  Zt t 2 S(t−τ)g(τ, u(ητ)) −g(τ, v(ητ))dBQ(τ)   p λ := R1 2+R2 2.(3.13) It follows from Lemma 2.1, H¨older’s inequality and assumptions (S1)-(S2) that R1 2≤22p−2α(t)CpEZt 2 0 AλS(t−τ)g(τ, u(ητ)) −g(τ, v(ητ)) 2 Qdτp 2 ≤22p−2α(t)CpCp λ,0t 2−pλ ×EZt 2 0 e−2(p−2) pδ(t−τ)e−4 pδ(t−τ) g(τ, u(ητ)) −g(τ, v(ητ)) 2 Qdτp 2(3.14) ≤22p−2α(t)CpCp λ,0ku−vkϑkL2kL∞(R+) ×t 2−pλ e−δpt/2 (2δ)p−2 2Zt 2 0 e−δ(t−2τ)(α(ητ))−1dτ, March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo Stochastic pantograph delay evolution equations 9 and R2 2≤22p−2α(t)CpEZt t 2 AλS(t−τ)g(τ, u(ητ)) −g(τ, v(ητ)) 2 Qdτp 2 ≤22p−2α(t)CpCp λ,0Zt t 2 (t−τ)−2λe−2δ(t−τ)dτp−2 2 ×Zt t 2 (t−τ)−2λe−2δ(t−τ)E g(τ, u(ητ)) −g(τ, v(ητ)) p Qdτ ≤22p−2CpCp λ,0(2δ)2λ−1Γ(1 −2λ)p 2ku−vkϑkL2kL∞(R+) α(t) α(ηt/2). (3.15) Inserting (3.14)-(3.15) into (3.13) gives R2≤22p−2CpCp λ,0t 2−pλ e−δpt/2 (2δ)p−2 2Zt 2 0 e−δ(t−2τ)(α(ητ))−1dτ +(2δ)2λ−1Γ(1 −2λ)p 2α(t) α(ηt/2)kL2kL∞(R+)ku−vkϑ. (3.16) Then, by assumptions (3.2) and (S0), in view of (3.12) and (3.16), we can take T sufficiently large such that for any t≥0, R2<1 2ku−vkϑ.(3.17) This together with (3.11) and (3.6) implies that e Tis contractive on the space Cp,λ ϑ. Step 2. By (3.5) we have ϑ(t)E (e Tu)(t) p λ ≤3p−1ϑ(t)Cp 0e−δptEku0kp λ+ 6p−1ϑ(t)EZt 0 S(t−τ)f(τ, 0) λdτp + 6p−1ϑ(t)E  Zt 0 AλS(t−τ)g(τ, 0)dBQ(τ)   p + 6p−1ϑ(t)EZt 0 S(t−τ)f(τ, u(ητ)) −f(τ, 0) λdτp + 6p−1ϑ(t)E  Zt 0 AλS(t−τ)g(τ, u(ητ)) −g(τ, 0)dBQ(τ)   p := 3p−1ϑ(t)Cp 0e−δptEku0kp λ+R3+R4+R5+R6. (3.18) Following similar calculations as in (3.8)-(3.10) and (3.13)-(3.15), we conclude that R5+R6≤3p−1kukϑ,for t≥T, (3.19) March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo 16 Liu, Wang & Caraballo (II) supt∈REk˜u∗(t)kp λ<∞. On account of assumption (S1), we have E  Zt −∞ S(t−τ)ζ(τ)dτ   p λ ≤Cp λ,0EZt −∞ e−p−1 pδ(t−τ)(t−τ)−p−1 pλζ(τ)e−1 pδ(t−τ)(t−τ)−1 pλdτp ≤Cp λ,0δλ−1Γ(1 −λ)psup t∈R Ekζ(t)kp.(4.13) Thanks to Lemma 2.1, by a similar reasoning as in (4.13), we obtain that E  Zt −∞ S(t−τ)ψ(τ)dBQ(τ)   p λ ≤CpCp λ,0(2δ)2λ−1Γ(1 −2λ)p 2sup t∈R Ekψ(t)kp Q.(4.14) By (4.13) and (4.14), the assertion follows from (4.8). (III) The process ˜u∗(t) satisfies (4.5). ˜u∗(t) = Zt0 −∞ S(t−t0)S(t0−τ)ζ(τ)dτ +Zt0 −∞ S(t−t0)S(t0−τ)ψ(τ)dBQ(τ) +Zt t0 S(t−τ)ζ(τ)dτ +Zt t0 S(t−τ)ψ(τ)dBQ(τ) (4.15) =S(t−t0)˜u∗(t0) + Zt t0 S(t−τ)ζ(τ)dτ +Zt t0 S(t−τ)ψ(τ)dBQ(τ). Step 3. The H¨older regularity, exponential stability and uniqueness of ˜u∗(t). Now we show that ˜u∗(t) is continuous in time. It follows from (4.8) that, for each h > 0,  ˜u∗(t+h)−˜u∗(t) Lp(Ω;Hλ) ≤  Zt −∞ S(t+h−τ)−S(t−τ)ζ(τ)dτ  Lp(Ω;Hλ) +  Zt −∞ S(t+h−τ)−S(t−τ)ψ(τ)dBQ(τ)  Lp(Ω;Hλ) +  Zt+h t S(t+h−τ)ζ(τ)dτ  Lp(Ω;Hλ) +  Zt+h t S(t+h−τ)ψ(τ)dBQ(τ)  Lp(Ω;Hλ) := R12 +R13 +R14 +R15.(4.16) March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo Stochastic pantograph delay evolution equations 17 To deal with the term R13, let us consider Zt+h t  Zt −∞ AS(s−τ)ψ(τ)dBQ(τ)  Lp(Ω;Hλ)ds. Thanks to Lemma 2.1, in view of assumption (S1) and H¨older’s inequality, we deduce that Zt+h t  Zt −∞ AS(s−τ)ψ(τ)dBQ(τ)  Lp(Ω;Hλ)ds =Zt+h tE  Zt −∞ A1+λS(s−τ)ψ(τ)dBQ(τ)   p1 pds ≤(Cp)1 pZt+h tEZt −∞  A1+λS(s−τ)ψ(τ) 2 Qdτp 21 p ds ≤(Cp)1 pC1+λ,0Zt+h tEZt −∞ e−2δ(s−τ)(s−τ)−2(λ+1)kψ(τ)k2 Qdτp 21 p ds ≤C(p, λ)Zt+h tZt −∞ e−2δ(s−τ)(s−τ)−2(λ+1)dτp−2 2p ×Zt −∞ e−2δ(s−τ)(s−τ)−2(λ+1)Ekψ(τ)kp Qdτ1 pds ≤C(p, λ)sup t∈R Ekψ(t)kp Q1 pZt+h t (s−t)−λ−1 2ds =C(p, λ)sup t∈R Ekψ(t)kp Q1 ph1 2−λ.(4.17) Then, applying the stochastic Fubini theorem to R13 gives R13 ≤  Zt −∞ Zt+h t AS(s−τ)ψ(τ)dsdBQ(τ)  Lp(Ω;Hλ) =  Zt+h tZt −∞ AS(s−τ)ψ(τ)dBQ(τ)ds  Lp(Ω;Hλ) ≤C(p, λ)sup t∈R Ekψ(t)kp Q1 ph1 2−λ.(4.18) By making use of Lemma 2.1, assumption (S1) and H¨older’s inequality, we deduce that R15 =E  Zt+h t AλS(t+h−τ)ψ(τ)dBQ(τ)   p1 p ≤(Cp)1 pCλ,0EZt+h t AλS(t+h−τ)ψ(τ) 2 Qdτp 21 p ≤C(p, λ)Zt+h t e−2δ(t+h−τ)(t+h−τ)−2λdτp−2 2p ×Zt+h t e−2δ(t+h−τ)(t+h−τ)−2λEkψ(τ)kp Qdτ1 p March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo 18 Liu, Wang & Caraballo ≤C(p, λ)sup t∈R Ekψ(t)kp Q1 ph1 2−λ.(4.19) In view of assumption (S1) and H¨older’s inequality, we have R12 =  Zt+h tZt −∞ AS(s−τ)ζ(τ)dτds  Lp(Ω;Hλ) ≤Zt+h tZt −∞ E A1+λS(s−τ)ζ(τ) p1 pdτds ≤C1+λ,0sup t∈R Ekζ(t)kp1 pZt+h tZt −∞ e−δ(s−τ)(s−τ)−(λ+1)dτds ≤C(λ)sup t∈R Ekζ(t)kp1 ph1−λ,(4.20) and R14 ≤Zt+h t S(t+h−τ)ζ(τ) Lp(Ω;Hλ)dτ ≤Cλ,0Zt+h t e−δ(t+h−τ)(t+h−τ)−λdτp−1 p ×Zt+h t e−δ(t+h−τ)(t+h−τ)−λEkζ(τ)kpdτ1 p ≤C(λ)sup t∈R Ekζ(t)kp1 ph1−λ.(4.21) Substituting (4.18)-(4.21) into (4.16) yields that ˜u∗(t) is mean-pH¨older continuous. If ˜%(t) is any solution of (4.6) satisfying Ek˜%(t0)kp λ<∞, then ˜%(t) = S(t−t0)˜%(t0) + Zt t0 S(t−τ)ζ(τ)dτ +Zt t0 S(t−τ)ψ(τ)dBQ(τ).(4.22) It follows immediately from (4.15), (4.22) and assumption (S1) that E ˜u∗(t)−˜%(t) p λ≤Cp 0e−pδ(t−t0)E ˜u∗(t0)−˜%(t0) p λ,(4.23) which implies that ˜u∗is exponentially stable. Finally, we show that ˜u∗(t) is unique. Let v(t) be another solution such that supt∈REkv(t)kp λ<∞. By Definition 4.1 and the assumption (S1), we obtain that for arbitrary r≤t, E ˜u∗(t)−v(t) p λ≤Cp 0e−pδ(t−r)E ˜u∗(r)−v(r) p λ≤Ce−pδ(t−r).(4.24) Letting r→ −∞, we have E ˜u∗(t)−v(t) p λ= 0 for all t∈R.(4.25) Using Markov’s inequality, we deduce that for each t∈Rand any ε > 0, Pkv(t)−˜u∗(t)kλ> ε≤1 εpEkv(t)−˜u∗(t)kp λ,(4.26) March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo Stochastic pantograph delay evolution equations 19 and consequently Pkv(t)−˜u∗(t)kλ= 0 for all t∈Q∗∩R= 1,(4.27) where Q∗denotes the rational numbers. Since the mapping t→ kv(t)−˜u∗(t)kλis continuous with probability 1, we conclude that Pkv(t)−˜u∗(t)kλ= 0 for all t∈R= 1.(4.28) Therefore, the uniqueness of ˜u∗(t) is confirmed. The proof of this theorem is complete. 4.2. Nonlinear version The following theorem shows the existence, uniqueness and α-type stability of the solution u∗to problem (4.1). Theorem 4.2. Suppose that p≥2,λ∈(0,1 p)and assumptions (S2)-(S3)hold for t∈R. Let us further assume that assumptions (S0)-(S1)hold, and the Lipschitz constants L1, L2in assumption (S2)are sufficiently small such that 22p−2Cp λ,0δλ−1Γ(1 −λ)pkL1kL∞(R) +Cp(2δ)2λ−1Γ(1 −2λ)p 2kL2kL∞(R):= K1<1, 2p−1Cp λ,0Cp(2δ)2λ−1Γ(1 −2λ)p 2kL2kL∞(R) +δλ−1Γ(1 −λ)pkL1kL∞(R):= K2<1,(4.29) and 2pCp λ,0δλ−1Γ(1 −λ)pkL1kL∞(R)<1, 2pCpCp λ,0(2δ)2λ−1Γ(1 −2λ)p 2kL2kL∞(R)<1, 6pc∗Cp λ,0δλ−1Γ(1 −λ)pkL1kL∞(R)<1, 6pc∗CpCp λ,0(2δ)2λ−1Γ(1 −2λ)p 2kL2kL∞(R)<1. (4.30) Then, problem (4.1) has a unique solution u∗(t)in the sense of Definition 4.1which is mean-pH¨older continuous in t∈R, i.e., sup t∈R ku∗(t+h)−u∗(t)kLp(Ω;Hλ)≤Ch1 2−λ,for each h > 0. Moreover, the solution u∗(t)is α-type stable, that is, lim t→∞ log Eku∗(t)−%(t)kp λ log α(t)<0,(4.31) where %(t)is any solution of problem (1.1)-(1.2) in the sense of Definition 3.1. Proof. March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo 20 Liu, Wang & Caraballo Let us construct a sequence of stochastic processes {un}which converges to the solution u∗. Let u0≡0. For n≥0, define un+1(t) as dun+1(t) = −Aun+1(t)dt +f(t, un(ηt))dt +g(t, un(ηt))dBQ(t).(4.32) Notice that sup t∈R E f(t, un(ηt)) p≤2p−1kl1kL∞(R)+ 2p−1kL1kL∞(R)sup t∈R Ekun(t)kp λ, sup t∈R E g(t, un(ηt)) p Q≤2p−1kl2kL∞(R)+ 2p−1kL2kL∞(R)sup t∈R Ekun(t)kp λ.(4.33) By using Theorem 4.1, we obtain the unique solution un+1(t) satisfying sup t∈R Ekun+1(t)kp λ<∞,(4.34) and un+1(t) = Zt −∞ S(t−τ)f(τ, un(ητ))dτ +Zt −∞ S(t−τ)g(τ, un(ητ))dBQ(τ).(4.35) Step 1. The sequence {un(t)}converges to the process u∗(t) and the process u∗(t) is a solution in the sense of Definition 4.1. (1) supt∈RkunkLp(Ω;Hλ)is bounded which is independent of n. It follows directly from (4.35) that Ekun+1(t)kp λ≤2p−1E  Zt −∞ S(t−τ)f(τ, un(ητ))dτ   p λ + 2p−1E  Zt −∞ S(t−τ)g(τ, un(ητ))dBQ(τ)   p λ := R16 +R17.(4.36) By applying Lemma 2.1, assumptions (S1)-(S3), H¨older’s inequality and (4.33), we obtain R17 ≤2p−1CpEZt −∞  AλS(t−τ)g(τ, un(ητ)) 2 Qdτp 2 ≤2p−1CpCp λ,0EZt −∞ e−2δ(t−τ)(t−τ)−2λkg(τ, un(ητ))k2 Qdτp 2 ≤2p−1CpCp λ,0Zt −∞ e−2δ(t−τ)(t−τ)−2λdτp−2 2 ×Zt −∞ e−2δ(t−τ)(t−τ)−2λEkg(τ, un(ητ))kp Qdτ ≤22p−2CpCp λ,0(2δ)2λ−1Γ(1 −2λ)p 2 ×kl2kL∞(R)+kL2kL∞(R)sup t∈R Ekun(t)kp λ.(4.37) March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo Stochastic pantograph delay evolution equations 21 Similar to (4.13), we deduce that R16 ≤22p−2Cp λ,0δλ−1Γ(1 −λ)p ×kl1kL∞(R)+kL1kL∞(R)sup t∈R Ekun(t)kp λ.(4.38) Inserting (4.37)-(4.38) into (4.36) gives sup t∈R Ekun+1(t)kp λ≤ K0+K1sup t∈R Ekun(t)kp λ.(4.39) Then we derive from (4.39), assumption (4.29) and the recursive method that sup t∈R Ekun(t)kp λ≤K0 1− K1 ,(4.40) where we have used the notation K0:= 22p−2Cp λ,0δλ−1Γ(1 −λ)pkl1kL∞(R) +Cp(2δ)2λ−1Γ(1 −2λ)p 2kl2kL∞(R). (2) The sequence {un}is convergent. Arguing as in (3.7) and (3.12), it follows from (4.35) that Ekun+1(t)−un(t)kp λ ≤2p−1E  Zt −∞ S(t−τ)f(τ, un(ητ)) −f(τ, un−1(ητ))dτ   p λ + 2p−1E  Zt −∞ S(t−τ)g(τ, un(ητ)) −g(τ, un−1(ητ))dBQ(τ)   p λ ≤2p−1CpCp λ,0(2δ)2λ−1Γ(1 −2λ)p 2kL2kL∞(R) ×sup t∈R Ekun(t)−un−1(t)kp λ(4.41) + 2p−1Cp λ,0δλ−1Γ(1 −λ)pkL1kL∞(R)sup t∈R Ekun(t)−un−1(t)kp λ, which implies that sup t∈R Ekun+1(t)−un(t)kp λ≤ K2sup t∈R Ekun(t)−un−1(t)kp λ.(4.42) Using the recursive method again, in view of (4.40) and the assumption K2<1, we obtain that sup t∈R kun(t)−um(t)kLp(Ω;Hλ) ≤ n−1 X j=m sup t∈R kuj+1(t)−uj(t)kLp(Ω;Hλ)= n−1 X j=m sup t∈REkuj+1(t)−uj(t)kp λ1 p ≤ n−1 X j=msup t∈R Ekuj+1(t)−uj(t)kp λ1 p≤sup t∈R Eku1(t)kp λ1 p n−1 X j=m (K2)j p March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo 22 Liu, Wang & Caraballo ≤K0 1− K11 p n−1 X j=m 1 2j p →0,as n, m → ∞.(4.43) This means that un(t) is a Cauchy sequence, and thus there exists a limiting function u∗(t) such that sup t∈R Ekun(t)−u∗(t)kp λ→0,as n→ ∞.(4.44) Combining (4.40) and (4.44) results in Eku∗(t)kp λ≤K0 1− K1 ,for each t∈R.(4.45) Since the sequence {un}is Ft-measurable for each t∈R, the process u∗(t) is Ft-measurable as a limit of {un}. (3) The process u∗(t) satisfies (4.5) and has continuous trajectories with probability 1. By similar calculations as in (4.15), it follows from (4.35) that un+1(t) = S(t−t0)un+1(t0) + Zt t0 S(t−τ)f(τ, un(ητ))dτ +Zt t0 S(t−τ)g(τ, un(ητ))dBQ(τ). (4.46) To show that u∗(t) satisfies (4.5), we need to pass to the limit in the above identity. It follows from Markov’s inequality and (4.44) that, for each ε > 0, Pkun+1(t)−u∗(t)kλ> ε≤1 εpEkun+1(t)−u∗(t)kp λ n→∞ −→ 0,(4.47) which implies that, for each t∈R, un+1(t)→u∗(t) in probability, as n→ ∞.(4.48) Due to the fact that S(t−t0) is a bounded operator, we obtain that S(t−t0)un+1(t0)−→ S(t−t0)u∗(t0) in probability, as n→ ∞.(4.49) Arguing as in (3.12), in view of Markov’s inequality, we deduce that P  Zt t0 S(t−τ)g(τ, un(ητ)) −g(τ, u∗(ητ))dBQ(τ)  λ> ε ≤1 εpE  Zt t0 S(t−τ)g(τ, un(ητ)) −g(τ, u∗(ητ))dBQ(τ)   p λ(4.50) ≤1 εpCpCp λ,0(2δ)2λ−1Γ(1 −2λ)p 2kL2kL∞(R)sup t∈R Ekun(t)−u∗(t)kp λ, which together with (4.44) implies Zt t0 S(t−τ)g(τ, un(ητ))dBQ(τ) March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo Stochastic pantograph delay evolution equations 23 n→∞ −→ Zt t0 S(t−τ)g(τ, u∗(ητ))dBQ(τ),in probability. (4.51) In a similar way as in (4.50), we find that Zt t0 S(t−τ)f(τ, un(ητ))dτ n→∞ −→ Zt t0 S(t−τ)f(τ, u∗(ητ))dτ, in probability. (4.52) Finally, by (4.48), (4.49) and (4.51)-(4.52), we can conclude that for all t∈R, u∗(t) = S(t−t0)u∗(t0) + Zt t0 S(t−τ)f(τ, u∗(ητ))dτ +Zt t0 S(t−τ)g(τ, u∗(ητ))dBQ(τ) a.s. (4.53) i.e. u∗(t) satisfies (4.5). The continuity of the first two terms can be checked straightforwardly, and the continuity of the third term follows from the factorization formula (2.5) and Proposition 6.1. Hence the process u∗(t), defined by (4.2), has continuous trajectories with probability 1. Step 2. The process u∗is H¨older continuous in t∈R. By similar arguments as in (4.16)-(4.21) and (4.33), we obtain that for each h > 0,  u∗(t+h)−u∗(t) Lp(Ω;Hλ) ≤  Zt −∞ S(t+h−τ)−S(t−τ)f(τ, u∗(ητ))dτ  Lp(Ω;Hλ) +  Zt −∞ S(t+h−τ)−S(t−τ)g(τ, u∗(ητ))dBQ(τ)  Lp(Ω;Hλ) +  Zt+h t S(t+h−τ)f(τ, u∗(ητ))dτ  Lp(Ω;Hλ) +  Zt+h t S(t+h−τ)g(τ, u∗(ητ))dBQ(τ)  Lp(Ω;Hλ) ≤C(λ)sup τ∈R Ekf(τ, u∗(ητ))kp1 ph1−λ+C(p, λ)sup τ∈R Ekg(τ, u∗(ητ))kp Q1 ph1 2−λ ≤C(λ)kl1k 1 p L∞(R)+kL1k 1 p L∞(R)sup t∈R Eku∗(t)kp λ1 ph1−λ +C(λ, p)kl2k 1 p L∞(R)+kL2k 1 p L∞(R)sup t∈R Eku∗(t)kp λ1 ph1 2−λ,(4.54) which means that u∗is H¨older continuous in time. Step 3. The process u∗is α-type stable in the sense of pth moment. March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo 24 Liu, Wang & Caraballo The assertion of this step can be proved by applying the Banach fixed point theorem. Since the proofs of the case t0≥0 is simpler than the case t0<0, we assume that t0<0. Consider the abstract phase space Cp,λ ϑ∗=Cϑ∗t0,∞;Lp(Ω; Hλ)with the norm kukϑ∗= sup t∈[t0,∞) ϑ∗(t)Eku(t)kp λ, u ∈Ct0,∞;Lp(Ω; Hλ), where ϑ∗(t) = (α(T), t ∈[t0, T], α(t), t ≥T, with T > 0 given later. Then Cp,λ ϑ∗,k·kϑ∗is a Banach space. Put b%(t) = %(t)−u∗(t),(4.55) where %(t) is any solution of problem (1.1)-(1.2) in the sense of Definition 3.1. Define the mapping T∗by (T∗b%)(t) = S(t−t0)b%(t0) +Zt t0 S(t−τ)f(τ, b%(ητ) + u∗(ητ)) −f(τ, u∗(ητ))dτ +Zt t0 S(t−τ)g(τ, b%(ητ) + u∗(ητ)) −g(τ, u∗(ητ))dBQ(τ). (4.56) Now we show that T∗is contractive and bounded on Cp,λ ϑ∗. (I) T∗is a contraction mapping. Due to (4.56), we have that for any b%1,b%2∈Cp,λ ϑ∗, ϑ∗(t)E (T∗b%1)(t)−(T∗b%2)(t) p λ ≤2p−1ϑ∗(t)E  Zt t0 S(t−τ)f(τ, b%1(ητ) + u∗(ητ)) −f(τ, b%2(ητ) + u∗(ητ))dτ   p λ + 2p−1ϑ∗(t)E  Zt t0 S(t−τ)g(τ, b%1(ητ) + u∗(ητ)) −g(τ, b%2(ητ) + u∗(ητ))dBQ(τ)   p λ := R28 +R29.(4.57) It follows from the assumptions (S1)-(S2), H¨older’s inequality and (4.55) that for t∈[t0, T], R28 ≤2p−1α(T)Cp λ,0EZt t0 e−δ(t−τ)(t−τ)−λ × f(τ, b%1(ητ) + u∗(ητ)) −f(τ, b%2(ητ) + u∗(ητ)) dτp ≤2p−1α(T)Cp λ,0Zt t0 e−δ(t−τ)(t−τ)−λdτp−1 March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo Stochastic pantograph delay evolution equations 25 ×Zt t0 e−δ(t−τ)(t−τ)−λE f(τ, b%1(ητ) + u∗(ητ)) −f(τ, b%2(ητ) + u∗(ητ)) pdτ ≤2p−1Cp λ,0δλ−1Γ(1 −λ)pkL1kL∞(R)kb%1−b%2kϑ∗.(4.58) For t≥T, R28 ≤6p−1α(t)EZ0 t0 S(t−τ)f(τ, b%1(ητ) + u∗(ητ)) −f(τ, b%2(ητ) + u∗(ητ)) λdτp + 6p−1α(t)EZt 2 0 S(t−τ)f(τ, b%1(ητ) + u∗(ητ)) −f(τ, b%2(ητ) + u∗(ητ)) λdτp + 6p−1α(t)EZt t 2 S(t−τ)f(τ, b%1(ητ) + u∗(ητ)) −f(τ, b%2(ητ) + u∗(ητ)) λdτp := R1 28 +R2 28 +R3 28.(4.59) Using again the assumptions (S1)-(S2), H¨older’s inequality and (4.55), we deduce that R1 28 ≤6p−1Cp λ,0α(t)Z0 t0 e−δ(t−τ)(t−τ)−λ × f(τ, b%1(ητ) + u∗(ητ)) −f(τ, b%2(ητ) + u∗(ητ)) dτp ≤6p−1Cp λ,0α(t)t−pλZ0 t0 e−δ(t−τ)dτp−1 ×Z0 t0 e−δ(t−τ)E f(τ, b%1(ητ) + u∗(ητ)) −f(τ, b%2(ητ) + u∗(ητ)) pdτ ≤6p−1Cp λ,0 1 δp−1kL1kL∞(R)α(t)t−pλe−pδt Z0 t0 eδτ E b%1(ητ)−b%2(ητ) p λdτ ≤6p−1Cp λ,0 (α(T))−1 δpkL1kL∞(R)kb%1−b%2kϑ∗α(t)t−pλe−pδt.(4.60) For terms R2 28 and R3 28, by a similar way as in (3.9) and (3.10), we obtain that R2 28 ≤6p−1α(t)Cp λ,0kb%1−b%2kϑ∗kL1kL∞(R) ×t 2−pλ e−δpt/2 δp−1Zt 2 0 e−δ(t/2−τ)(α(ητ))−1dτ, (4.61) and R3 28 ≤6p−1Cp λ,0δλ−1Γ(1 −λ)pkb%1−b%2kϑ∗kL1kL∞(R) α(t) α(ηt/2).(4.62) Hence by (4.58) and (4.59)-(4.62), in view of the assumption (4.30), we can take T sufficiently large such that for any t≥t0, R28 <1 2kb%1−b%2kϑ∗.(4.63) March 14, 2023 22:54 WSPC/INSTRUCTION FILE liu˙wang˙caraballo 32 Liu, Wang & Caraballo 2. 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