Almost Automorphic Solutions For Stochastic Differential Equations

In this paper, we attempt to investigate the existence, uniqueness and asymptoticstability of almost automorphic solutions for stochastic diferential equations. In addi-tion to continuous noise, things may be afected by noise, which is not continuous inthe real world, such as earthquakes, tsunamis, typhoons, the fnancial crisis and theoutbreak of epidemics. So far, stochastic diferential equation driven by Gaussian noiseis widely applied to various continuous noise models by a large number of researchers.For more accurate describing reality of models, researchers began to study stochasticdiferential equations driven by Poisson processes; further, they investigate stochasticdiferential equations driven by semimartingales, which are more complex than Poissonprocesses. Lèvy processes form an important class of semimartingales, and they alsoinclude Wiener processes and Poisson processes as important special cases. Hence, thestochastic diferential equation driven by Lèvy noise provides us with a more powerfulmodel and a useful tool. When the diferential equation or the system is perturbed bya jump process or, more generally, a Lèvy process, the small size jump should afectslightly the almost automorphic property, but the more interesting question is whetherthe large size jump will appear to destroy almost automorphy. In Chapter2, we provethe existence and uniqueness of almost automorphic in distribution mild solutions forsemilinear stochastic diferential equations driven by Lèvy noise, and this solution isglobally asymptotically stable in square-mean sense. In Chapter3, we investigatethe existence and uniqueness of bounded solution for semilinear stochastic diferentialequations driven by Lèvy noise with exponential dichotomy. From the results of Chap-ters2and3, we know that the large size jump will not afect the almost automorphyunder some suitable conditions. In Chapter4, we study the existence of solutions forlinear stochastic diferential equations driven by Gaussian noise, and prove that thestochastic diferential equation with almost automorphic (respectively, almost period-ic) coefcients admits at least one almost automorphic (respectively, almost periodic) in distribution solution.if it admits at least one bounded solutioon.and all bounded solutions of the corresponding homogeneous equation are homclinic to zero.The main results of this paper are as follows.Throughout this paper,we assume that(H,||·||)and(U,|·|u)are two real separable Hilbert spaces,(Ω,F,P)is a probability space,and the space C.(P,H)stands for the space of all H-valued random variables Y such that whose norm is denoted by Let L(U,H)be the space of al bounded linear operators from U to H equipped with the norm‖·‖L(U,H)1. Stochastic differential equations driven by Ldvy noiseDefinition1A stochastic process Y: Râ†'L2(p,H) is said to be L2-continuous if for any s∈R, It is L2-bounded if supt∈R‖Y(t)‖2<∞.Definition2(1) An L2-continuous stochastic process x:Râ†'L2(P,H) is said SI to be square-mean almost automorphic if for every sequence of real numbers {S’n}, there exist a subsequence {Sn} and a stochastic process y:Râ†'L2(P,H) such that hold for each t∈R. The collection of all square-mean almost automorphic stochastic processes x: Râ†'L2(P,H) is denoted by AA(R;L2(P,H)).(2) A function g: R×L2(P,H)â†'L(U,L2(P,H)),(t,Y)â†'g(t,Y) is said to be square-mean almost automorphic in t∈R for each Y∈L2(P,H) if g is continuous in the following sense and that for every sequence of real numbers {S’n} there exist a subsequence {Sn} and a function g:R×L2(P,H)â†'L(U,L2(P,H)) such that lira Z‖g(t+Sn, Y)-g(t, Y)‖2L(U,L2(P,H)=0and lim E‖g(t-Sn,Y)-g(t,Y)‖2L(U,L2(P,H))=0 hold for each t∈R and each Y∈L2(P, H).(3) A function F:R×L2(P,H)×Uâ†'L2(P,H),(t,Y,x)â†'F(t,Y,x) is said to be Poisson square-mean almost automorphic in t∈L2for each Y∈L2(P, H) if F is continuous in the following sense and that for every sequence of real numbers {s’n}, there exist a subsequence {sn} and a function F:R×L2(P,H)×Uâ†'L2(P,H) with∫u E‖F(t,Y,x)‖2v,(dx)<∞such that and for each t∈R and each Y∈L2(P, H).Consider the following semilinear stochastic differential equations driven by Levy noise dY(t)=AY(t)dt+f(t,Y(t))dt+g(t,Y(t))dW(t)(1) where A is an infinitesimal generator which generates a dissipative C0-semigroup (T(t)t≥0) on H such that with K>0, ω>0; f: R×L2(P,H)â†'L2(P,H), g: R×L2(P,H)â†'L(U,L2(P,H)), F: R×L2(P,H)×Uâ†'L2(P,H), G: R×L2(P,H)×Uâ†'L2(P,H); W is the Q-Wiener process, N is the Poisson random measure and N is the compensated Poisson random measure of N. W, N and N are the Levy-Ito decomposition components of the two-sided Levy process L.Definition3An.Ft-progressively measurable stochastic process {Y(t)}t∈R is called a mild solution of (1) if it satisfies the corresponding stochastic integral equation for all t≥r and each r∈R.Theorem1Consider (1). Assume that A generates a dissipative C0-semigroup such that (2) holds, f, g are square-mean almost automorphic in t for each Y E L2(P,H); and F, G are Poisson square-mean almost automorphic in t∈R for each Y∈L2(P,H). Moreover f, g, F and G satisfy Lipschitz conditions in Y uniformly for t, that is, for all Y,Z E C2(P, H) and t E R, for some constant L>0independent oft. Then (6) has a unique L∈-bounded mild solution, provided Furthermore, this unique L2-bounded solution is almost automorphic in distribution, providedDefinition4A solution Y(t) of (1) is said to be stable in square-mean sense, if for arbitrary θ>0, there exists δ>0such that whenever E||d-Y(0)||2<δ, where Yd(t) stands for the solution of (2.3.1) with initial condition Yd(0)=d. The solution Y(t) is said to be asymptotically stable in square-mean sense if it is stable in square-mean sense and If(5) holds for any d∈L2(P, H), then Y(t) is said to be globally asymptotically stable in square-mean sense.Theorem2. Assume that the assumptions of Theorem1hold. Then the unique C2-bounded solution of (1) is globally asymptotically stable in square-mean sense if (3) is improved toIn particular, when (4) holds, this unique L2-bounded solution is both almost auto-morphic in distribution and globally asymptotically stable in square-mean sense.2. Stochastic differential equations driven by Levy noise with exponen-tial dichotomyDefinition5An evolution family{U(t, s):t≥s, t, s∈R} is said to have an exponential dichotomy, if there are projectors P(t), t∈R, being uniformly bounded and strongly continuous in t and two constants K≥1and ω>0such that1. P(t)U(t,s)=U(t,s)P(s);2. the restriction UJ(t,s):J(s)L2(P,H)â†'J(t)L2(P,H) ofU(t,s) is invertible (and we set U(s,t):=(UJ(t, s))-1);3.||U{t,s)P(s)||≤Ke-ω(t-s),fort≥s, and||U(t, s)J(s)||<Keω(t-s),for t<s, where J(s)=I-P(s).Consider the following semilinear stochastic differential equation driven by Levy noise dY(t)=A(t)Y(t)dt+f{t, Y(t))dt+g(t, Y(t))dW(t)(6) where f: R×L2(P,H)â†'L2(P,H), g: R×L2(P,H)â†'L2(P,H), F R×L2(P,H)×Uâ†'L2(P,H), G:R×L2(P,H)×Uâ†'L2(P,H) are continuous functions; W is the Q-Wiener process, N is the Poisson random measure and N is the compensated Poisson random measure of N. Note that W, N and N are the Levy-Ito decomposition components of the two-sided Levy process.We require the following basic assumptions. (H1) The evolution family{U(t, s):t≥s, t,s∈R} generated by A(t) is exponential dichotomy, that is, there exist projector P(t), some constants K≥1and ω>0such that where J(s)=I-P(s).(H2) Suppose A(t) satisfies the’Acquistapace-Terreni’conditions, U(t, s) is exponential dichotomy and R(λ0, A(·)) E AA(R; Lb(L2(P, H))). Then for every sequence of real numbers{s’n}n∈N, there exist a constant S>0and a subsequence{sn}n∈N such that for any∈>0, there exists an N∈N such that for all n> N and t> s, moreover for all n> N and t<s.(H3) Assume that f, g are square-mean almost automorphic in t for each Y∈L2(P, H), and F, G are Poisson square-mean almost automorphic in t∈R for each Y∈L2(P,H). Moreover f, g, F and G satisfy Lipschitz conditions in Y uniformly for t, that is, for all Y, Z∈L2(P,H) and t∈R, for some constant L>0independent of t.Definition6An An Ft-progressively measurable stochastic process {Y(t)}t∈R is called a mild solution of (6) if it satisfies the corresponding stochastic integral equation for all t≥r and each r∈R.Theorem3Let (H1) and (H3) be satisfied. Then (6) has a unique L2-bounded mild solution, providedTheorem4Let (H1),(H2) and (H3) be satisfied. Then the unique L2-bounded mild solution of (6) is almost automorphic in distribution, provided3. Linear stochastic differential equation driven by Gaussian noiseLet Lb(B) be the Banach space of all bounded linear operators on a Banach space B equipped with the operator norm. Let C(R, Lb(Rn)) be the space of all continuous operator-valued function A:Râ†'Lb(Rn) equipped with the compact-open topology. Consider the following linear stochastic differential equation driven by Gaussian noise and the corresponding homogenenous equation where A∈C(R,Lb(Rn)). f:Râ†'C(R,Rn), g:Râ†'C(R,Rn), h:Râ†'C(R,Rn). Along with (7) and (8) we also consider the H-class of (7) and (8), which is the family of equations and the corresponding homogenenous equation with (B,f’,g’,h’)∈H(A,f,g,h):={(AÏ„,fÏ„,gÏ„,hÏ„)|τ∈R}, where AÏ„(t):=A(t+Ï„), fÏ„(t):=f(t+Ï„), gÏ„(t):=g(t+Ï„), h;(t):=h(t+Ï„) and t∈R. Weset V:=H(d,f,g,h) and denote the dynamical system of shifts on H(A, f,g,h) by (V,{σt}∈R).Definition7A solution φ∈L2(P,Rn) of (7) is called compatible in distribution by the character of recurrence (or simply, compatible in distribution) if (?)(A,f,g,h)(?)(?)μφ, where#is the distribution of φ,(?)(A,f,g,h)={{tn}(?)R|(Atn,ftn,gtn,htn)â†'(A,f,g,h)} and (?)μφ={{tn}(?)R|β(μ(tn),μ)â†'0}.Remark1If φ is comparable with a∈V by the character of recurrence, and a is stationary (respectively, T-periodic, almost automorphic, Levitan almost periodic, recurrent, Poisson stable), then so is the point φ.Theorem5If (A, f, g, h) is almost automorphic (respectively, T-periodic, Levitan almost periodic, recurrent, Poisson stable). Suppose that the following conditions hold:1.(7) admits an L2-bounded solution φ(t, x0,(A,f,g,h));2. all the L2-bounded solutions on R of (8) tend to zero as the time tends to∞, i.e. if φ(t,x,(A,g)) is an L2-bounded solution.Then (7) admits at least one almost automorphic (respectively, T-periodic, Levitan almost periodic, recurrent, Poisson stable) in distribution solution.Corollary1Under the assumptions in Theorem5, if (A, f,g, h) is almost periodic, then (7) admits at least one almost automorphic in distribution solution.Definition8A solution φ∈L2(P,Rn) of (7) is called uniformly compatible in distribution by the character of recurrence (or simply, uniformly compatible in distribution) if (?)(A,f,g,h)(?)(?)μφ, where#is the distribution of φ,(?)(A,f,g,h)={{tn}(?) R] such that the sequence {(Atn,ftn,gtn,htn)} is convergent} and (?)μφ={{tn}(?)R|such that the distribution {μ(tn)} of the sequence {φtn} is weakly convergent}.Remark2If φ is uniformly comparable with a∈V by the character of recur-fence, and α is stationary (respectively, T-periodic, recurrent, almost periodic, almost automorphic, Poisson stable), then so is the point φ.Theorem6If (A, f, g, h) is almost periodic (respectively, T-periodic, recurrent). Suppose that the following conditions hold:1.(7) admits an L2-bounded solution φ(t, x0,(A,f,g,h)); 2.for all B∈H(A) the L2-bounded solutions on R of (10) tend to zero as the time tends to∞,i.e. if Φ(t,x,(B,g’))is an L2-bounded solution.Then (7) admits least one almost periodic (respectively,T-periodic,recurrent)in distributon solution.