Some Problems In Stochastic Differential Equations Driven By Lévy Noise

In this paper, we attempt to investigate the solutions of stochasticdiferential equations driven by L′evy noise, that is existence, uniquenessand asymptotic stability. Systerms in real world are inevitably perturbedby noises, and Gaussian noise is one of the most common noises. In realworld there are many continuous noises, while many noises are discontinu-ous, such as some sudden changes in nature (earthquake, hurricane, epidemic,etc.). These can be described as jump perturbations, or more general L′evynoise. So far, all the study on almost periodic solutions for stochastic dif-ferential equations focus on equations perturbed by Gaussian noise. Whena diferential equation or a system is perturbed by L′evy noise, is large sizejump harmful for the persistence of recurrent properties, or can the recur-rent property be destroyed completely? Intuitively, large jump may destroyalmost periodicity. In Chapter2, we prove that the stochastic diferentialequation perturbed by L′evy noise admits a unique bounded almost periodicin distribution solution and this unique almost periodic in distribution solu-tion is globally asymptotically stable and that any other solutions convergeto it exponentially fast. In Chapter3, we study the stochastic diferentialequations with exponential dichotomy driven by L′evy noise, and prove thatthere exists a unique bounded almost periodic in distribution solution. Fromthe results of Chapter2and3, we see that almost periodicity property maypersist under some suitable conditions. Indeed, we obtain the existence anduniqueness of bounded almost periodic in distribution solutions for stochas-tic evolution equations with almost periodic coefcients even in large jump case. When population system is perturbed by Levy noise, a stable positive solution means longtime survival. So the existence of a unique globally at-tractive positive solution is of great biological meaning. In Chapter4, we study the Logistic equation driven by Levy noise, and we prove the exis-tence and uniqueness of the solution to the Logistic equation with initial condition. When the initial amount is less than environmental capacity, the unique positive solution is globally attractive under some suitable conditions.The main results of this paper are as follows.Throughout the paper, we assume that (H,‖·‖) and (U,|·|u) are real separable Hilbert spaces. We denote by L(U,H) the space of all bounded linear operators from U to H. Note that L(U,H) is a Banach space, and we denote the norm by‖·‖L(U,H).We assume that (Ω,F, P) is a probability space, and L2(P,H) stands for the space of all H-valued random variables Y such that E‖Y‖2=∫‖Y‖2dP<∞For Y∈L2(P,H),letThen L2(P,H) is a Hilbert space equipped with the norm‖·‖2. The Levy processes we consider in Chapter1-3are U-valued.1. Stochastic differential equations driven by Levy noise Definition1A stochastic process Y:Râ†'L2(P,H) is said to be C2-continuous if for any s∈R, It is called L2-bounded if Difinition2(1) An L2-continuous stochastic process x: Râ†'L2(P,H) is said to be square-mean almost periodic if for every sequence of real numbers {s’n}, there exists a subsequence {sn} and an L2-continuous stochastic process x: IRâ†'L2(P,H) such that The collection of all square-mean almost periodic stochastic processes x: Râ†'L2(P,H) is denoted by AP(IR; L2(P,H)).(2) A continuous function R×L2(P,H)â†'L(U,L2(P,H));(t,Y)â†'g(t,Y) is said to be square-mean almost periodic in t G R uniformly in bounded or compact subsets of L2(P,H) if for every sequence of real numbers {s’n} and any bounded or compact subset K∈L2(P,H); there exists a subse-quence {sn} and a continuous function g:IR×L2(P,H)â†'L(U, L2(P,H)) such that The collection of all square-mean almost periodic functions uniformly on bounded or compact sets g: R×L2(P,H)â†'L(U, L2(P,H)) is denoted byAP(R×L2(P,H);L(P,L2(P,H))).Definition3A function F: R×L2(P,H)×Uâ†'L2(P,H);(t,Y,x) F(t,Y,x) is said to be Poisson square-mean almost periodic in t E R uni-formly in bounded or compact subsets of 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),(t, Y, x)â†'F(t, Y, x) continuous in the sense above such that for any bounded or compact subset K∈L2(P,H). The collection of all Poisson square-mean almost periodic functions F: R×L2(P,H)×Uâ†'L2(P,H) is denoted by PAP(R×L2(P,H)×U;L2(P,H)).Denote by V(H) the space of all Borel probability measures on H endowed with the β metric. Note that P(H) is complete under the metric β and that a sequence {μn} weakly converges to μ if and only if β(μn,μ/)â†'0as nâ†'∞. Definition4An H-valued stochastic process Y(t) is said to be almost peri-odic in distribution if its law μ(t) is a V(H)-valued almost periodic mapping, i.e. μ(?) is continuous and that, for any sequnce {s’n} of real numbers, there exists a subsequence {sn} and a continuous V(H)-valued mapping μ(t) such thatWe consider the semilinear stochastic differential equation driven by Levy noise dY(t)=AY(t-)dt+f(t,Y(t-))dt+g(t,Y(t-))dW(t)(1) where f: R×L2(P,H)â†'L2(P,H), g: R×L2(P,H)â†'L(U, L2(P,H)), F,G: R×L2(P,H)×Uâ†'L2(P,H), and W(t) as well as N are the components of Levy-Ito decomposition. We assume that A generates a C0-semigroup (T(t)≥0) on H, such that‖T(t)‖≤Ke-ωt, all t>0with K>0, ω>0.We have the following results for the semilinear stochastic differential equation (1). Theorem1Assume that f, g are square-mean almost periodic in t uni-formly in bounded subsets of L2(P,H); and F, G are Poisson square-mean almost periodic in t∈IR uniformly in bounded subsets of 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, E‖∫f(t,Y)-f(t,Z)‖2≤LE‖Y-Z‖2, for some constant L>0independent of t. Then (1) has a unique L2-bounded mild solution, provided Furthermore, this unique L2-bounded solution is almost periodic in distribu-tion ifTheorem2Assume that the assumptions of Theorem1hold and that (2) is improved tothen the unique L2-bounded solution of (1) is globally asymptotically stable in square-mean sense. If, in addition,(3) holds also, then the L2-bounded almost periodic in distribution solution is globally asymptotically stable in square-mean sense.2. Stochastic differential equation with exponential dichotomy driven by Levy noise We consider the stochastic differential equation with exponential di-chotomy driven by Levy noise, and introduce the concept of exponential dichotomy.Let A be a linear operator on H, and D(A), R(A) and ker(A) represent the domain, range and kernel of A, respectively. Definition5A semigroup {T(t)}t≥0is hyperbolic, that is, there exist a projection P and constants M,δ>0such that each T(t) commutes with P, ker(P) is invariant with respect to T(t), T(t):R(J)â†'R(J) is invertible and‖T(t)Px‖≤M exp(-δt)‖x‖, fort≥0,‖T(t)Jx‖≤M exp(δt)‖x‖, fort≤0, where J=I-P, and for t≤0, T(t)=(T(-t))-We consider the following semilinear stochastic differential equation with exponential dichotomy driven by Levy noisedY(t)=AY(t-)dt+f(t,Y(t-))dt+g(t,Y(t-))dW(t)(4)where 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,U),G:R×L2(P,H)×Uâ†'L2(P,H),andW(t) are the Levy-Ito decomposition components of the two-sided Levy process L. We assume that A generates a hyperbolic and C0-semigroup (T(t)≥0) on H, such that‖T(t)P‖≤K exp(-ωt), t≥0;‖T(t)J‖≤K exp(ωt), t≤0with K>0,ω>0, J=I-P and for t≤0, T(t)=(T(-t))-1.We have the following results for the semilinear stochastic differential equation (4). Theorem3Assume that T(t) is a hyperbolic C0-semigroup, f, g are square-mean almost periodic in t uniformly in bounded subsets of L2(P,H); and F, G are Poisson square-mean almost periodic in t G R uniformly in bounded subsets of 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 l∈R, E‖f(t,Y)-f(t,Z)‖2≤LE‖Y-Z‖2, for some constant L>0independent of t. Then (4) has a unique L2-bounded mild solution, provided Furthermore, this unique L2-bounded solution is almost periodic in distribu-tion provided3. Logistic equation driven by Levy noiseWe study the following form Logistic equation driven by Levy noisewhere W(t) is standard one-dimensional Wiener process, N is a Poisson ran-dom measure independent of W(t) and N is the associated compensated Poisson random measure of N, constants α,β,γ denote the intensity of per-turbations, respectively.Theorem4Assume that|βH(t,x)≤η<1and v(B1)<∞; where B1represents the ball centered at0with radius1. Then for arbitrary initial value X(0)=X0with0<X0<K, equation (5) admits a unique solution X(t) such that0<X(t)<K almost surely for all t≥0.Theorem5Assume that0<βH(t,x)<1,0<γD(t,x)<1and v{B1)<∞. If [r-α2/2-v(B1)]>0, then the unique solution of (5) with initial value X(0)=X0,0<X0<K is global attractive to K.