Stochastic Differential Equations With Infinite Delay And Applications

This paper investigates the existence-and-uniqueness, moment stability and pathwise stability of solution for both stochastic functional differential equations and neutral stochastic functional differential equations with infinite delay. As a special case of the stochastic functional differential equations, stochastic Kolmogorov equations with infinite delay are studied. Some conditions are obtained for the existence-and-uniqueness of global positive solution and its asymptotic properties. Finally, the paper discusses in detail population models with different perturbation.This paper consists of five chapters.The first charter mainly introduces development and current situation of the functional differential equations and stochastic functional differential equations, underlines the literature of the phase space theory for the infinite delay functional differential equations. Moreover, some concrete differential equations for dealing with the real problem are presented. For example, viscoelasticity, population models, economic problem and Pantograph equations.Chapter 2 discusses the stochastic functional differential equations with infinite delay in an abstract phase space. Considering infinite delay, we introduce the theory of the abstract phase space B and some concrete phase spaces, such as C~b,C_φ,C_h. Under the uniform Lip-schitz condition and linear growth condition, we derive the existence-and-uniqueness theorem of solution for stochastic functional differential equations with infinite delay in abstract phase space, and give the moment estimate of the solution. Under suitable conditions, the existence-and-uniqueness of the local solution and the global solution are obtained respectively. Finally, Razumikhin-type theorems on exponential stability are given in bounded continuous function space C~b.Chapter 3 mainly investigates neutral stochastic functional differential equations with infinite delay, and obtains the existence-and-uniqueness theorems of the solution in abstract phase space B. Under some conditions, Razumikhin-type theorems on p-th moment exponential stability are established. Moreover, we obtain some results on pathwise exponential stability via three ways.Chapter 4 first studies the existence, the boundedness and pathwise estimation of the global positive solution for stochastic Kolmogorov equations with infinite delay in phase space C_+~b. We obtain two classes of sufficient conditions to guarantee these properties. These two classes of the conditions show that deterministic part or stochastic part play a dominant role in the Kolmogorov equations. By the M matrix technique, we obtain the conditions more convenient in applications. Furthermore, the nice positive property provides us with a great opportunity to discuss the asymptotic properties, such as moment bounedness, asymptotic property of pathwise.The last chapter discusses in detail stochastic Lotka-Volterra equations which are presented in population models. Under different perturbation, we study three type stochastic Lotka-Volterra equations with infinite delay in phase space C_r. Some algebraic conditions on the coefficient matrices are obtained to guarantee the existence of global positive solutions and their asymptotic properties. On the other hand, we reveal that the environmental noise can suppress a potential population explosion in such models.