Stochastic Partial Differential Equations With Levy Jump

Partial differential equations and ordinary differential equations are used to describe development discipline in the real world. A special equation describe a certain phenomenon. When we study a process, some factors are considered to be minor factors. Then is this appropriate? That is to say the factors ignored whether will influent the property of the solution or not. There is one method to check, We study the stability of the equation. Let the stochastic term tend to zero, if the influence on the solution tend to zero, we say the equation is stable, and that the factors are ignored is reasonable, deterministic analysis is true. Furthermore, there is other important reason that we introduce stochastic noise. A deterministic equation under the stochastic noise will produce new phenomenon, and we hope that this can In this dissertation, we discuss several kinds of stochastic partial differential equations with Levy jump. The article is composed of five parts.In chapter 1, we cite some conclusions used in our article, which come from Soblov space, functional analysis and so on.In chapter 2, we studied stochastic generalized porous medium equations with Levy jump. Porous medium equations is a kind of classical partial differential equa-tion, and it has a lot of application in physics. It is used to describe diffusion and heat conduction. And it play very important role in biomathematics, filtration of stream flow and boundary layer theory so on. Porous medium equations perturbed by stochastic noise are called stochastic generalized porous medium equations. We take advantage of randomness to obtain ergodicity of phenomenon. It is the macro-scopic behaviour of our real world and also foundation of FluidMechanics. we get the existence,uniqueness and markov property of stochastic generalized porous medium equations with Levy jump.In chapter three, we get the uniqueness in law of stochastic differential equa-tions with Levy jump, and then we get the ergodicity of porous midium equation with Levy noise.In chapter four, we studied stochastic generalized porous medium equations with Levy jump In the practise case, phenomenon is perturbed by continuous noise and discontinuous noise. So in mathematics, we need non-local operators to describe it, this bright great difficulty to analysis method. we use stochastic analysis to study stochastic differential equations perturbed by continuous noise and discontinuous noise, we get the ergodicity of the equation.In chapter five, we prove the existence and uniqueness of mild solution to stochastic equations with jumps, establish a stochastic Fubini theorem and a type of Burkholder-Davis-Gundy inequality which are fundamental tools, and then we use them to study the regularity property of the mild solution of a general stochas-tic evolution equation perturbed by Levy process.