The Stochastic Pollution Models Related To The Spatial Location And Their Analysis

The water pollution in our country has become more and more serious in recentyears, especially in areas where chemical industries and heavy industries areconcentrated. To solve the problem of pollution, we have to conduct a keenobservation and a reasonable analysis on its situation so as to find a solution to theproblem.The main trait of the pollutions around chemical industry zones is that thepollution concentrations are reltated to stochastic elements as well as time andspatial positions. Based on the analysis of this problem, the paper sets up amathematical model expressed by the initial boundary-valued problems of stochasticpartial differential equations(SPDE). The characteristic of the mathematical model isthat the driving process is a stochastic process related to the position.First, the paper introduces the conception of Hilbert space-valued weakBrownian motion, by choosing the proper space and defining of linear and nonlinearoperators, the initial boundary-valued problems of stochastic partial differentialequation is transferred into the initial boundary-valued problems of semilinearabstract stochastic differential equations in Hilbert space. Real-valued Brownianmotion is a continuous martingale, the conditional expectation is linear, the weakBrownian motion in Hilbert space is proved to be a martingale and the mean of theincrement is zero. We infer from the weak continuity of the weak Brownianmovement that the norm is weakly lower semicontinuous. By Hahn-Banach theorem,the weak Brownian motion in Hilbert space is proved to be a bounded infinite-dimensional vector-valued continuous martingale. On this basis we give a definitionof stochastic integral similar to the Ito integral. This integral has both theoreticaland practical significance in the infinite-dimensional stochastic analysis.Second, by introducing the Hilbert space-valued Ito integral and Borel-Cante-lli lemma, the strong solution of the initial boundary-valued problems of semilinearabstract stochastic differential equation in Hilbert space is transferred intoequivalent solution expressed by Ito integral and Bochner integral. This paper partlylays the foundation for the application of the fixed point theorem to prove theexistence and uniqueness of the solution.