Study On A Few Kinds Of Stochastic Partial Differential Equations And Kolmogorov Equations

With the continuous development of stochastic partial differential equations, a great deal of applied research has been devoted to the Kolmogorov equations, and they play an important role in many fields, such as fluid dynamics, chemistry, population biology, mathematical finance, distributed parameter control systems theory and so on. The main contents of this thesis is to consider the Kolmogorov equations associated to stochastic partial differential equations with multiplicative noise in infinite dimensional and related problems. We divide our Ph.D. thesis into six chapters.In Chapter1, we introduce the historical background and the current situation of stochastic partial differential equations and Kolmogorov equations, and then give the main results of our thesis.In Chapter2, we list some conclusions used in our Ph.D. thesis, which consist of some notions in probability and stochastic processes, some basic knowledge of Markov semigroups, important results on stochastic analysis in infinite dimension and other related knowledge.Chapter3is intended to study the existence of the classical solution for the Kol-mogorov equation. We firstly consider the stochastic reaction-diffusion equation with multiplicative noise with Dirichlet boundary conditions, under the conditions of the existence and uniqueness of the mild solution, and further introduce the corresponding transition semigroup and the associated Kolmogorov equation. By applying a general-ized Gronwall inequality we obtain the estimate of the regularity of the mild solution, then combining with Galerkin approximation and the results of the finite dimensional case, the existence of the classical solution for the Kolmogorov equation is concluded. Finally, we prove the properties for its transition semigroup and the existence of the invariant measure. We extend and improve some known results.Chapter4is devoted to investigate the existence and uniqueness of the solutions to the Fokker-Planck equation (i.e. the dual of the Kolmogorov equation). Firstly the general stochastic partial differential equations in separable Hilbert space is considered, with the guaranteed of existence and uniqueness of its mild solution, the corresponding transition semigroup and the associated Kolmogorov operator are defined. Later on, by the analysis of their properties, the relationship between the infinitesimal generator of the transition semigroup in a suitable weighted space and the Kolmogorov operator are given. Last, the existence and uniqueness of the solutions to the associated Fokker-Planck equation are obtained. This method is different from the original ones, and some known results are generalized.Chapter5is concerned with stochastic Burgers equation with multiplicative cylin-drical Wiener process in the interval [0,1]. By applying factorization formula and the regularity analysis of the stochastic convolution term in the mild solution, we obtain the moment estimates of invariant measure for the transition semigroup corresponding to its mild solution, then the boundedness of the invariant measure in Hilbert space and interpolation space is obtained. At last we prove the dissipativity of the Kolmogorov operator. The result is new.In the final Chapter6, we give a brief summary of our works and propose the issues of further research directions.