Researches On Value Distribution Of Random Analytic Functions And Complex Equations

The theory of value distribution is an important subject in complex analysis.Much work has been taken on this issue.In this dissertation,we mainly study the value distribution of random analytic functions in the complex plane and in the unit disk,respectively,based on Nevanlinna theory and probability theory.With these results in hand,some problems of the uniqueness of random analytic functions are dealt with.Then,the applications of Nevanlinna theory in complex equations and some new exact meromorphic solutions of some non-linear partial differential equations have been given.The structure of this dissertation is as follows:In Chapter 1,the background and research status of this topic are introduced.At the same time,the main research work of this dissertation is given.In Chapter 2,some definitions and theorems of Nevanlinna theory,uniqueness and random series are given to prepare for following chapters.Chapter 3 is devoted to study the value distribution of a family of random entire functions perturbed by the transcendental entire function f,denoted as y*,which includes Gaussian,Rademacher and Steinhaus entire functions.Thus,we can deal with these three classes of famous random entire functions all together.Then,several inequalities concerning the maximum modulus M(r,f),?(r,f)and the counting function N(r,a,f?)for the random entire functions in the family y*are established.These inequalities show that the counting function N(r,a,f?)of almost all randomly perturbed function f? is close to the maximum modulus of f,up to an error term.In particular,the error terms in these inequalities are also carefully dealt with.As a by-product of our main results,a second main theorem for random entire functions is proved.Thus,the characteristic function of almost all functions in the family is bounded above by a counting function,rather than by two counting functions in the classical Nevanlinna theory.In Chapter 4,the uniqueness problems for random entire functions in family y*are studied by using the results obtained in the last chapter.In particular,when two random entire functions in this family share two distinct complex numbers counting multiplicities,then they are identically equal almost surely.In Chapter 5,the value distribution of random analytic functions in the unit disk is investigated.Represented by Rademacher analytic functions f?,the relationship between counting-zero function of f? and the maximum modulus M(r,f)is investigated.In Chapter 6,two kinds of extended Kadomtsev-Petviashvili equations and Jimbo-Miwa equations are investigated by using some results from complex analysis.Then some new meromorphic exact solutions of these equations are obtained,which contain rational solutions,exponential function solutions and elliptic function solutions.The properties of these results are also shown by some figures.In the last chapter,the research content of this dissertation is summarized and the future research topics are expected.