The Local Existence Of The Mild Solution Of Stochastic Evolution Equations

These years, stochastic analysis has been improved not only in theory but also in application, which has become one of the most active branches in probability theory. The method of stochastic analysis has been used widely in many fields, such as natural science, engineering and also in the modeling of economical management. Scholars have studied stochastic analysis deeply and comprehensively, the paper is just discussing one branch-the local existence of the mild solution of semilinear stochastic evolution equations in Hilbert spaces.In this paper, we mainly study the local existence of the mild solution of semilinear stochastic evolution equations in Hilbert spaces. The functional analysis has been used widely, including semigroup, operator, strong measure, weak measure, Pettis integral, Bochner integral and the definition of Winer process in Hilbert spaces and the stochastic integral with respect to the process. The definition of Strong solution and mild solution and their relationship have introduced in the linear stochastic evolution equations. At last, the conditions of the local existence of the mild solution of semilinear stochastic evolution equations in Hilbert spaces have been presented, which are the most important results in these paper.