Kramers Exit Problem In N-dimensional Space And Its Application For The Advertising Model

In this paper we mainly discuss the Brown particle in n-dimensional potential by means ofsingular perturbation expansion, which escape trap in the random force over the barrier to a deeperand more stable potential. So far, research about using the method of asymptotic analysis has beenrarely seen to n-dimensional Kramers system, merely numerical simulation or experimental ideas.More importantly, n-dimensional Kramers system is more suitable for the motion of compoundnucleus in nuclear fission processes of quantum physics than the low-dimensional system inresearch. In addition, the promotion of exclusive monopoly of the market by advertising model isextended from one-dimensional to two-dimensional competitive model,which makes it moresuitable for actual market situation; Kramers theory is used to analysis the two-dimensional modelof competitive advertising goodwill volatility circumstances, to give businesses and corporatedecision-making support.We use the qualitative theory of ordinary differential equations to analyze equilibrium pointcorresponding deterministic dynamical system problems, continue to analyze the nature ofdemarcation of domains and domain of attraction; The computing of the mean first passage timeand the point of probability distributions is transformed into a corresponding PDE with smallparameter.It translates into a singular perturbation problem; asymptotic analysis method to solveabove problems has been related to asymptotic expressions.Integrated use of ODE,SDE,PDE,theoryof singular perturbation, methods of nonlinear dynamics and modern economy, such as theadvertising model, we not only fully enrich Kramers theory, but also meet the research of nuclearmotion in nuclear fission process, two-dimensional competitive advertising model,which gives greathelp for future practical applications.The full text divides into four chapters:First, the dimension of the space is added from low dimension to high dimension. The equationof the Brownian particle activated in n-dimensional space is a PDE with two-order of n-dimension.We can treat it as a SDE with one order of2n-dimension. And the sort of the criticalpoints,boundary domain and the domain of attraction are analyzed. Second, with the knowledge ofSDE, PDE and singular perturbation theory, we discuss the system trajectory from the domain ofattraction of a stable equilibrium point,calculate mean first passage time,exit probabilitydistribution function, based on the asymptotic analysis of the steady Fokker-Planck equation. Third,the tail trajectory which satisfies different dynamic from the original trajectories is considered. Thetails is achieved by Bayes formula and conditional probability.Fourth,the Kramers exit theory isapplied to advertising profit model.It is improved from one-dimensional monopolist Nerlove-Arrowmodel to two-dimensional competitive model,which is based on. Random variables are introducedto establish a stochastic dynamic system and asymptotic analysis of the system of the correspondingsteady Fokker Planck equation is carried out.The boundary layer is fully discussed and exitprobability distribution function is obtained.