Brown, Motion, And Brown, Motion Of The First Distribution

The theory of stochastic processes is an important branch of Maths , however, the investigation of Brown motion and its functional analysis is one of the primary contents of stochastic processes . Brown motion has many elegant and elaborate analysis properties , so that it becomes vital vehicle of the study in the other science areas such as economics , communication theory , biology , management science , mathematical statistics , and so on .As a physical phenomenon the Brown motion was firstly discovered by the English botanist Brown in 1827 when he observed the irregular motion of flower particles on the surface of the liquid . A mathematical description of this phenomenon was derived from the laws of physics by Einstein in 1905 for the first time . After then the sample paths properties of Brown motion was further perfected by Wiener , Levy and the others . since 1918 Wiener put forward to define measure and integration in paths space and gradually developed the conception of Wiener Space .Up to now the basic frame investigation about the Brown motion theory have been comparablly perfect ,but there exist some important questions at several local areas . For instance ,the evident solution formula to joint distribution of the first hitting time and point of Brown motion , dependent Brownian Motion , and Brown motion with drift as special diffusion processes with regard of some regular compact setsIt is well known that the distribution of radom variable decides the law of its varation , but not all the radom variables we are interested in have explicitly solution formula . We can verify the first hitting time and sojourn time of the diffusion processes with regard of any bounded compact set are both a kind of special radom time -stopping time . The corresponding questions of the first hitting time , last existing time and sojourn time of Brown motin to rectangle or cubes , circle cylinder , hiperplane and so on have been solved now , especially the same questions with regard of ball , my professor Mr.Yin obtained comparablly good results by heat conducting equation , the formula of ItO , the change of measure , martingale ,radom integral , partial differential equation and some special functions .This article includes three parts . In part one we give out the general distribution formula of the first hitting time of the Brownian Motion in rectangle regionTheorem 1.1:Let we have:and we obtain the distribution of the first time with regard of the ellipse cylinder as a corollary . LetTheorem 1.2:where are the positive roots of the equation Secondly , we give out the first hitting place with regard of ecllipse cylinder by the transition density and the strong Markov property . Theorem 1.3:then we obtain:Further perfect we obtain the joint distribution of the first hitting tune and place of the Brown motion with regard of ellipse cylinder . And as a corollary we give out the solution of the corresponding heat conducting equation .Theorem 1.4:Letthenis the distribution of the first hitting place of the Brown motion at boundary of the ellipse cylinder starting at one point in it.where let (11,0:2) = x, (y\,yz) = y thenCorollary 1.1:D D\ x D'i described as before and letthen (x, t) is the unique solution of the following heat conducting equation.where M = In the scecond part we will verify the existence of the dependent Brown motion and of its first hitting time and place with regard of a sort of slant ellipse by orthogonal change , the transition semi group and infinitesimal general operator Theory of the Markov processes and stochastics differential equation. Meanwhile we solve the problem of Dirichlet.Theorem 2.1 :Given the fixed matrix with (0 < r < 1)we can obtain unique dependent Brown motion which matrix of covariance is A.Theorem 2.2 : LetThe transition density of the killed dependent Brown motion in D is:Theorem 2.3 .- Letthen the joint density of the first h...