The First Exit Time Of A Brownian Motion With Increasing Dimensions From An Unbounded Convex Domain

In probability,the first exit time or the stopping of Brownian Motion plays a key role in many filed,such as the probabilistic solution to the Dirichlet problem.More and more mathe-maticians have realized the importance of the first exit time.In the last few years,a large number of mathematicians have studied the first exit time of a Brownian motion from various domains,which was also widely applied in mathematics and physics.In addition,in many fields,such as finance,insurance and informatics,as time goes by,the information which we face becomes more and more complex,and the number of factors which we consider is increasing rapidly.However,the classical stochastic process with fixed dimensions cannot be used to describe these phenomena very well.Thus,the research of the stochastic process with increasing dimensions is important not only to theoretical interest but also to practical applications.In our article,we consider the first exit time of a Brownian motion with increasing dimensions from an unbounded convex domain,namely the fixed dimension d in Li?2003?^[1]is replaced by a function d?s?.The second part of our paper is to consider the small ball probability with increasing di-mensions.Combining some new properties of Bessel function derived by us,the results in Li?2003?^[1]and Gaussian techniques,we obtain very general upper and lower estimates for the small ball probability of a Brownian motion with increasing dimensions.In the third part,based on the property in the second part,we obtain very general upper and lower estimates for the probability of the first exit time of a Brownian motion with increasing di-mensions.Furthermore,in some specific domains,we prove that the upper and lower estimates are asymptotically equivalent.