The First Exit Time Of Brownian Motion From Unbounded Domains With Variable Dimension

More and more mathematicians have realized the importance of the first exit time. The first exit time or the stopping time plays a key role in the probabilistic solution to the Dirichlet problem. In the last few years, there has been a large number of mathematicians to study the first exit time of a Brownian motion from various domains, it was also widely applied in mathematics, biology and physics. Up to now, to the best of our knowledge, most researchers only consider that the dimension of Brownian motions d is a constant. Little is known about the variable dimension. Therefore, in this paper, we mainly consider the exit probabilities of d(t)-dimensional Brownian motion, here the dimension changes over time t. The dimension is no longer the fixed constant d as previous studies.In the second chapter, we considered the sum of a number of Brownian motions in the case of Bessel function in the unbounded domain. Wenbo Li provided general upper and lower estimates for the asymptotic tail distribution of the exit time Ï„D on Rd+1. It is based on a powerful Gaussian technique, Slepian’s inequality, after putting pieces into context. The way has been extended to the sum of the number of Brownian motions in the unbounded domain. It’s also considered the exit probabilities with moving boundary.In the third chapter, we estimated the upper and lower bounds of the first exit time of Brownian motion from a parabolic domain with variable dimension. Consider a Brownian motion with variable dimension starting at an interior point of a general parabolic domain Dt in Rd(t)+1, d(t)≥1is an increasing integral function as tâ†'∞, d(t)â†'∞, and let Ï„Dt denote the first time the Brownian motion exits from Dt. Upper and lower bounds with exact constants for the asymptotics of log P(Ï„Dt>t) are given as t tâ†'∞, depending on the shape of the domain Dt. The methods of the proof are based on the results of the small ball probability in Gaussian field.In the fourth chapter, first we introduced that Lifshits and Shi considered the first exit time of Brownian motion from a parabolic domain. Li got the upper and lower bounds of the first exit time of Brownian motion from a convex unbounded domain, however the lower and upper estimates were not asymptotically equivalent. Then Lifshits and Shi got an asymptotic estimate from a parabolic domain. Based on the result of Lifshits and Shi we consider the exit probability of Brownian motion from a parabolic domain with variable dimension, here the dimension changes over time t. In the fifth chapter, consider a Brownian motion with variable dimension starting at an interior point of the minimum or maximum parabolic domains Dtmax and Dtmin in Rd(t)+2, d(t)>1is an increasing integral function as tâ†'∞, d(t)â†'∞, and let Ï„Dtmax and Ï„Dtmin denote the first time the Brownian motion exits from Dtmax and Dtmin, respectively. Upper and lower bounds with exact constants for the asymptotics of log P(Ï„Dtmax>t) and log P(Ï„Dtmin>t) are given as tâ†'∞, depending on the shape of the domain Dtmax and Dtmin. The methods of proof are based on Gordon’s inequality and early works of Li, Lifshits and Shi in the single general parabolic domain case.In the sixth chapter, using the theory of small ball estimate to study the biological population for keeping ecological balance in an ecosystem, We considered a Brownian motion with a regular variation from a parabolic domain with variable dimension. The dimension of Brownian motion can be thought as the number of different species, and we describe the population changes for all the species by Brownian motion B(t). The problem is motivated by the early results of Lifshits and Shi, Li, Lu in the exit probabilities. The methods of proof are based on the calculus of variations and early works of Lifshits and Shi, Li, Shao in the exit probabilities of Brownian motion.