摘要
Using the theory of small ball estimate to study the biological population for keeping ecological balance in an ecosystem, we consider a Brownian motion with variable dimen- sion starting at an interior point of a general parabolic domain Dt in Rd（t）＋1 where d（t） ≥ 1 is an increasing integral function as t →∞, d（t） →∞. Let TOt denote the first time the Brownian motion exits from Dr. Upper and lower bounds with exact constants of log P（rDt 〉 t） are given as t →∞, depending on the shape of the domain Dr. The problem is motivated by the early results of Lifshits and Shi, Li, Lu in the exit proba- bilities. 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.
