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 gen...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.展开更多
文摘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.
