THE EXIT PROBABILITIES OF BROWNIAN MOTION WITH VARIABLE DIMENSION APPLYING TO THE CONTROL OF POPULATION GROWTH

Abstract

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 dimension 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 τDt denote the first time the Brownian motion exits from Dt. Upper and lower bounds with exact constants of log P(τDt > t) are given as t → ∞, depending on the shape of the domain Dt. 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.