Upper tail probabilities of integrated Brownian motions

We obtain new upper tail probabilities of m-times integrated Brownian motions under the uniform norm and the Lp norm. For the uniform norm, Talagrand's approach is used, while for the Lp norm, Zolotare's approach together with suitable metric entropy and the associated small ball probabilities are used. This proposed method leads to an interesting and concrete connection between small ball probabilities and upper tail probabilities(large ball probabilities) for general Gaussian random variables in Banach spaces. As applications,explicit bounds are given for the largest eigenvalue of the covariance operator, and appropriate limiting behaviors of the Laplace transforms of m-times integrated Brownian motions are presented as well.

Chung＇s Law of the Iterated Logarithm for Subfractional Brownian Motion

Let XH = {xH（t）,t ∈ R＋} be a subfractional Brownian motion in Rd. We provide asufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that XH has the property of strong local nondeterminism. Applying this property and a stochastic integral representation of XH, we establish Chung＇s law of the iterated logarithm for XH.

A Functional LIL for Integrated α Stable Process

Let {X（t）,t ∈ R＋} be an integrated α stable process. In this paper, a functional law of the iterated logarithm （LIL） is derived via estimating the small ball probability of X. As a corollary,, the classical Chung LIL of X is obtained. Furthermore, some results about the weighted occupation measure of X（t） are established.

A Functional LIL for m-Fold Integrated Brownian Motion

Let {Xm（t）, t∈R＋} be an m-Fold integrated Brownian motion. In this paper, with the help of small ball probability estimate, a functional law of the iterated logarithm （LIL） for Xm（t） is established. This extends the classic Chung type liminf result for this process. Furthermore, a result about the weighted occupation measure for Xm（t） is also obtained.

Chung-type Law of the Iterated Logarithm on l^p-valued Gaussian Processes

By estimating small ball probabilities for l^P-valued Gaussian processes, a Chung-type law of the iterated logarithm of l^P-valued Gaussian processes is given.

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

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.