Small ball probabilities for integrals of weighted Brownian motion

Let , where W(t), t[greater-or-equal, slanted]0, is a standard Brownian motion and [psi] is a weight function. We determine the rate of , for a large class of weight functions. The methods of our proofs are general and can be applied to many other problems. As an application, a Chung-type law of the iterated logarithm is given for X[psi](t) with .