A large deviation principle for the Brownian snake

We consider the path-valued process called the Brownian snake, conditioned so that its lifetime process is a normalised Brownian excursion. This process denoted by ((W-s,zeta(s),); s is an element of [0, 1]) is closely related to the integrated super-Brownian excursion studied recently by several authors. We prove a large deviation principle for the law of ((epsilon W-s(zeta(s)), epsilon(2/3)zeta(s)); s is an element of [0, 1]) as epsilon down arrow 0. In particular, we give an explicit formula for the rate function of this large deviation principle. As an application we recover a result of Dembo and Zeitouni.