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 ((Ws, [xi]s); s [set membership, variant] [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 (([var epsilon]Ws([zeta]s), [var epsilon]2/3[zeta]s); s [epsilon] [0, 1]) as [var epsilon][downwards 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.