Integral Functionals for the Exponential of the Wiener Process and the Brownian Bridge: Exact Asymptotics and Legendre Functions

We prove results concerning the exact asymptotics of the probabilities
$$ \mathsf{P}\biggl\{\int_0^1 e^{\varepsilon\xi(t)}\,dt<b\biggl\},\qquad \mathsf{P}\biggl\{\int_0^1 e^{\varepsilon|\xi(t)|}\,dt<b\biggl\} $$
as $\varepsilon \to 0$ and $0<b<1$ for two Gaussian processes $\xi(t)$, the Wiener process and the Brownian bridge. We also obtain asymptotic formulas for integrals of Laplace type. Our study is based on the Laplace method for Gaussian measures in Banach spaces. The calculations of the constants are reduced to the solution of an extremal problem for the action functional and to the study of the spectrum of a second-order differential operator of Sturm–Liouville type using the Legendre functions.