Negative-order moments for $L^p$-functionals of Wiener processes: exact asymptotics

We prove theorems on the exact asymptotics as $T \to \infty$ of the integrals $\mathsf{E}\bigl[\frac{1}{T}\!\int_0^T\!|\eta(t)|^pdt\bigr]^{-T}$, $p>0$, for two stochastic processes $\xi(t)$, the Wiener process and the Brownian bridge, as well as for their conditional versions. We also obtain a number of related results. We shall use the Laplace method for the occupation times of homogeneous Markov processes. We write the constants in our exact asymptotic formulae explicitly in terms of the minimal eigenvalue and corresponding eigenfunction for the Schrödinger operator with a potential of polynomial type.