  In this paper, we consider the continuous parabolic Anderson model (PAM)
driven by a time-independent log-correlated Gaussian field (LGF). We obtain an
asymptotic result of
  $$\mathbb{E}\exp\Bigg\{\frac{1}{2}\sum\limits_{
j,k=1}^N\int_0^t\int_0^t\gamma(B_j(s)-B_k(r))drds\Bigg\}\qquad(N\rightarrow
  \infty)$$ which is composed of the independent Brownian motions $\{B_j(s)\}$
and the function $\gamma$ approximating to a logarithmic potential at $0$, such
as the covariances of massive free field and Bessel field. Based on the
asymptotic result, we get the precise high moment asymptotics for Feynman-Kac
formula of the PAM with LGF.

Probability