  In this paper, a Feynman-Kac formula is established for stochastic partial
differential equation driven by Gaussian noise which is, with respect to time,
a fractional Brownian motion with Hurst parameter $H<1/2$. To establish such a
formula, we introduce and study a nonlinear stochastic integral from the given
Gaussian noise. To show the Feynman--Kac integral exists, one still needs to
show the exponential integrability of nonlinear stochastic integral. Then, the
approach of approximation with techniques from Malliavin calculus is used to
show that the Feynman-Kac integral is the weak solution to the stochastic
partial differential equation.
