  In this work we study the one-dimensional stochastic Kimura equation
$\partial_{t}u\left(z,t\right)=z\partial_{z}^{2}u\left(z,t\right)+u\left(z,t\right)\dot{W}\left(z,t\right)$
for $z,t>0$ equipped with a Dirichlet boundary condition at $0$, with $\dot{W}$
being a Gaussian space-time noise. This equation can be seen as a degenerate
analog of the parabolic Anderson model. We combine the Wiener chaos theory from
Malliavin calculus, the Duhamel perturbation technique from PDEs, and the
kernel analysis of (deterministic) degenerate diffusion equations to develop a
solution theory for the stochastic Kimura equation. We establish results on
existence, uniqueness, moments, and continuity for the solution
$u\left(z,t\right)$. In particular, we investigate how the stochastic potential
and the degeneracy in the diffusion operator jointly affect the properties of
$u\left(z,t\right)$ near the boundary. We also derive explicit estimates on the
comparison under the $L^{2}-$ norm between $u\left(z,t\right)$ and its
deterministic counterpart for $\left(z,t\right)$ within a proper range.
