  We prove the existence and uniqueness of the mild solution for a nonlinear
stochastic heat equation defined on an unbounded spatial domain. The
nonlinearity is not assumed to be globally, or even locally, Lipschitz
continuous. Instead the nonlinearity is assumed to satisfy a one-sided
Lipschitz condition. First, a strengthened version of the Kolmogorov continuity
theorem is introduced to prove that the stochastic convolutions of the
fundamental solution of the heat equation and a spatially homogeneous noise
grow no faster than polynomially. Second, a deterministic mapping that maps the
stochastic convolution to the solution of the stochastic heat equation is
proven to be Lipschitz continuous on polynomially weighted spaces of continuous
functions. These two ingredients enable the formulation of a Picard iteration
scheme to prove the existence and uniqueness of the mild solution.
