  The parabolic Anderson model (PAM) is one of the most interesting and
challenging SPDEs related to various physical phenomena, and can be described
mathematically as a stochastic heat equation driven by linear multiplicative
noise. In this paper, we consider PAM with initial condition given by a signed
Borel measure on $R^d$. The forcing term under investigation is examined in two
cases: (i) the regular noise, with the spatial covariance given by the Riesz
kernel of order $\alpha \in (0, d)$ in spatial dimension $d \ge 1$; (ii) the
rough noise, which is a fractional noise in space with Hurst index $H < 1/2$
and $d = 1$. In both cases, the noise is assumed to be colored in time and we
consider a general initial condition. The objective of this article is to show
that the solution is continuous in law with respect to the spatial noise
parameter. The similar problem for the constant initial condition has been
recently studied in Balan and Liang (2023).
