Parabolic Anderson model with rough initial condition: continuity in law of the solution

Abstract: 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).