This is joint work with Davar Khoshnevisan and Kunwoo Kim, which continues our study of the parabolic Anderson model, also known as the stochastic heat equation (SHE). \(\partial_tu=\partial_x^2u+\sigma(u)\dot{W}(t,x)\) Here \(t>0,x\in[0,J]\) and \(u(t,x)\) satisfies periodic boundary conditions. The initial data \(u(0,x)\) is nonnegative and not identically 0. \(\sigma(u)\) is comparable to \(u\), so we are working with a generalization of the usual parabolic Anderson model for which \(\sigma(u)=u\).

Let \(m(t)=\inf_xu(t,x)\) and \(M(t,x)=\sup_xu(t,x)\). We show that \(m(t),M(t)\) tend to 0 at the same exponential rate. Secondly, given two solutions starting from different initial conditions, the exponential rates of decrease of \(m(t),M(t)\) are the same for the two solutions. In the case \(\sigma(u)=\lambda u\) for some constant \(\lambda>0\), and \(x\in\mathbf{R}\) and for certain specific initial data, these facts were known if we redefine \(m(t)=M(t)=u(t,x_1)\) and \(x_1\) is fixed, using the machinery of integrable probability. In that case the rate of decay was also explicitly known, while in our case we do not explicitly know the exponential rate.