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This is joint work with Davar Khoshnevisan and Kuwoo Kim. The parabolic Anderson model with nonnegative drift is a widely studied linear SPDE which is related to the KPZ equation and many other topics. We consider this equation on the torus $[−1, 1]$ with end-points identified. It was previously shown that for certain choices of parameters and for positive initial data, solutions $(u(t, ·))$ tend to $0$ in $L^\infty$, as $t\to 0$, and this convergence occurs at an exponential rate. We study certain perturbed versions of this equation. Under appropriate conditions, we show that solutions $(w(t, ·))$ also approach $0$ at an exponential rate, and furthermore $w(t, ·)$ closely tracks $u(t, ·)$ on a logarithmic scale as $t\to \infty$. Thus one can read off properties of w, a solution to a nonlinear equation, from the solutions u of a linear equation.

An invariance principle for the almost linear stochastic heat equation