On the support of solutions to nonlinear stochastic heat equations

We discuss the support properties of solutions to nonlinear stochastic heat equations. We prove that when the nonlinearity is sufficiently large near zero, the solution has compact support at all positive times, provided the initial datum has suitable spatial decay. In contrast, when the nonlinearity is sufficiently small near zero, the solution is strictly positive everywhere at all positive times. This is based on joint work with Beom-Seok Han and Kunwoo Kim.