Chen, LeDalang, Robert C.2016-02-162016-02-162016-02-16201510.1214/14-Aop954https://infoscience.epfl.ch/handle/20.500.14299/124127WOS:000367416100003We study the nonlinear stochastic heat equation in the spatial domain R, driven by space-time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on R, such as the Dirac delta function, but this measure may also have noncompact support and even be nontempered (e.g., with exponentially growing tails). Existence and uniqueness of a random field solution is proved without appealing to Gronwall's lemma, by keeping tight control over moments in the Picard iteration scheme. Upper bounds on all pth moments (p >= 2) are obtained as well as a lower bound on second moments. These bounds become equalities for the parabolic Anderson model when p = 2. We determine the growth indices introduced by Conus and Khoshnevisan [Probab. Theory Related Fields 152 (2012) 681-701].Nonlinear stochastic heat equationparabolic Anderson modelrough initial datagrowth indicestext::journal::journal article::research article