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We study the nonlinear stochastic heat equation in the spatial domain$\mathbb {R}$, driven by space-time white noise. A central special case is theparabolic Anderson model. The initial condition is taken to be a measure on$\mathbb {R}$, such as the Dirac delta function, but this measure may also havenoncompact support and even be nontempered (e.g., with exponentially growingtails). Existence and uniqueness of a random field solution is proved withoutappealing to Gronwall's lemma, by keeping tight control over moments in thePicard iteration scheme. Upper bounds on all $p$th moments $(p\ge2)$ areobtained as well as a lower bound on second moments. These bounds becomeequalities for the parabolic Anderson model when $p=2$. We determine the growthindices introduced by Conus and Khoshnevisan [Probab. Theory Related Fields 152(2012) 681-701].

Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions