Asymptotic properties of stochastic partial differential equations in the sublinear regime

In this paper we investigate stochastic heat equation with sublinear diffusion coefficients. By assuming certain concavity of the diffusion coefficient, we establish nontrivial moment upper bounds and almost sure spatial asymptotic properties for solutions. These results shed light on the smoothing intermittency effect under weak diffusion (i.e., sublinear growth) previously observed by Zeldovich et al. (Proc. Natl. Acad. Sci. USA 84 (1987) 6323–6325). The sample-path spatial asymptotics obtained in this paper partially bridge a gap in earlier works of Conus et al. (Ann. Probab. 41 (2013) 2225–2260; Probab. Theory Related Fields 156 (2013) 483–533), which focused on two extreme scenarios: a linear diffusion coefficient and a bounded diffusion coefficient. Our approach is highly robust and applicable to a variety of stochastic partial differential equations, including the one-dimensional stochastic wave equation.