This talk consists of two parts. In the first part, we make a general introduction of stochastic partial differential equations (SPDEs) from our own perspective. This part is intended to be accessible to a general audience.

In the second part of the talk, we present a recent work jointwork with Panqiu Xia from Auburn University (preprint availabe at arXiv:2306.06761). In this paper, we investigate stochastic heat equation with sublinear diffusion coefficients. By assuming certain concavity of the diffusion coefficient, we establish non-trivial moment upper bounds for the solution. These moment bounds shed light on the smoothing intermittency under weak diffusion (i.e., sublinear growth) previously observed by Zeldovich et al. The method we employ is highly robust and can be extended to a wide range of stochastic partial differential equations, including the one-dimensional stochastic wave equation.