On Besov regularity and local time of the stochastic heat equation

Sharp Besov regularities in time and space variables are investigated for $(u(t,x),t\in [0,T],x\in \mathbb{R})$, the mild solution to the stochastic heat equation driven by space-time white noise. Existence, H"{o}lder continuity, and Besov regularity of local times are established for $u(t,x)$ viewed either as a process in the space variable or time variable. Hausdorff dimensions of their corresponding level sets are also obtained.