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Sharp Besov regularities in time and space variables are investigated for$\left(u(t,x),\; t\in [0,T],\; x\in \mathbb{R}\right)$, the mild solution tothe 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.

On Besov regularity and local time of the stochastic heat equation