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We consider the stochastic heat equation on $[0,\,1]$ with periodic boundaryconditions and driven by space-time white noise. Under various naturalconditions, we study small ball probabilities for the H\"older semi-norms ofthe solutions, and provide near optimal bounds on these probabilities. As anapplication, we prove a support theorem in these H\"older semi-norms.

Small ball probability estimates for the H\"older semi-norm of the stochastic heat equation