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In this paper, we prove a sample-path comparison principle for the nonlinearstochastic fractional heat equation on $\mathbb{R}$ with measure-valued initialdata. We give quantitative estimates about how close to zero the solution canbe. These results extend Mueller's comparison principle on the stochastic heatequation to allow more general initial data such as the (Dirac) delta measureand measures with heavier tails than linear exponential growth at $\pm\infty$.These results generalize a recent work by Moreno Flores [25], who proves thestrict positivity of the solution to the stochastic heat equation with thedelta initial data. As one application, we establish the full intermittency forthe equation. As an intermediate step, we prove the H\"older regularity of thesolution starting from measure-valued initial data, which generalizes, in somesense, a recent work by Chen and Dalang [6].

On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations