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