  The aim of this paper is to study the Stanley depth of symbolic powers of a
squarefree monomial ideal. We prove that for every squarefree monomial ideal
$I$ and every pair of integers $k, s\geq 1$, the inequalities ${\rm sdepth}
(S/I^{(ks)}) \leq {\rm sdepth} (S/I^{(s)})$ and ${\rm sdepth} (I^{(ks)}) \leq
{\rm sdepth} (I^{(s)})$ hold. If moreover $I$ is unmixed of height $d$, then we
show that for every integer $k\geq1$, ${\rm sdepth}(I^{(k+d)})\leq {\rm
sdepth}(I^{{(k)}})$ and ${\rm sdepth}(S/I^{(k+d)})\leq {\rm
sdepth}(S/I^{{(k)}})$. Finally, we consider the limit behavior of the Stanley
depth of symbolic powers of a squarefree monomial ideal. We also introduce a
method for comparing the Stanley depth of factors of monomial ideals.
