  This paper addresses the problem of comparing minimal free resolutions of
symbolic powers of an ideal. Our investigation is focused on the behavior of
the function depth R/I^(t) = dim R - pd I^(t) - 1, where I^(t) denotes the t-th
symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and
pd denotes the projective dimension.
  It has been an open question whether the function depth R/I^(t) is
non-increasing if I is a squarefree monomial ideal. We show that depth R/I^(t)
is almost non-increasing in the sense that depth R/I^(s) \ge depth R/I^(t) for
all s \ge 1 and t \in E(s), where
  E(s) = \cup_{i \ge 1} {t \in N| i(s-1)+1 \le t \le is}
  (which contains all integers t \ge (s-1)^2+1). The range E(s) is the best
possible since we can find squarefree monomial ideals I such that depth R/I^(s)
< depth R/I^(t) for t \not\in E(s), which gives a negative answer to the above
question.
  Another open question asks whether the function depth R/I^(t) is always
constant for t \gg 0. We are able to construct counter-examples to this
question by monomial ideals. On the other hand, we show that if I is a monomial
ideal such that I^(t) is integrally closed for t \gg 0 (e.g. if I is a
squarefree monomial ideal), then depth R/I^(t) is constant for t \gg 0 with
lim_{t \to \infty} depth R/I^(t) = dim R - dim \oplus_{t \ge 0} I^(t)/m I^(t).
  Our last result (which is the main contribution of this paper) shows that for
any positive numerical function \phi(t) which is periodic for t \gg 0, there
exist a polynomial ring R and a homogeneous ideal I such that depth R/I^(t) =
\phi(t) for all t \ge 1. As a consequence, for any non-negative numerical
function \psi(t) which is periodic for t \gg 0, there is a homogeneous ideal I
and a number c such that pd I^(t) = \psi(t) + c for all t \ge 1.
