Let
X
=
(
X
t
)
t
≥
0
\mathbb {X}=(\mathbb {X}_t)_{t\geq 0}
be the subdiffusive process defined, for any
t
≥
0
t\geq 0
, by
X
t
=
X
ℓ
t
\mathbb {X}_t = X_{\ell _t}
where
X
=
(
X
t
)
t
≥
0
X=(X_t)_{t\geq 0}
is a Lévy process and
ℓ
t
=
inf
{
s
>
0
;
K
s
>
t
}
\ell _t=\inf \{s>0; \mathcal {K}_s>t \}
with
K
=
(
K
t
)
t
≥
0
\mathcal {K}=(\mathcal {K}_t)_{t\geq 0}
a subordinator independent of
X
X
. We start by developing a composite Wiener-Hopf factorization to characterize the law of the pair
(
T
a
(
b
)
,
(
X
−
b
)
T
a
(
b
)
)
(\mathbb {T}_a^{({b})}, (\mathbb {X} - {b})_{\mathbb {T}_a^{({b})}})
where
T
a
(
b
)
=
inf
{
t
>
0
;
X
t
>
a
+
b
t
}
\begin{equation*}\mathbb {T}_a^{({b})} = \inf \{t>0; \mathbb {X}_t > a+ {b}_t \} \end{equation*}
with
a
∈
R
a \in \mathbb {R}
and
b
=
(
b
t
)
t
≥
0
{b}=({b}_t)_{t\geq 0}
a (possibly degenerate) subordinator independent of
X
X
and
K
\mathcal {K}
. We proceed by providing a detailed analysis of the cases where either
X
\mathbb {X}
is a self-similar or is spectrally negative. For the later, we show the fact that the process
(
T
a
(
b
)
)
a
≥
0
(\mathbb {T}_a^{({b})})_{a\geq 0}
is a subordinator. Our proofs hinge on a variety of techniques including excursion theory, change of measure, asymptotic analysis and on establishing a link between subdiffusive processes and a subclass of semi-regenerative processes. In particular, we show that the variable
T
a
(
b
)
\mathbb {T}_a^{({b})}
has the same law as the first passage time of a semi-regenerative process of Lévy type, a terminology that we introduce to mean that this process satisfies the Markov property of Lévy processes for stopping times whose graph is included in the associated regeneration set.