Consider the stochastic partial differential equation
∂
∂
t
u
t
(
x
)
=
−
(
−
Δ
)
α
2
u
t
(
x
)
+
b
(
u
t
(
x
)
)
+
σ
(
u
t
(
x
)
)
F
˙
(
t
,
x
)
,
t
≥
0
,
x
∈
R
d
,
\begin{equation*} \frac {\partial }{\partial t}u_t(\boldsymbol {x})= -(-\Delta )^{\frac {\alpha }{2}}u_t(\boldsymbol {x}) +b\left (u_t(\boldsymbol {x})\right )+\sigma \left (u_t(\boldsymbol {x})\right ) \dot F(t, \boldsymbol {x}), \quad t\ge 0,\: \boldsymbol {x}\in \mathbb R^d, \end{equation*}
where
−
(
−
Δ
)
α
2
-(-\Delta )^{\frac {\alpha }{2}}
denotes the fractional Laplacian with power
α
2
∈
(
1
2
,
1
]
\frac {\alpha }{2}\in (\frac 12,1]
, and the driving noise
F
˙
\dot F
is a centered Gaussian field which is white in time and has a spatial homogeneous covariance given by the Riesz kernel. We study the detailed behavior of the approximation of the temporal gradient
u
t
+
ε
(
x
)
−
u
t
(
x
)
u_{t+{\varepsilon }}(\boldsymbol {x})-u_t(\boldsymbol {x})
at any fixed
t
>
0
t > 0
and
x
∈
R
d
\boldsymbol {x}\in \mathbb R^d
, as
ε
↓
0
{\varepsilon }\downarrow 0
. As applications, we deduce Khintchin’s law of iterated logarithm, Chung’s law of iterated logarithm, and a result on the
q
q
-variations of the temporal process
{
u
t
(
x
)
}
t
≥
0
\{u_t(\boldsymbol {x})\}_{ t \ge 0}
of the solution, where
x
∈
R
d
\boldsymbol {x}\in \mathbb R^d
is fixed.