Consider the following stochastic differential equation (SDE): $$ X_{t}=x+ \int _{0}^{t}b(s,X_{s})\,ds+ \int _{0}^{t}\sigma (s,X_{s}) \,dB_{s}, \quad 0\leq t\leq T, x\in \mathbb{R}, $$
X
t
=
x
+
∫
0
t
b
(
s
,
X
s
)
d
s
+
∫
0
t
σ
(
s
,
X
s
)
d
B
s
,
0
≤
t
≤
T
,
x
∈
R
,
where $\{B_{s}\}_{0\leq s\leq T}$
{
B
s
}
0
≤
s
≤
T
is a 1-dimensional standard Brownian motion on $[0,T]$
[
0
,
T
]
. Suppose that $q\in (1,\infty ]$
q
∈
(
1
,
∞
]
, $p\in (1,\infty )$
p
∈
(
1
,
∞
)
, $b=b_{1}+b_{2}$
b
=
b
1
+
b
2
, $b_{1}\in L^{q}(0,T;L^{p}(\mathbb{R}))$
b
1
∈
L
q
(
0
,
T
;
L
p
(
R
)
)
such that $1/p+2/q<1$
1
/
p
+
2
/
q
<
1
and $b_{2}$
b
2
is bounded measurable, with $\sigma \in L^{\infty }(0,T;{\mathcal{C}}_{u}(\mathbb{R}))$
σ
∈
L
∞
(
0
,
T
;
C
u
(
R
)
)
there being a real number $\delta >0$
δ
>
0
such that $\sigma ^{2}\geq \delta $
σ
2
≥
δ
. Then there exists a weak solution to the above equation. Moreover, (i) if $\sigma \in \mathcal{C}([0,T];\mathcal{C}_{u}(\mathbb{R}))$
σ
∈
C
(
[
0
,
T
]
;
C
u
(
R
)
)
, all weak solutions have the same probability law on 1-dimensional classical Wiener space on $[0,T]$
[
0
,
T
]
and there is a density associated with the above SDE; (ii) if $b_{2}=0$
b
2
=
0
, $p\in [2,\infty )$
p
∈
[
2
,
∞
)
and $\sigma \in L^{2}(0,T;{\mathcal{C}}_{b}^{1/2}({\mathbb{R}}))$
σ
∈
L
2
(
0
,
T
;
C
b
1
/
2
(
R
)
)
, the pathwise uniqueness holds.
