
If V
()
)
00x =
, then (3.30) will be reduced to (3.2).
If the synthetic gene network is free of the external disturbances, i.e.,
0v =
, then we have
()
()
0
<0
T
ExQxdtVx
∞
∫
(3.31)
For some positive constant V
()
)
0x
, it means
0x →
in probability as
t →∞
.
Appendix 3.2: Proof of proposition 3.2
Let us choose a Lyapunov function V
()
0x >
for the stochastic gene network
in (3.7). By the Ito formula (Chen and Hsu 1995), we get
() ()
()
()
()
()
()
()
()
()
()
2
2
1
2
2
1
1
2
1
2
T
d
m
TT
idi iid
i
T
d
m
TT
idi iid
i
dV x V x
EE Nfxxv
dt x
Vx
gxxM Mgxx
x
Vx
Nfx x
x
Vx
gxxM Mgxx
x
=
=
⎧
∂
⎛⎞
⎪
=++
⎡⎤
⎨
⎜⎟
⎣⎦
∂
⎝⎠
⎪
⎩
∂
++ +
∂
∂
⎛⎞
+Δ+
⎜⎟
∂
⎝⎠
⎫
∂
⎪
+Δ+ Δ+
⎬
∂
⎪
⎭
∑
∑
(3.32)
By the fact that
()
()
() ()
()()
() ()
()()
2
1
4
1
4