Biomedical Engineering Reference
In-Depth Information
2
d
t
F
0
N
)
i
=
1
|
g
(
X
k
(
t
T
1
N
2
=
C
1
E
(
t
)
,
v
0
)
|
0
2
d
t
F
0
N
)
i
=
1
|
∇
v
g
(
x
,
v
)
|
(
t
T
2
N
2
σ
+
C
2
E
0
sup
t
F
0
C
T
2
2
≤
N
(
g
+
∇
g
)E
T
T
N
(
t
)
,
1
≤
C
T
<
N
.
(42)
E
sup
)
|
2
From Eq. (
42
),
t
≤
T
|
M
N
[
Q
N
,
W
](
t
tends to zero, as
N
increases to infinity.
Hence, from
E[
M
N
(
t
)] =
0
,
the martingale Eq. (
17
) vanishes in probability, i.e.,
uniformly in
[
0
,
T
]
,
P
−→
[
,
](
)
.
M
N
Q
N
W
t
0
This is the substantial reason of the deterministic limiting behavior of the process
Q
N
,as
N
increases to infinity.
Let us consider now a heuristic derivation of a continuum dynamics, by assuming
that, as
N
→
∞
(
)
(
)
, the measure
Q
N
t
admits a limit measure
Q
t
, i.e., formally, we
take
(
)(
(
,
))
→
Q
∞
(
)(
(
,
)) =
(
,
,
)
,
Q
N
t
d
x
v
t
d
x
v
p
t
x
v
d
x
d
v
(43)
(
)
(
)
, i.e.,
T
∞
(
)(
)=
then we would have limit measures also for
T
N
t
and
V
N
t
t
d
x
(
,
)
d
x
;and
V
∞
(
)(
)=
(
,
)
,
p
t
x
t
d
x
w
t
x
d
x
where
p
(
t
,
x
)=
p
(
t
,
x
,
v
)
d
v
,
w
(
t
,
x
)=
vp
(
t
,
x
,
v
)
.
(44)
Then system (
16
)-(
20
) becomes
d
s
d
x
d
v
σ
2
2
Δ
v
g
t
g
(
x
,
v
)
p
(
t
,
x
,
v
)
d
x
d
v
=
p
(
s
,
x
,
v
)
(
x
,
v
)
B
0
B
(
C
+
∇
x
g
(
x
,
v
)
v
+
g
(
x
,
v
)
α
h
(
s
,
x
))
δ
{
v
0
}
(
v
)
kv
F
C
)
,
f
−
∇
v
g
(
x
,
v
)
−
(
t
,
x
(
t
,
x
)
(45)
∂
∂
t
C
d
1
C
)
−
η
C
(
t
,
x
)=
c
1
δ
A
(
x
)+
(
t
,
x
(
t
,
x
)
w
(
t
,
x
)
;
(46)
∂
∂
t
f
)
f
(
t
,
x
)=
β
p
(
x
,
t
)
−
γ
m
(
x
,
t
(
t
,
x
)
;
(47)
∂
∂
t
m
(
t
,
x
)=
ε
1
m
(
t
,
x
)+
ν
1
p
(
x
,
t
)
−
ν
2
m
(
t
,
x
)
.
(48)
Search WWH ::
Custom Search