Biomedical Engineering Reference
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(
)
Furthermore, we obtain the random empirical distribution of tips X N
t
as a marginal
of the measure Eq. ( 13 )
N ( t )
i = 1 ε X k ( t ) = Q N ( t )( ·× R
1
N
d
X N (
t
)=
) .
(14)
From the Lagrangian description above, Ito's formula applied to a function g
∈
d
d
of Z k N (
C b (R
, provides the time evolution equation
of the empirical measure Q N . Indeed, by Ito's formula [ 4 ], from system ( 8 ), we get
× R
)
t
)
,forany k
=
1
,...,
N
(
t
)
X k
v k
X k
v k
X k
v k
d X k
d g
((
(
t
) ,
(
t
))) =
d g
((
(
0
) ,
(
0
)))+ ∇ x g
((
(
t
) ,
(
t
)))
(
t
)
X k
v k
d v k
+ ∇ v g
((
(
t
) ,
(
t
)))
(
t
)
1
2 Δ v g
X k
v k
d v k
2
+
((
(
t
) ,
(
t
)))(
(
t
))
X k
v k
X k
v k
v k
=
d g
((
(
0
) ,
(
0
)))+ ∇ x g
((
(
t
) ,
(
t
)))
(
t
)
X k
v k
− ∇ v g
((
(
t
) ,
(
t
)))
kv k
F C
d t
X k
X k
×
(
t
) −
(
t
,
(
t
)) ,
f
(
t
,
(
t
))
2
2 Δ v g
+ σ
X k
v k
((
(
t
) ,
(
t
)))
d t
X k
v k
d W k
+ σ∇ v g
((
(
t
) ,
(
t
)))
(
t
) .
(15)
By summing up into Eq. ( 15 ), we may obtain evolution equations for the random
measure Q N ,
as follows. For any B
∈B R d × R d
g
(
x
,
v
)
Q N (
t
)
d
(
x
,
v
)=
g
(
x
,
v
)
Q N (
0
)
d
(
x
,
v
)
B
B
t
+
∇
x g
(
x
,
v
)
v
+
g
(
x
,
v
) α
h
(
C
(
s
,
x
)) δ
(
v
)
v 0
0
B
− ∇ v g
(
x
,
v
)[
kv
−
F
(
C
(
t
,
x
) ,
f
(
t
,
x
))]
Q N (
2
2 Δ v g
+ σ
M N (
(
x
,
v
))
t
)(
d
(
x
,
v
))
d s
+
t
) ,
(16)
where the last term
t
M N (
t
)=
n g
(
s
,
x
)[ Φ (
d s
,
d x
) −
N
α
h
(
C
(
s
,
x
))
X N (
s
)(
d x
)
d s
]
0
R
N
)
k = 1 ∇ v g (( X k
(
t
t
2 N
v k
d W k
+
(
t
) ,
(
t
)))
(
t
)
(17)
0
 
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