A Stochastic Model of Crystallization in an Emulsion - Laboratoire de ...

Considering (7), one has, for any integer n,
and so,
where ε( t) goes to 0 in H with t
.
Moreover, ( J( u ) (1 ϕ )) ∈ is a bounded sequence in H,
as well as ( b( u , ϕ )) ∈ ; so,
there exists a subsequence of ( ϕ ) , still noted ( ϕ ) , and, J and B in H,
such that :
from H to H,
one has
n
1 n
1
k k
k=0
k=0
ϕ ϕ = t J( u ) (1 ϕ ) + b( u , ϕ )( w w )
n
0
( n 1)t
( n 1)t
N N
ϕ ( n t) ϕ (0) =
N N
J( u ) (1 ϕ ) ds +
N N
b( u , ϕ ) dw( s)+
t J( u ) (1 ϕ ) + b( u , ϕ )( w w ) .
Thus, for any t in [0 , T ] and n such that ( n 1)t < t nt, 0
0 0 0 0 1 0
t t
N N
ϕ ( t) ϕ (0) =
N N
J( u ) (1 ϕ ) ds +
N N
b( u , ϕ ) dw( s)+
Since and b are bounded functions,
0 0
t J( u ) (1 ϕ ) + b( u , ϕ )( w w )
0 0 0 0 1 0
t t
n n
J( u ) (1 ϕ ) ds +
n n
b( u , ϕ ) dw( s ) .
( n1) t
( n1)t t t
N N
ϕ ( t) ϕ (0) =
N N
J( u ) (1 ϕ ) ds +
N N
b( u , ϕ ) dw( s) + ε( t)
0 0
N N
N
N N
N∈
N∈
N N
J( u ) (1 ϕ ) ⇀ J ,
N N
b( u , ϕ ) ⇀ B.
t t
Remarking that the applications u ↦→ u ds and u ↦→ u dw( s)
are continuous linear functions
0 0
t t t t
N N
J( u ) (1 ϕ ) ds +
N N
b( u , ϕ ) dw( s) ⇀ J ds + B dw( s ) .
t t
ϕ( t) ϕ(0) = J ds + B dw( s) in H;
for any t (since the Itô integral implies that ϕ can be chosen continuous).
2at
2
Then, for any a > 0 , by applying the Itô formula with f( t, x) = e || x||
, we get :
2at
2
e E|| ϕ( t) || E|| ϕ || =
k k k+1 k
0 0 0 0
So, as N goes to innity, we get,
0 0
10
0 2
0
N N N
(9)
(10)