To finish the proof of our theorem we have to show that there exists only one process that can be the weak limit of the family{Xε(t)}t≥0 asε↓0. We prove this with the help of Strook-Varadhan martingale identification theorem. We start with the following lemma : Lemma 4.7.1 (cf. [32], Lemma 13, p. 260) For any γ ∈(0,1), M ∈N, ψ: (Rd)M →R+ The second term above is equal to
**
The first term of the right hand side of (4.24) may be rewritten in the form ME
The last integral above is of orderε. SinceME[Vi(ρ)] = 0 we may estimate as we already did in the previous section :
Ent(εγT−2)
Lemma 4.7.2 Under the assumptions of Lemma 10, for any ε >0 we have To finish the proof of the lemma we are left with the proof of the fact that the terms III andIV in (4.25) are of magnitudeo(εγ). Once again we may find aV−∞,0-measurable random variable Ψ such thatIII may be rewritten in the form
**
1
Continue this way and we get that (4.27) is equal 1
sinceE[Vi(ρ)] = 0. The first term above may be estimated with help of Lemma 4.5.1. We have 5QkΨY5Lq ≤ kC2 for any q ∈ [1,2), k ∈ N, the constant C >0 depending only on q
This ends the proof of Lemma 4.7.2 !
Lemma 4.7.3 For any γ ∈(0,1) and 0< γ( < γ there exists a constant C >0 such that get ii)−iv) by Schwartz inequality. But v) is obvious. To prove i) we have to find an appropriate estimate from above of the term
ε4
and this is exactly the result of Lemma 15 of [32] ! Now let f ∈C0∞(Rd). Let ψ be as in Lemma 11 and let tm =s+mεγ. From the Taylor expansion formula we have
ME[(f(Xε(t))−f(Xε(s)))Ψ] = /
m:s<tm<t
ME{@[Xε(tm+1)−Xε(tm)],(∇f)(Xε(tm))AΨ}+ 1
2 /
m:s<tm<t
ME{@[Xε(tm+1)−Xε(tm)]⊗[Xε(tm+1)−Xε(tm)],(∇ ⊗ ∇f)(Xε(tm))AΨ}+ 1
6 /
m:s<tm<t
ME{@[Xε(tm+1)−Xε(tm)]⊗[Xε(tm+1)−Xε(tm)]⊗[Xε(tm+1)−Xε(tm)], (∇ ⊗ ∇ ⊗ ∇f)(Xε(θm))AΨ},
where θm is a point in the segment [Xε(tm),Xε(tm+1)]. The first term of the sum above is estimated with help of Lemma 4.7.1 by the term of magnitude o(εγ). The third one, by H¨older inequality and Lemma 4.7.3, may be estimated from above byCε3γ2!. Choosing carefullyγ ∈(0,1) andγ( ∈(0, γ) we get again the term of magnitude o(εγ). The second term we may rewrite in the form
1 2
/
m:s<tm<t
ME{@[Xε(tm+1)−Xε(tm)]⊗[Xε(tm+1)−Xε(tm)]−Dε,(∇ ⊗ ∇f)(Xε(tm))AΨ}+ 1
2 /
m:s<tm<t
ME{@Dε,(∇ ⊗ ∇f)(Xε(tm))AΨ}, where
Dε=ME[[Xε(tm+1)−Xε(tm)]⊗[Xε(tm+1)−Xε(tm)]].
And this, by Lemma 4.7.2, up to the terms of ordero(εγ) is equal to : 1
2 /
m:s<tm<t
/d
i,j=1
εγDijME[∂ij2f(Xε(tm))Ψ] +o(εγ)
. (4.30)
Thus we get that the limit of the family of processes {Xε(t)}t≥0 as ε ↓ 0 must have the lawµ on the spaceC([0,+∞),Rd) such that for any f ∈C0∞(Rd),
f(x(t))−1 2
/d
i,j=1
Dij
@ t
0
∂ij2f(x(ρ))dρ, t≥0
is a martingale underµ. By the theorem of Strook and Varadhan (see e.g. [46]) it may be identified as a diffusion with the covariance matrixD= [Dij]. This is the end of the proof
of our theorem !
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