ON THE APPROXIMATE SOLUTIONS OF THE STRATONOVITCH EQUATION

We present new methods for proving the convergence of the classical approximations of the Stratonovitch equation. We especially make use of the fractional Liouville-valued Sobolev space W r;p ( J (cid:11);p ). We then obtain a support theorem for the capacity c r;p .


Introduction
This paper is a contribution to the study of the approximations of solutions of the Stratonovitch equation Many authors and especially Ikeda-Watanabe [8] have studied this problem by means of piecewise linear approximations of the Brownian motion. Here we introduce a method which simplifies and shortens the calculations in three ways. a) We use the notion of (strong) approximate solution of (S), which eliminates the need to have simultaneously the approximate solution and the exact solution in the calculations. b) We use the Liouville space J α,p , where it turns out that the calculations are simpler even than with uniform convergence. The main point is the isomorphism J α,p (L p ) ≈ L p (J α,p ). Moreover this isomorphism is a sharpening of the Kolmogorov lemma (cf. [5,6]). c) With the classical regularity conditions on σ and β, we prove convergence of approximate solutions in each space W r,p (J α,p ) for suitable values of α and p. Without using truncation property, this improves some results of [8].
d) The p-admissibility (cf. [4]) of the vector-valued Sobolev space W r,p (J α,p ) allows us to obtain easily convergence in the space L 1 (Ω, c r,p , J α,p ) which is the natural space of J α,p -valued quasi-continuous functions on the Wiener space Ω (cf. Cor.4 below). This is a sharpening of the preceding known results ( [3,8,10,12]). As a corollary we not only see that the image measure X(µ) is carried by the closure of the skeleton X(H) in the space J α,p , but that this is also true for the image measure X(ξ) for every measure ξ majorized by the capacity c r,p . In fact, in the same way as for the Hölder support theorem (cf. [1,2,3,7,9,10,11,14]), we obtain a support theorem for capacity : the support of the image capacity X(c r,p ) is exactly the closure of the skeleton.

I. Preliminaries
Let f : [0, 1] → IR a Borel function. For 0 < α ≤ 1 the Liouville integral is Recall that I α (L p (dx)) ⊂ L p (dx) and that I α is one to one (cf. [5,13]). The range J α,p = I α (L p ) is a separable Banach space under the norm where N p stands for the L p -norm. Denote H α the space of α-Hölder continuous functions vanishing at 0 with its natural norm. This is not a separable space.
Particularly, taking B = L p (Ω, µ) where µ is a measure, we get the Kolmogorov theorem: if (X t ) t∈[0,1] is a IR m -valued process satisfying N p (X t − X s ) ≤ c|t − s| α , then for α > β > 1/p, this process has a modification with (β − 1/p)-Hölder continuous trajectories. Indeed, it suffices to point out that X − X 0 belongs to the space Proof : First it is easily seen that N p (X t − X n t ) converges to 0 uniformly with respect to t as n → ∞. Take α such that α > α > β > 1/p and η > 0. We get Hence, convergence holds in the space H α (L p ) ⊂ J β,p (L p ), and we are done.
2 Remarks: a) One can only assume that Lim n→∞ N p (X t − X n t ) = 0 for every t in a dense subset D ⊂ [0, 1]. b) We can prove more precisely the estimate X − X n H α (L p ) ≤ Kε 1−α /α n . This gives a criterion for the convergence of the series Σ n (X − X n ). Now assume that (Ω, µ) is a Gaussian vector space, and let W r,p (Ω, µ) be the (r, p) Sobolev space endowed with the norm f r,p = N p (I − L) r/2 f where L is the Ornstein-Uhlenbeck operator. Recall that we have the isomorphism W r,p (Ω, J β,p ) ≈ U r (L p (Ω, J β,p )) where U = (I − L) −1/2 according to [4], th.25 (p-admissibility of the space J β,p which is a closed subspace of an L p -space). In view of the above proposition, we obtain 3 Proposition: (The Sobolev-Kolmogorov-Ascoli lemma) For p ∈]1, +∞[ and r ∈]0, +∞[, we have J α,p (W r,p (Ω, µ)) ≈ W r,p (Ω, µ, J α,p ) as above. Moreover, let X n ∈ H α (W r,p ) a sequence of processes. Assume that X n t − X n s r,p ≤ c|t − s| α and that Lim n→∞ X t − X n t r,p = 0 for every t. Then X n converges in the space The first isomorphism is obvious (cf. [5]). Put Y n t = (I − L) r/2 X n t and apply the previous proposition to X n and Y n .
4 Corollary: Under the same conditions, the process X n converges to X in the space L 1 (Ω, c r,p , J β,p ).

II The Stratonovitch equation.
a Stratonovitch SDE. In this formula, W t is the -dimensional Brownian motion, •dW s stands for the Stratonovitch differential, σ(x) is an (m, )-matrix, β(x) an (m, 1)-column, σ and β are Lipschitz, and x 0 ∈ IR m .
If X is a Borel process, we denote X its predictable projection. Note that we have Let ε > 0, we say that a Borel process X is an ε-approximate solution in L p of (S) if we have for every t ∈ [0, 1]. In this formula, the stochastic integral is to be taken in the Ito sense. Moreover ϕ = σ ·σ is the contracted tensor product ϕ i j, = Σ k (∂ k σ i j )σ k and Tr ϕ stands for the convenient vector-valued trace Σ j,k (∂ k σ i j )σ j k .
In fact we will suppose in the following that β = 0. Indeed, the case β = 0 does not bring any other difficulty.
5 Proposition: Let ε n be a sequence tending to 0, and let X n be a sequence of ε n -approximate solutions in L p . Assume that σ and ϕ are Lipschitz. Then, X n t converges in L p towards the solution of (S).
In addition suppose that we have With this additional condition, we say that X n is a sequence of strong ε n -approximate solutions.
Proof : Burkholder's inequality gives so that by Gronwall's lemma we have Now under the additional hypothesis, in view of the Kolmogorov-Ascoli lemma, the convergence holds in the space L p (J α,p ) for 1/p < α < 1/2.

Searching approximate solutions
Now the problem is to find a sequence of strong ε n -approximate solutions of (S) with ε n → 0.
Let W π t be the linear interpolation defined by For t ∈ π we also have We remark that X is not an adapted process, we only have X t ∈ F t for t ∈ π so that X is an adapted process.
so that we obtain the following inequalities The next lemma proves that if as δ → 0 X is an approximate solution of (S), then it defines a strong approximate solution.
6 Lemma: If σ is Lipschitz, and p > 1, there exists a constant K such that N p (X t − X s ) ≤ K|σ(x 0 )| √ t − s, for every s, t ∈ [0, 1] and every π. Proof : First, for a, b ∈ π we get from (3) By Burkholder's and Cauchy-Schwarz inequalities and by Gronwall's lemma, as σ is Lipschitz In view of (2) this last inequality extends to every a, b ∈ [0, 1], with a constant K independent of π.
7 Lemma: If σ and ϕ are Lipschitz, we have for t ∈ π where the symbol ϕ(X t i )·(∆W i ) (2) stands for k, ϕ j k, (X t i )∆W k i ∆W i Proof : First by the fundamental theorem of calculus we get Tr ϕ( X s ) ds It is easy to see that H t is the martingale with the symmetrized ϕ, so that by Burkholder's inequality we get Proof : In view of the preceding lemmas, this is obvious for t ∈ π. If t ∈ [t i , t i+1 [ we have to add a term which is (by formulas (1)- (3)) easily seen to be smaller than K|σ(x 0 )| √ t − t i , and use the inequality Then, as δ → 0 we see that X t is a convergent sequence of approximate solutions of (S) (it is strong by lemma 6).
10 Theorem: If σ and ϕ are Lipschitz then we have so that as δ → 0, X t is a sequence of approximate solutions of (S). Moreover we also have so that the convergence holds in L p (J α,p ) for 1/p < α < 1/2.
and to bring it up in the inequality of proposition 5. The second inequality is exactly lemma 6.
11 Remarks: a) The theorem extends to the case where β = 0 and β Lipschitz, as noted before proposition 5. b) By remark 2b we can calculate the rate of decrease of a sequence δ n in order to have for the corresponding series Σ n (X n − X n+1 ) to converge normally.

III. Approximate solutions in the Sobolev space
By Meyer's theorem, the Sobolev space W 1,p (Ω, µ) exactly is the space of functions f ∈ L p such that the weak derivative f (x, y) ∈ L p (Ω × Ω, µ ⊗ µ) with the norm Recall that the solution of an ODE. Its derivative Y t (ω, ) = X t (ω, ) in W 1,p (Ω, µ) satisfies the following ODE This is a linear equation, we then have 12 Lemma: Assume that σ and σ are Lipschitz and bounded. Then R t and R −1 t are bounded in L p independently of π. Proof : For t ∈ π, write as above. Burkholder's inequality yields for t ∈ π By Gronwall's lemma we get as above (2). The same result holds for R −1 t , as it satisfies 13 Proposition: Assume that σ is bounded with all of its derivatives up to order 3. Then R t converges in L p (J α,p ) for every 1/2 > α > 1/p as π refines indefinitely.
Proof : As in lemma 7, we get and as in proposition 9 As in theorem 10 we infer that R t is an approximate solution of the Stratonovitch SDE where X t is the solution of the corresponding SDE.
14 Remark: The analogous result holds for R −1 t . As in theorem 10, the convergence holds in the space L p (J α,p ) for every 1/2 > α > 1/p.
15 Theorem: Assume that σ is bounded with all of its derivatives up to order 3. Then as π refines indefinitely, Y t (ω, ) converges in every L p (J α,p ). In the same way X t (ω) converges in every W 1,p (J α,p ).
Proof : It suffices to remark that for almost every ω, Y t (ω, ) converges to a Wiener integral in . The convergence takes place in L p (µ ⊗ µ, J α,p ).

Higher order derivatives
Now, assume that σ is bounded with all of its derivatives. For a partition π compute As in the preceding proof we can write where R t denotes the resolvent of this linear ODE, which is the same as in proposition 13.
16 Lemma: Let π be a partition of [0,1], let v t be a continuous process which is π-adapted, that is v t ∈ F t for t ∈ π. Assume that where K p does not depend on π. Consider the following processes If for every t u t and v t converges in every L p , then s t converges in the space L p (J α,p ) for 1/p < α < 1/2.
Proof : As in lemma 7, by the fundamental theorem of calculus, we get for t ∈ π As in lemma 7 and lemma 8 where Tr stands for a suitable tensor-contraction of v t . As π refines indefinitely, we get the convergence of s t in every L p , for every t. It remains to prove the convergence in the space L p (J α,p ). Replacing [0, t] with [t i , t j ] and using Burkholder's inequality yield Then for s < t i < t j < t we get Applying the Kolmogorov-Ascoli lemma (prop. 1) gives the result.
Proof : First, for the second derivative, it suffices to apply the preceding lemma to the processes which is an L p (dω 1 ⊗ dω 2 )-valued process.
It is straightforward to verify that the lemma applies in the same way at any order of derivation.
19 Remark: As proved in [4], th.27, we find again that X has a modification X : J α,p → J α,p which is c r,p -quasi-continuous for every (r, p).

Application to the support theorem
Assume that the hypotheses of theorem 15 hold, that is σ and β are bounded with all of its derivatives up to order 3. Denote X n the solution of the ODE where dW n s stands for the ordinary differential associated to the subdivision of maximum length δ n . We have seen that X n converges in the space L p (J α,p ) for 1/2 > α > 1/p. Let X be the limit, which is the solution of the Stratonovitch SDE. We also denote X(h) the solution of the ODE where h(t) = t 0 h (s) ds belongs to the Cameron-Martin space H l = W 1,2 0 ([0, 1], dx, IR l ). Recall that h → X(h), which is a map from H l into H m , is the so-called skeleton of X. Of course X n = X(ω n ) where ω n is the piecewise linear approximation of ω. Then we have an improvement of the classical support theorem 20 Theorem: The support of the image capacity X(c 1,p ) is the closure in Ω = J α,p of the skeleton X(H).
Proof : Let ϕ be a continuous function on J α,p which vanishes on X(H). Then ϕ(X n ) vanishes for every n. We can assume that X n converges c 1,p -q.e. to X, so that as n converges to +∞, ϕ(X n ) = 0 converges to ϕ(X). Then ϕ(X) = 0 q.e., so X(c 1,p ) is carried by the closure of the skeleton. Now notice that by the result of [3,10], any point in the skeleton belongs to the support of X(µ) in the space H γ . As H γ ⊂ J α,p for 1/2 > γ > α > 1/p, (cf. [5,6]), so by the obvious inclusion Supp (X(µ)) ⊂ Supp (X(c 1,p )), we get the result. Now let ξ a measure belonging to the dual space of L 1 (Ω, c 1,p ). Then X is ξmeasurable and we have 21 Corollary: The image measure X(ξ) is carried by the closure of the skeleton X(H).
22 Remark: If σ is bounded with all of its derivatives, we can replace c 1,p with c r,p in the preceding results.