Materiales plásticos

Of course one could now formulate a theorem that takes expectations of (3.1) at times 0 and t under conditions such that it is a true martingale. In the semigroup case for example, this yields a stochastic representation of∇dPtf (v, w) with terms that are perfectly symmetric in Ks and Ls, the first one starting in v and the second one in w.

For applications this result is too complicated, and we find a shorter formula if we addi-tionally assume that Ks and Ls vary on disjoint (stochastic) time intervals.

Theorem 3.4 (Non-intrinsic representation formula for the Hessian). fulfill the following properties:

i) K0 = v, Kt = 0, ˙Ks = 1_{]σ,τ ]}(s) ˙Ks (hence Ks = v for s ≤ σ and then decays to 0 order of integrability compared to Theorem 2.19).

Under these assumptions we derive the following stochastic representation formula for the Hessian Hess_xF₀(v, w)≡ ∇dF0(v, w):

In the heat semigroup case the result modifies to

∇dPtf (v, w) = − E

We discuss briefly how the results of [E-L 1] on the Hessian of heat semigroups are con-tained in our formulae.

Remark 3.5. In the case of compact M (such that ζ =∞) and f ∈ bC²(M ) for σ = ₂^t as

3.1. Non-intrinsic calculations 53

This is almost the same as Elworthy-Li’s result in [E-L 1], Theorem 3.1, except for their permutation of w and v in the first term of the right hand side. This is due to a slight error in the proof of Theorem 2.3 in that paper. One could also use the attempt of Remark 3.3 to use processes K_s and L_s both starting at 0 and varying to v and w within [0,₂^t] and

which now differs from the Elworthy-Li result by integration over the second time period in the last two terms instead of integrating over the first part.

Proof of Theorem 3.4. We apply our prerequisites on K_s and L_s to reduce the local mar-tingale n_s from (3.1) to the form

ns=∇dFs(X_s∗Ks, X_s∗Ls) + (dFs)_Xs(x)∇T Xs(Ks, Ls)

because integrals over ˙K need only be taken from time σ on, and this implies that L already has vanished.

Application of Theorem 2.14 for the pathwise integrals Rs

0 X_r∗⁻¹∇T Xr(v, ˙Lr)dr as well as R_s

0 X_r∗⁻¹∇A(Xr∗v)A(Xr(x))^∗Xr∗L˙rdr instead of Ks leads to the local martingale Ns

(because it differs from n_s only by two local martingales of first order type) N_s=∇dFs(X_s∗K_s, X_s∗L_s) + (dF_s)_Xs(x)∇T Xs(K_s, L_s)

− Fs(Xs(x)) Z σ∧s

0

h∇T Xr(v, ˙Lr), A(Xr(x))dZri

− Fs(X_s(x))

In the following, we verify that this new local martingale Ns is a true one on [0, τ ] by verifying that{N%: 0≤ % ≤ τ stopping time} is uniformly integrable.

Due to the continuity of F and its derivatives on the compact set [0, t]× D2, we are given finite constants C₀ := sup_{(s,y)∈[0,t]× ¯D2}|Fs(y)| as well as C1 := sup_{(s,y)∈[0,t]× ¯D2}kd(Fs)_yk and finally C2 := sup(s,y)∈[0,t]× ¯D2k∇d(Fs)yk (maximum norm of a bilinear map on TxM× TxM ).

As in the proof of Theorem 2.19 we make use of g1 := sup_{s∈[0,τ ]}kTxXsk ∈ L^p(P) as well as g₂ := sup_{s∈[0,τ ]}k∇TxX_sk ∈ L^p(P) for arbitrary 1 ≤ p < ∞ (because this holds for diffusions on compact manifolds and we may modify M outside D₂).

By choosing upper bounds C4 for|Ks| and C5 for|Ls| on [0, τ ] we get for any 0 ≤ % ≤ τ

The first two terms of the right hand side have finite expectation since g₁ and g₂ raised to any power are integrable. Moreover, the same Burkholder-Gundy and H¨older argument as in the proof of 2.19 provides that for some c > 0

E integrability ofRτ

0 k ˙Lrk²dr assumed in the theorem.

The analogous computation yields

E

3.1. Non-intrinsic calculations 55 any power is sufficient for this part.

For the fifth term of the right hand side of (3.17) according to the Schwartz inequality

E Again by Burkholder-Gundy we compute

E Obviously, by another H¨older argument with respect to the partition 1 = _1+α^α + _1+α¹ we split the right hand side into two expectations and the assumptionRt

0k ˙Lrk²dr in L^1+α(P) provides that this is finite.

Now we focus on the last term for which H¨older yields E

and reasoning as in the case just before proves that both factors are finite. So (Ns)0≤s≤τ

is a true martingale.

Comparing expectations for s = 0 and s = τ yields the Bismut formula (3.15), where we used∇T X0 = 0 since X0 = idM and the martingale property of Fs◦ Xs(x) for s ≤ τ to derive E[Fτ◦ Xτ(x)|Fσ^Z] = E[Fσ◦ Xσ(x)].

In particular, for the Hessian of the heat semigroup P_tf we have the preliminary result for τ = ˜τ ∧ (t − ε) and Ks≡ 0 on [t − ε, t], where ε < ₂^t:

The extension to general finite energy processes K_s works as in the first order case by approximating with suitable K_s^ε.

If our goal for second order derivatives was just to prove Theorem 3.4, we could replace the computations in the preceding section by the shorter following argument:

Theorem 3.6. Fix x ∈ M and v, w ∈ TxM . Let (L_s) be a (Fs^Z)-adapted process with sample paths in H([0, t], TxM ) and L0 = w. Then we have the local martingale

ns:=∇dFs(Xs∗v, Xs∗Ls) + (dFs)_Xs(x)∇T Xs(v, Ls)

− (dFs)_Xs(x)Xs∗

Z s 0

X_r∗⁻¹∇T Xr(v, ˙Lr)dr + (dF_s)_Xs(x)X_s∗

Z s 0

X_r∗⁻¹∇A(Xr∗v)A(X_r(x))^∗X_r∗L˙_rdr

− (dFs)_Xs(x)Xs∗v Z s

0

hXr∗L˙r, A(Xr(x))dZri.

(3.18)

Obviously, this result is formula (3.1) in the particular case of ks = 0 and therefore all terms depending linearly on ˙K_s vanish. In other words, one drops the dependence on ε the except for the deterministic one of the initial value H₀^ε,0(x) fulfilling _∂ε^∂ 

ε=0H₀^ε,0(x) = v = K0 ≡ Ks, 0 ≤ s ≤ t. We give a short proof based on the observation that in this special case we can simply take the covariant derivative in direction v of the first order local martingale (depending on L_s instead of K_s).

Proof of Theorem 3.6. We compute the covariant differential in direction v of the first order local martingale from (2.12) (with Ks replaced by Ls)

∇v



(dF_s)_Xs(x)X_s∗L_s− Fs(X_s(x)) Z s

0

hXr∗L˙_r, A(X_r(x))dZ_ri



=∇dFs(X_s∗v, X_s∗L_s) + (dF_s)_Xs(x)∇T Xs(v, L_s)

− (dFs)_Xs(x)X_s∗v Z _s

0

hXr∗L˙_r, A(X_r(x))dZ_ri

− Fs(Xs(x)) Z s

0 h∇T Xr(v, ˙Lr), A(Xr(x))dZri

− Fs(X_s(x)) Z _s

0

hXr∗L˙_r,∇A(Xr∗v)dZ_ri,

which is again a local martingale. The step to the claimed formula consists of the same integration by parts arguments as in the proof of 3.4 above (where we derived ns from Ns).

Remark 3.7. It should be obvious that and which way it is possible to prove Theorem 3.4 by means of this local martingale instead of (3.1) except for the last term.

For that purpose under the assumptions of 3.4 we consider the two local martingales O¹_s := (dF_s)_Xs(x)X_s∗(K_s− v) − Fs◦ Xs(x)

Z s 0

hXr∗K˙_r, A(X_r(x))dZ_ri, O²_s :=

Z s 0

hXr∗L˙r, A(Xr(x))dZri.

3.1. Non-intrinsic calculations 57

We obtain d(O¹_sO²_s) = dO^{m 1}_sdO²_s = 0 because of O¹ = 0 on [0, σ] and O² being constant on ]σ, τ ]. According to the assumptions on K and L, (again by a H¨older and Burkholder-Gundy argument) O¹O² is a true martingale on [0, τ ], which provides E[MτOτ] = 0 and therefore, as K_τ = 0,

E



(dFτ)_Xτ(x)X_τ∗v Z σ

0

hXr∗L˙r, A(Xr(x))dZri



=−E



Fτ◦ Xτ(x) Z τ

σ

hXr∗K˙r, A(Xr(x))dZri Z σ

0

hXr∗L˙r, A(Xr(x))dZri

 .

(3.19)