# Malliavin Calculus of Bismut Type for Fractional Powers of Laplacians in Semi-Group Theory

Authors: Rémi Léandre
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

We translate into the language of semi-group theory Bismut's Calculus on boundary processes (Bismut (1983), Lèandre (1989)) which gives regularity result on the heat kernel associated with fractional powers of degenerated Laplacian. We translate into the language of semi-group theory the marriage of Bismut (1983) between the Malliavin Calculus of Bismut type on the underlying diffusion process and the Malliavin Calculus of Bismut type on the subordinator which is a jump process.


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## Body

## 1. Introduction

LetX01,X11,…,X1m,X02,X12,…,X2m be 2m+2 vector fields on ℝd with bounded derivatives at each order. Let(1.1)L1=∂∂s+X01+12∑i>0(Xi1)2
be an Hoermander's type operator on ℝ1+d. Let(1.2)L2=∂∂s+X02+12∑i>0(Xi2)2
be a second Hoermander's operator on ℝ1+d. Bismut [1] considers the generator(1.3)A=-12-2L1-12-2L2
and the Markov semi-group exp[t𝔸]. This semi-group has a probabilistic representation. We consider a Brownian motion t→zt independent of the others Brownian motions Bti. Bismut introduced the solution of the stochastic differential equation starting at x in Stratonovitch sense:(1.4)dxt(x)=Izt<0(X01(xt(x))dt+∑i>0Xi1(xt(x))dBti)+Izt>0(X02(xt(x))dt+∑i>0Xi2(xt(x))dBti),
where t→Bti are m independent Brownian motions.Let us introduce the local timet→Lt associated with  t→zt and its right inverse t→At (see [2, 3]). Then,(1.5)exp⁡[tA]f(0,x)=E[f(At,xAt(x))].
Such operator is classically related to the Dirichlet Problem [3].Classically [4],(1.6)exp⁡[tL1]f(x)=E[f(xt1(x))],
where xt1(x) is the solution of the Stratonovitch differential equation starting at x:(1.7)dxt1(x)=X01(xt1(x))dt+∑Xi1(xt1(x))dBti,
The question is as following: is there an heat-kernel associated with the semi-group exp[t𝕃1]? This means that(1.8)exp⁡[tL1]f(x)=∫Rdf(y)pt(x,y)dy.
There are several approaches in analysis to solve this problem, either by using tools of microlocal analysis or tools of harmonic analysis. Malliavin [5] uses the probabilistic representation of the semi-group. Malliavin uses a heavy apparatus of functional analysis (number operator on Fock space or equivalently Ornstein-Uhlenbeck operator on the Wiener space, Sobolev spaces on the Wiener space) in order to solve this problem.Bismut [6] avoids using this machinery to solve this hypoellipticity problem. In particular, Bismut’s approach can be adapted immediately to the case of the Poisson process [7]. The main difficulty to treat in the case of a Poisson process is the following: in general the solution of a stochastic differential equation with jumps is not a diffeomorphism when the starting point is moving (see [8–10]).The main remark of Bismut in [1] is that if we consider the jump process t→xAt1(x), then it is a diffeomorphism almost surely in x. So, Bismut mixed the tools of the Malliavin Calculus for diffusion (on the process t→xt1(x)) and the tools of the Malliavin Calculus for Poisson process (on the jump process t→At) in order to show that this isthe problem if(1.9)E[f(At,xAt(x))]=∫R1+dqt(s,y)f(s,y)dsdy.
Developments on Bismut’s idea was performed by Léandre in [9, 11]. Let us remark that this problem is related to study the regularity of the Dirichlet problem (see [1, page 598]) (see [12–14] for related works).Recently, we have translated into the language of semi-group theory the Malliavin Calculus of Bismut type for diffusion [15]. We have translated in semi-group theory a lot of tools on Poisson processes [16–22]. Especially, we have translated the Malliavin Calculus of Bismut type for Poisson process in semi-group theory in [17]. It should be tempting to translate in semi-group theory Bismut’s Calculus on boundary process. It is the object of this work.On the general problematic on this work, we refer to the review papers of Léandre [23–25]. It enters in the general program to introduce stochastic analysis tools in the theory of partial differential equation (see [26–28]).


## 2. Statements of the Theorems

Let us recall some basis on the study of fractional powers of operators [29]. Let 𝕃 be a generator of a Markovian semi-group Ps. Then,(2.1)--L=C∫0∞s-3/2(Ps-I)ds.
The results of this paper could be extended to generators of the type(2.2)A=∫0∞g(s)(Ps-I)ds,
where ∫0∞g(t)∧1dt<∞ and g≥0, but we have chosen the operator of the type (1.3) to be more closely related to the original intuition on Bismut’s Calculus on boundary process. Let be 𝔼d=ℝ1+d×𝔾d×𝕄d where 𝔾d is the space of invertible matrices on ℝd and 𝕄d the space of symmetric matrices on ℝd. (s,x,U,V) is the generic element of 𝔼d. V is called the Malliavin matrix.On𝔼d, we consider the vector fields:(2.3)X̂i1=(0,Xi,DXi1(x)U,0),Ŷ1=(0,0,0,∑i=1m〈U-1Xi,⋅〉2).
We consider the Malliavin generator 𝕃̂1 on 𝔼d:(2.4)L̂1=∂∂s+X̂01+12∑i>0(X̂i1)2+Ŷ1.
We consider the Malliavin semi-group P̂t1 associated and -L̂1.We perform the same algebraic considerations on𝕃2. We get L̂2, P̂t2, and -L̂2. Let us consider the total generator(2.5)Â=--L̂1--L̂2
and the Malliavin semi-group exp[t𝔸̂].We get a theorem which enters in the framework of the Malliavin Calculus for heat-kernel.Theorem 2.1.
Let one suppose that the Malliavin condition inx is checked:
(2.6)exp⁡⁡[tÂ][det⁡V-p](0,x,I,0)<∞
holds for all p, then
(2.7)exp⁡[tA]f(0,x)=∫R1+df(s,y)qt(s,y)dsdy,
where qt(s,y) is the density of a probability measure on ℝ1+d.Theorem 2.2.
If the quadratic form(2.8)∑i>0〈Xi1(x),⋅〉2+∑i>0〈Xi2(x),⋅〉2
is invertible in x, then the Malliavin condition holds in x.Remark 2.3.
We give simple statements to simplify the exposition. It should be possible by the method of this paper to translate the results of [9, part III], got by using stochastic analysis as a tool.


## 3. Integration by Parts on the Underlying Diffusion

We consider the vector fields onℝ1+d+1,(3.1)Xi,s,tj,1=(0,Xij(x),Zi,s,tj),
where Zi,s,tj=〈ϕ(x),hs,tj〉i (ϕ(x) is a convenient matrix on ℝm which depends smoothly on x and whose derivatives at each order are bounded. (s,t)→hs,tj does not depend on x, and hs,tj belong to ℝm). Let f̃ be a smooth function on ℝ1+d+1, D̃f̃ denotes its gradient, and D̃2f̃ denotes its Hessian.We consider the generator𝕃s,tj,1 acting on smooth functions on ℝ1+d+1,(3.2)Ls,tj,1f̃=∂∂sf̃+〈X0j(x),D̃f̃〉+12∑i>0〈DXij(x)Xij(x),D̃f̃〉+12∑i>0〈Xi,s,tj,1,D̃2f̃,Xi,s,tj,1〉.
In (3.2), the generator is written under Itô’s form. It generates a time inhomogeneous in the parameter s semi-group Ps,tj,1.  We can consider(3.3)--L⋅,tj,1=C∫0∞s-3/2(Ps,tj,1-I)ds.
We put(3.4)At1=--L⋅,tj,1--L⋅,tj,2.
It generates a semi-group Pt1.Let us consider the Hoermander’s type generator associated with the smooth Lipschitz vector fields onℝ1+d+d((s,x,U) on ℝ1+d+d):(3.5)Xij,2=(0,Xij,DXijU),Y0,s,tj,2=(0,0,∑Xij(x)Zi,s,tj)=(0,0,Yi,s,tj),Ls,tj,2=X0j,2+12∑i>1(Xij,2)2+Y0,s,tj,2.
We consider the heat semi-group associated with 𝕃s,tj,2(3.6)∂∂sPs,tj,2f̃=Ls,tj,2Ps,tj,2f̃.
Let us recall [15, Theorem 2.2] that(3.7)Ps,tj,1[uf](s0,x0,0)=Ps,tj,2[〈Df,U〉](s0,x0,0),
where f depends only on (s,x). In the left-hand side of (3.7), we apply the enlarged semi-group to the test function (s,x,u)→f(s,x)u and in the right-hand side we apply the semi-group to the test function (s,x,U)→〈Df,U〉. u belongs to ℝ and U belongs to ℝd. From this, we deduce the following.Lemma 3.1.
One has the relation(3.8)-L⋅,tj,1[uf](s0,x0,0)=-L⋅,tj,2[〈Df,U〉](s0,x0,0).Let us consider the semi-groupPt2 associated with(3.9)At2=--L⋅,t1,2--L⋅,t2,2.
We get, with the same notations for (s,x,u,U) the following.Theorem 3.2.
Forf bounded continuous with compact support in (s,x), one has the following relation:
(3.10)Pt2[〈Df,U〉](s0,x0,0)=Pt1[fu](s0,x0,0).Proof.
For the integrability conditions, we refer to the appendix.
We remark that∂/∂u commute with 𝔸t1, therefore with Pt1. We deduce that
(3.11)Pt1[fu](s0,x0,u0)=u0exp⁡⁡[tA][f](s0,x0)+Pt1[fu](s0,x0,0).
By the method of variation of constants,
(3.12)Pt1[fu](s0,x0,0)=∫0texp⁡⁡[(t-s)A][As1[uexp⁡⁡[sA][f](⋅,⋅,0)]](s0,x0)ds.
In order to show that, we follow the lines of (2.17) and (2.18) in [15]. We apply 𝔸t1 to (3.11).
By Lemma3.1,
(3.13)As1[uexp⁡⁡[sA][f](⋅,⋅)](s1,x1,0)=As2[〈D(exp⁡[sA]),U〉](s1,x1,0).
Let us consider the vector fields on ℝ1+d×𝔾d,
(3.14)Xij,3=(0,Xij,DXijU).
We consider the Hoermander’s type operator associated with these vector fields:
(3.15)Lj,3=X0j,3+12∑i>0(Xij,3)2.
We consider the generator
(3.16)At3=--Ls,t1,3--Ls,t2,3.
It generates a semi-group Pt3. By lemma 3.2 of [15], we have
(3.17)Dexp⁡⁡[sA][f](s1,x1)=Ps3[DfV](s1,x1,I).
By [15, Equation (3.18)],
(3.18)Ps,tj,2[Pt3[DfU](⋅,I)V](s1,x1,0)=∑i∫0sPs-v,tj[∑i〈Yi,v,tj,Pv,tj,3[Pt3[DfU](⋅,I)]〉](s1,x1,0).
In [15, Equation (3.18)], we consider the semi-group P¯t′ instead of the semi-group Ps,tj,2 and the test function Df instead as of the test function Pt3[DfU](·,I) here. Yi,v,tj is considered as an element of ℝ and not as a one-order differential operator:
(3.19)∂∂sPs,tj,3f̃=Ls,tj,3Ps,tj,3f̃.
Therefore,
(3.20)At2[Pt3[DfV](⋅,I)V](s1,x1,0)=∑i,jC∫0∞s-3/2∫0sPs-v,tj[〈∑iYi,v,tj,Pv,tj,3[Pt3[DfU](⋅,I)]〉](s1,x1,0)dvds.
We write
(3.21)At2=At3+Ãt3,
where
(3.22)Ãt3[fU](s0,x0,U0)=∑jC∫0∞s-3/2(Ps,tj,2-Ps,tj,3)[fU](s0,x0,U0)ds.
The Volterra expansion (see [15, Equation (3.17)]) if it converges gives the following formula:
(3.23)Ps,tj,2[fU](s0,x0,U0)=∑∫0<s1<s2<⋯<sn<tds1⋯dsnPs1j,3∑Yi,s1,tj⋯Psn-sn-1j,3×∑Yi,sn,tj⋯Pt-snj,3[fU](s0,x0,U0).
But u0→Ps,tj,3[fU](s0,x0,U0) is linear in u0. Therefore:
(3.24)Ps,tj,2[fU](s0,x0,U0)=Ps,tj,3[fU](s0,x0,U0)+∫0sPvj〈∑iYi,v,tj,Ps-v,tj,3[fU]〉(s0,x0,U0)dv.
In this last formula, Yi,s,tj are considered as differential operators.
Therefore,Ãt3[fU](s0,x0,U0) does not depend on U0 and is equal to
(3.25)∑i,jC∫0∞s-3/2∫0sPvj〈∑iYi,v,tj,Ps-v,tj,3[fU](s0,x0,I)〉dsdv,
where Yi,s,tj are considered as elements of ℝd. We deduce as in [15, Equation (3.17)],
(3.26)Pt2[fU](s0,x0,0)=∫0texp⁡[(t-s)A]Ãs3Ps3[fU](s0,x0,0)ds.
But U0→Ps3[fU](s0,x0,U0) is linear. Therefore,
(3.27)Ãt3Pt3[fU](s0,x0,0)=∑i,jC∫0∞s-3/2∫0sPvj〈∑iYi,v,tj,Ps-v,tj,3[Pt3[fU]](s0,x0,I)〉dsdv.
It remains to replace f by Df in this last equation and to compare (3.26) with (3.13) and (3.20).We consider the Malliavin generator𝔸̂. We can perform the same algebraic construction as in Theorem 3.2. We get two semi-groups P̂t2 and P̂t1. Ŷi,s,tj and Ẑi,s,tj are smooth with bounded derivatives in x̂=(x,U,U-1,V). We get by the same procedure the following.Theorem 3.3.
Iff̂ is bounded with bounded derivatives and with compact support in s, then one gets
(3.28)P̂t2[〈Df̂,Û〉](s0,x̂,0)=P̂t1[f̂û](s0,x̂,0),
where one take does not derivative in the direction of s in Df̂.We can perform the same improvements as in [15, page 512]. We define on ℝd×ℝd1×⋯×ℝdk some vectors fields:(3.29)Xij,tot=(Xij,1(x1),…,Xij,l(x1,…,xl),Xij,k(x1,…,xk)),
where(3.30)Xij,l(x1,…,xl)=X1,ij,l(x1,…,xl-1)xl∂∂xl+X2,il(x1,…,xl)∂∂xl+X3,il(x1,…,xl-1),
where X1,ij,l,X2,ij,l have derivatives bounded at each order and X3,ij,l has derivative with polynomial growth.We can consider the generator𝔸̂tot associated with these vector fields and perform the same algebraic computations as in Theorem 3.2. We get two semi-groups P̂t2,tot and P̂t1,tot·Ŷi,s,tj and Ẑi,s,tj are smooth with bounded derivatives in x̂=(x,U,U-1,V). We get by the same procedure the following.Theorem 3.4.
Iff̂tot is bounded with bounded derivatives and with compact support in s, then one gets
(3.31)P̂t2,tot[〈Df̂tot,Û〉](s0,x̂tot,0)=P̂t1,tot[f̂totû](s0,x̂tot,0),
where Df̂tot does not include derivative in the direction of s.We refer to the appendix for the proof and the subsequent estimates.Remark 3.5.
Let us show from where come these identities, by using (1.4): we consider a time interval [At-,At]. On this random time interval, we do the following translation on the leading Brownian motion Bsi:(i)
ifzs>0 on this time interval, then dBsi is transformed in dBsi+λ〈ϕ(xs),hs,t2〉ids for a small parameter λ,(ii)
ifzs<0 on this time interval, then dBsi is transformed in dBsi+λ〈ϕ(xs),hs,t1〉ids for a small parameter λ.According to the fact thatf has compact support (this means that we consider bounded values of At), the transformed Brownian motion has an equivalent law through the Girsanov exponential to the original Brownian motions. The term in u in Theorem 3.2 come that from the fact we take the derivative in λ=0 of the Girsanov exponential. When we do this transformation, we get a random process xtλ(x). Derivation of it in λ=0 is done classically according to the stochastic flow theorem, which leads to the study of generators of the type 𝕃s,tj,2 and of the type 𝕃j,3.


## 4. Integration by Parts on the Subordinator

Let us consider diffusion type generator of the previous part:(4.1)L=Y0+12∑Yi2,Lt=(t)2Y0+12∑i>0(tYi)2.
Let us consider the semi-group(4.2)∂∂tPt=LPt
and the semi-group(4.3)∂∂sPst=LtPst.
We have classically(4.4)Pt=P1t,
where the smooth vector fields are Lipschitz.Therefore, we can write(4.5)--L=C∫0∞s-3/2(P1s-I)ds.
We consider a diffeomorphsim fλ(s) of [0,∞[ with bounded derivative of first order in λ equal to s if s<ϵ and equals to s if s>2 (we suppose λ small). We can write(4.6)-Aλ=C∫0∞(fλ(s))-3/2P1fλ(s)-s-3/2I)ds.
We do this operation on the two operators on ℝ1+d giving 𝔸. We get a generator 𝔸λ.According the line of stochastic analysis, we consider a generator𝔸λ,1 on ℝ1+d+1. If 𝕃1 is a generator on ℝ1+d with associated semi-group Ps, then we consider 𝔸λ,1 the generator on ℝ1+d+1,(4.7)Aλ,1f(s0,x0,u0)=∑jC∫0∞(fλ(s)-3/2[P1j,fλ(s)f(s0,x0,u0Jλ(s))]-s-3/2f(s0,x0,u0))ds,
where Jλ(s) is the Jacobian of the transformation s→fλ(s). By doing this procedure in (1.3), we deduce a global generator 𝔸λ,1 and a semi-group Ptλ,1 associated with it.It is not clear thatPtλ,1 is a Markovian semi-group. We decompose(4.8)Aλ,1=Aλ,1,ϵ+Aλ,1,ϵc,
where(4.9)Aλ,1,ϵf(s0,x0,u0)=∑jC∫0ϵ(s-3/2P1j,sf(s0,x0,u0)-s-3/2f(s0,x0,u0))ds.𝔸λ,1,ϵ generates a Markovian semi-group Ptλ,1,ϵ. But 𝔸λ,1,ϵc is a bounded operator on the set of bounded continuous functions on ℝ1+d+1 endowed with the uniform norm. The Volterra expansion converges on this set:(4.10)Ptλ,1f(s0,x0,u0)=Ptλ,1,ϵf(s0,x0,u0)+∑n∫0<s1<⋯<sn<tPs1λ,1,ϵAλ,1,ϵcPs2-s1λ,1,ϵ⋯Aλ,1,ϵcPt-snλ,1,ϵds1⋯dsn.Theorem 4.1 (Girsanov).
Forf with compact support in (s,x), one has
(4.11)Ptλ,1[uf](s0,x0,1)=exp⁡⁡[tA][f](s0,x0).Proof.
By linearity,(4.12)Ptλ,1[uf](s0,x0,u0)=u0Ptλ,1[uf](s0,x0,1).
But by an elementary change of variable,
(4.13)Aλ,1[uPtλ,1[uf](⋅,⋅,1)]=A[Ptλ,1[uf](⋅,⋅,1)].
The result holds by the unicity of the solution of the parabolic equation associated with 𝔸. To state the integrability of u, we refer to [16].Remark 4.2.
Let us show from where this formula comes. In the previous part, we have done a perturbation of the leading Brownian motionBti. Here, we do a perturbation of ΔAs into fλ(ΔAs)=ΔAsλ. By standard result on Levy processes, the law of the Levy process Atλ is equivalent to the law of At. Moreover, Atλ and Bti are independents. Therefore, the result.
Bismut’s idea to state hypoellipticity result is to take the derivative inλ of(4.14)Ptλ,1[uf](s0,x0,1)=exp⁡⁡[tA][f](s0,x0)
in order to get an integration by parts.
First of all, let us compute(∂/∂λ)Ptfλ(s)f in λ=0. It is fulfilled by the next considerations. Let us consider a generator written under Hoermander’s form:(4.15)Lλ=gλY0+12gλ2∑i>0Yi2,
where gλ are smooth and where the vector fields Yi are smooth Lipschitz on ℝd̃. We consider the semi-group Ptλ,· associated with it. Let us introduce the vector fields on ℝd̃+d̃:
(4.16)Yiλ,1=(gλYi,gλDYiU+ddλgλYi).
Let us consider the diffusion generator
(4.17)Lλ,1=Y0λ,1+12∑i>0(Yiλ,1)2.
Associated with it there is a semi-group Ptλ,1,·.Proposition 4.3.
Forf smooth with compact support, one has
(4.18)∂∂λPtλ,⋅[f](x̃)=Ptλ,1,⋅[〈df,U〉](x̃,0).Proof.
Let us introduce the vector fields onℝd̃+d̃:
(4.19)Yiλ,2=(gλYi,gλDYiU)
and the generator
(4.20)Lλ,2=Y0λ,2+12∑i>0(Yiλ,2)2.
Associated with it there is a semi-group Ptλ,2,·.
If the Volterra expansion converges, then(4.21)Ptλ,1,⋅[〈df,U〉](x̃,0)=∑∫0<s1<⋯<sn<tds1⋯dsnPs1λ,2,⋅(Lλ,1-Lλ,2)×Ps2-s1λ,2,⋅(Lλ,1-Lλ,2)⋯(Lλ,1-Lλ,2)Pt-snλ,2,⋅[〈df,U〉](x̃,0).
But Ũ0→Ptλ,2,·[〈df,U〉](x̃,Ũ0) is linear in Ũ0 and therefore the quantity (𝕃λ,1-𝕃λ,2)Pt-snλ,2,·[〈df,U〉](x̃,Ũ0) does not depend on Ũ0. Therefore the Volterra expansion reads
(4.22)Ptλ,1,⋅[〈df,U〉](x̃,0)=∫0tPsλ(Lλ,1-Lλ,2)Pt-sλ,2,⋅[〈df,U〉](x̃,0).
Let us compute 𝕃λ,1-𝕃λ,2. It is equal to
(4.23)∑i>0gλgλ′〈DYiYi,DU〉+∑i>0gλgλ′〈Yi,DUDX,Yi〉+gλ′〈Y0,DU〉.
We use the relation (see [15, Lemma 3.2])
(4.24)DXPtλ,⋅f(x̃)=〈Ptλ,2,⋅[〈Df,U〉](x̃,I),⋅〉
and the relation
(4.25)DUPtλ,2,⋅[〈df,U〉](X̃,0)=Ptλ,2,⋅[〈df,U〉](X̃,Ĩ).
Therefore,
(4.26)(Lλ,1-Lλ,2)Ptλ,2,⋅[〈Df,U〉](x̃0,U0)=gλ′〈Y0,DPtλ,⋅〉+∑i>0gλgλ′〈DYiYi,DPtλ,⋅〉+∑i>0gλgλ′〈Yi,D2Ptλ,⋅,Yi〉.
We insert this formula in the right-hand side of (4.23) and we see that Ptλ,1,·(〈Df,U〉)(x̃,0) satisfies the same parabolic equation as (∂/∂λ)Ptλ,·f(x).Remark 4.4.
Let us show from where this formula comes. Classically,(4.27)Ptλ,⋅[f](x)=E[f(xtλ(x))],
where xtλ is the solution of the Stratonovitch equation starting at x:
(4.28)dxsλ(x)=gλY0(xsλ(s))ds+∑i>0gλYi(xsλ(s))dBsi.
Therefore, Us=(∂/∂s)xsλ(x) is solution starting at 0 of the Stratonovitch differential equation:
(4.29)dUs=gλ′Y0(xsλ(s))ds+∑i>0gλ′Yi(xsλ(s))dBsi+gλ〈DY0(xsλ(x)),Usλ〉ds+∑i>0gλ〈DYi(xsλ(x)),Usλ〉dBsi
which can be solved classically by using the method of variation of constant [4].
Let us introduce the generator onℝ1+d+1+1+d𝔸λ,2:(4.30)Aλ,2f(s0,x0,u0,v0,U0)=∑jC∫0∞(fλ(s)-3/2P1j,fλ(s),2,⋅f(s0,u0,u0Jλ(s),v0,U0)-s-3/2f(s0,x0,u0,v0,U0))ds.
It generates a semi-group Ptλ,2. In order to see that, we split the generator by keeping the values od s〈ϵors〉ϵ and we proceed as for 𝔸λ,1 (see (4.10)).
We get the following.Theorem 4.5.
Forf smooth with compact support in s and with derivatives of each order bounded, one has the relation if one takes only derivatives in (s0,x0) of the considered expressions:
(4.31)DPtλ,1[f](s0,x0,u0)=Ptλ,2[〈Df,v,U〉](s0,x0,u0,1,I).Proof.
We have(4.32)∂∂tDPtλ,1=∑jC∫0∞(fλ(s)-3/2DP1j,fλ(s)Ptλ,1[f](s0,u0,u0Jλ(s))-s-3/2DPtλ,1f(s0,x0,u0))ds.
But by [15, Lemma 3.2.]:
(4.33)DP1j,fλ(s)f(s0,u0,u0Jλ(s))=P1j,fλ(s),2,⋅[〈Df,v,U〉](s0,x0,u0Jλ(s),1,I)
Therefore DPtλ,1 satisfies the parabolic equation associated with Ptλ,2[〈df,v,U〉](s0,x0,u0,1,𝕀). Only the integrability of U puts any problem. It is solved by the appendix since f has compact support in s.Theorem 4.6.
Forf with compact support in x̃ in ℝd̃. 							(4.34)Ptλ,1,⋅[〈df,U〉](x̃0,Ũ0)=Ptλ,2,⋅[〈df,U〉](x̃0,Ũ0)+Ptλ,1,⋅[〈df,U〉](x̃0,0̃)
if Ũ, Ũ0 belong to ℝd̃.Proof.
If the Volterra expansion converges, then(4.35)Ptλ,1,⋅[〈df,U〉](x̃,Ũ0)=Ptλ,2,⋅[〈df,U〉](x̃,Ũ0)+∑∫0<s1<⋯<sn<tds1⋯dsnPs1λ,2,⋅(Lλ,1-Lλ,2)×Ps2-s1λ,2,⋅(Lλ,1-Lλ,2)⋯(Lλ,1-Lλ,2)Pt-snλ,2,⋅[〈df,U〉](x̃,Ũ0).
But Ũ0→Ptλ,2,·[〈df,U〉](x̃,Ũ0) is linear in Ũ0 and therefore the quantity (𝕃λ,1-𝕃λ,2)Pt-snλ,2,·[〈df,U〉](x̃,Ũ0) do not depend of Ũ0. Therefore the Volterra expansion reads
(4.36)Ptλ,1,⋅[〈df,U〉](x̃,Ũ0)=Ptλ,2,⋅[〈df,U〉](x̃,Ũ0)+∫0tPsλ(Lλ,1-Lλ,2)Pt-sλ,2,⋅[〈df,U〉](x̃,0)ds
But the last term in the right-hand side of (4.26) is equal to Ptλ,1,·[〈df,U〉](x̃,0) by the end of the proof of the Proposition 4.3.Remark 4.7.
Analogous formula works forDexp[t𝔸]f.
Let us computeαt=(∂/∂λ)Pt0,1[uf](s0,x0,1). We remark that(4.37)P1j,fλ(s)[uf](s0,x0,u0Jλ(s))=P1j,fλ(s)[f](s0,x0)u0Jλ(s).
Namely, the generator of Ptj,fλ(s) does not act on the u0 component such that the two sides of (4.37) satisfy the same parabolic equality.
Therefore,(4.38)dtαt=Aαt+∑jC∫0∞f0′(s)s-5/2P1j,s[exp⁡⁡[tA][f]]ds+∑jC∫0∞s-3/2J0′(s)P1j,s[exp⁡⁡[tA][f]]ds+∑jC∫0∞s-3/2∂∂λP1j,s[exp⁡⁡[tA][f]]ds=Aαt+a1(t)+a2(t)+a3(t),
where J0′(s)=(∂/∂λ)J0(s), Therefore,
(4.39)αt=∫0texp⁡[(t-s)A](a1(s)+a2(s)+a3(s))ds.a3(t) in the previous expression is the only term which contains a derivative of f, because by Proposition 4.3,
(4.40)∂∂λP1j,s[exp⁡tA][f](s0,x0)=P1j,s,1,⋅[〈Dexp⁡⁡[tA][f],u,U〉](s0,x0,0,0).
Let𝔸3 be the generator on ℝ1+d+1+d:(4.41)A3f(s0,x0,u0,U0)=C∑j∫0∞s-3/2(P1j,s,1,⋅[f](s0,x0,u0,U0)-f(s0,x0,u0,U0))ds.
It generates a semi-group, Pt3. We get  the following.Theorem 4.8.
Forf with compact support in s and with bounded derivatives at each order, we have
(4.42)Pt3[〈df,u,U〉](s0,x0,0,0)=∫0texp⁡⁡[(t-v)A][∑jC∫0∞s-3/2P1j,s,1,⋅[〈Dexp⁡⁡[vA][f],u,U〉](s0,x0,0,0)ds]dv.Proof.
If the Volterra expansion converges, then(4.43)Pt3[〈df,U〉](s0,x0,0,0)=∑n∫0<s1<⋯<sn<tds1⋯dsnPs12(A3-A2)⋯Psn-sn-12(A3-A2)×Pt-sn2[〈df,u,U〉](s0,x0,0,0).
But Pt-sn2[〈df,u,U〉](s0,x0,u0,U0) is linear in (u0,U0). Let us explain the details of that. We have to compute
(4.44)(P1j,s,1,⋅-P1j,s,2,⋅)Pt-sn2[〈df,u,U〉](s0,x0,u0,U0).
By the technique of the beginning of the proof of Proposition 4.3, it does not depend on (u0,U0). Therefore, the Volterra expansion reads:
(4.45)Pt3[〈df,U〉](s0,x0,0,0)=∫0<v<tPv2(A3-A2)⋅Pt-v2[〈df,u,U〉](s0,x0,0,0)dv.
But
(4.46)(A3-A2)⋅Pt-v2[〈df,u,U〉](s0,x0,u0,U0).
does not depend on (u0,U0). Therefore, the right-hand side of formula (4.45) is equal to
(4.47)∫0<v<texp⁡[vA](A3-A2)⋅Pt-v2[〈df,u,U〉](s0,x0,0,0)dv.
But
(4.48)A2⋅Pt-v2[〈df,u,U〉](s0,x0,0,0)=∑jC∫0∞s-3/2Psj,2,2,0Pt-v2[〈df,u,U〉](s0,x0,0,0)=O
because (u0,U0)→Pt-vj[〈df,u,U〉](s0,x0,u0,U0) is linear in (u0,U0) and because the vector fields which give the generator of Psj,s,2,· are linear in u0,U0. Therefore,
(4.49)∫0<v<texp⁡[vA](A3-A2)⋅Pt-v2[〈df,u,U〉](s0,x0,0,0)dv=∫0<v<texp⁡[vA]dv∑jC∫0∞s-3/2P1j,s,1,⋅[Pt-v2[〈df,u,U〉]](s0,x0,0,0)ds.
But by an analog of Theorem 4.5,
(4.50)Pt-v2[〈df,u,U〉](s0,x0,u0,U0)=〈Dexp⁡[tA],u0,U0〉.We can summarize the previous considerations in the next theorem.Theorem 4.9.
Iffλ(s) is a diffeomorphism of [0,∞[ equal to s if s∈[0,ϵ[ and if s>1, then one has the following integration by part formula if f is with compact support in s, bounded with bounded derivatives at each order:
(4.51)0=∑jC∫0tduexp⁡[(t-u)A][∫0∞f0′(s)s-5/2P1j,s[exp⁡⁡[tA][f]]](s0,x0)+∑jC∫0tduexp⁡[(t-u)A][∫0∞J0′(s)s-3/2P1j,s[exp⁡⁡[tA][f]]](s0,x0)+Pt3[〈df,u,U〉](s0,x0,0,0),
where J0′(s)=(∂/∂λ)J0(s).Theorem 4.10.
Let one suppose thatfλ(s)=s+λs5 near 0. Then, (4.51) is still true.Proof.
It is enough to show that wecan approximatefλ(s) by a function fλϵ(s) equal to s if s<ϵ. Let us give some details on this approximation. We consider a smooth function g from ℝ into [0,1] equal to zero if s≤1/2 and equal to 1 if s>1. We put
(4.52)fλ(s)=s+g(sϵ)λs5,∂∂λf0ϵ(s)=g(sϵ)s5,∂∂λJ0ϵ(s)=g′(sϵ)s5ϵ+5g(sϵ)s4.
We remark that(i)
ifs≤ϵ/2, then g′(s/ϵ)s5/ϵ=0,(ii)
ifs>ϵ, then g′(s/ϵ)s5/ϵ=0,(iii)
ifs∈[ϵ/2,ϵ], then |g′(s/ϵ)s5/ϵ|≤Cs4.
Pt3,ϵ is the semi-group associated with 𝔸3,ϵ where we replace in the construction of (4.41) fλ(s) by fλϵ(s):(4.53)Pt3,ϵ[〈df,u,U〉](s0,x0,0,0)⟶Pt3[〈df,u,U〉](s0,x0,0,0).
By the appendix,
(4.54)Ps3,ϵ[(|u|p+|U|p)h](s,x,u0,U0)<∞
if h is compact support in s. Let us consider the generator A3,ϵ associated with fλϵ. If g=〈df,u,U〉, then we have by Duhamel principle
(4.55)P13[g](s0,x0,0,0)=P13,ϵ[g](s0,x0,0,0)+∫01Ps3,ϵ[(A-A3,ϵ)P1-s3[g]](s0,x0,0,0).
By the proof of Theorem 4.8, P1-s3[g](s0,x0,u0,U0) is affine in (u0,U0). Namely, in the proof of this theorem, we have removed the P1-s2[g](s0,x0,u0,U0) which is equal to zero in u0=0,U0=0 because this expression is linear in u0,U0. Its component in (u0,U0) is smooth with bounded derivatives at each order. By Theorem 4.6, (𝔸3,ϵ-𝔸)P1-s3[g](s0,x0,u0,U0) is still affine in (u0,U0) and its components in (u0,U0) are smooth with bounded derivatives at each order. Moreover, if g1 is affine in (u0,U0) with components in (u0,U0) smooth with bounded derivatives at each order, then we get that, for s≤1,
(4.56)sup⁡s0,x0|(P1j,s,1,0-P1ϵ,j,s,1,⋅)[g1](s0,x0,u0,U0)|≤C(ϵ)s(|u0|+|U0|),
where C(ϵ)→0 when ϵ→0. This can be seen as an appliation of the Duhamel formula applied to the two semi-groups t→Ptj,s,1,·[g1] and t→Ptϵ,j,s,1,·[g1]. Then, the result arises from the Duhamel formula (4.55).We can consider vector fields at the manner of (3.30) and fλ(s)=s+λs5 in a neighborhood of 0. We get a generator 𝔸tot and semi-groups Psj,s,tot and Pt3,tot. We have with the extension of Theorem 4.10 the following.Theorem 4.11 (Bismut).
Iffλ(s)=s+λs5 in a neighborhood of 0 and is equal to 1 if s>1, then one has the following integration by parts: let ftot be a function with compact support in s, bounded with bounded derivatives at each order. Then,
(4.57)0=∑jC∫0tduexp⁡[(t-u)Atot][∫0∞f0′(s)s-5/2P1j,s,tot[exp⁡⁡[tAtot][ftot]]](s0,x0tot)+∑jC∫0tduexp⁡[(t-u)Atot][∫0∞J0′(s)s-3/2P1j,s,tot[exp⁡⁡[tAtot][ftot]]](s0,x0tot)+Pt3,tot[〈dftot,u,U〉](s0,x0tot,0,0).


## 5. The Abstract Theorem

The proof of Theorem2.1 follows the idea of Malliavin [5]. If there exist Cl such that, for function f with compact support in [0,1]×[0,l]d,(5.1)|exp⁡⁡[tA][Df](0,x)|≤Cl‖f‖∞
then the heat kernel qt(s,y) exists.There are two partial derivatives to treat:(i)
the partial derivative in the time of the subordinators,(ii)
the partial derivatives in the space of the underlying diffusionx.Let us begin by the most original part of Bismut's Calculus on boundary process, that is, the integration by parts in the times.We look at (4.42). We remark (see the next part) that(5.2)Pt3,tot[u-p](0,x,0,0)<∞
for all p. So, we take ftot(s,x,u)=f(s,x)1/u and we apply (4.42) for this convenient semi-group. We get(5.3)exp⁡⁡[tA][∂∂sf](0,x)=-Pt3,tot[〈Dxf,U〉1u](0,x,0,0)+RR can be estimated by using the appendix by Cl∥f∥∞ for f with compact support in [0,l]×[0,l]d and by (5.2).Lemma 5.1.
For a conveniently enlarged semi-group in the manner of Theorem3.4, one has for f with compact support in s(5.4)Pt2,tot[〈Dftot,U〉](s0,x0tot,0)=exp⁡⁡[tÂtot][〈Dftot,UV〉](s0,x0tot,I,0).Proof.
Iff̃ is a function with compact support depending only of s, xtot and V, we have
(5.5)Pt2,tot[f̃](s0,x0tot,0)=exp⁡⁡[tÂtot][f̃(⋅,UV)](s0,x0tot,I,0).
We do the change of variable U→U and V→UV on the Malliavin generator 𝔸̂tot. By using Lemma 3.7 of [15], it is transformed in 𝔸2,totf̃(s,xtot,U,V) where for 𝔸2,tot we consider the same type of operator as 𝔸2 but with the modified vector fields:
(5.6)Xij,2=(0,Xij,tot,DXijU,DXijV),Y0j,2=(0,0,0,∑(Xij)t(U-1Xij)).
It remains to use the appendix to show the Lemma.We considerZij=t(U-1Xij). By the previous Lemma and Malliavin hypothesis,(5.7)Pt2,tot[det⁡V-pg](0,x0tot,I,0)<∞
for all p if g(s) has compact support (V is a matrix). After we consider a test function of the type of Bismut, we consider the component ui of U in (5.3). We consider the Bismut function fV-1(ui/u). We integrate by parts as in Theorem 3.4. We deduce under Malliavin assumption that(5.8)|Pt3,tot[〈Dxf,U〉1u](0,x,0,0)|≤Cl‖f‖∞
if f has compact support in [0,l]×[0,l]d.By the same way, we deduce that iff has compact support in [0,l]×[0,l]dthen(5.9)|exp⁡⁡[tA][Dxf](0,x,0,0)|≤Cl‖f‖∞.
Therefore, the result is obtained.Remark 5.2.
We could do integration by parts to each order in order to show that the semi-groupexp[t𝔸] has a smooth heat-kernel under Malliavin assumption.


## 6. Inversion of the Malliavin Matrix

Proof of Theorem2.2.
Lets1<s2 and let ξ be of modulus 1. Then,
(6.1)exp⁡⁡[tÂ][I[0,s1]IV(ξ)≤ϵ](0,x,I,0)≥exp⁡⁡[tÂ][I[0,s1]IV(ξ)≤ϵ](0,x,I,0).
These two quantities are equal in t=0 when we consider the semi-group exp[t𝔸̂]. Let us compute their derivative in time t. The derivative of the left-hand side is bigger than the derivative of the right-hand side because
(6.2)Â[I[0,s1]IV(ξ)≤ϵ](s,x,U,V)≥Â[I[0,s2]IV(ξ)≤ϵ](s,x,U,V).
(These two quantities are negative.)
By the result of the appendix,(6.3)exp⁡⁡[tÂ][I[0,t]{I|U-1-I|>C+I|U-I|>C+I|⋅-x|>C+IV>C}](0,x,I,0)≤C(p)tp
for all p.Lemma 6.1.
If|ξ|=1, then there exist C and C0 independent of ξ such that
(6.4)exp⁡⁡[ϵÂ][I|Vξ|<C0ϵI[0,ϵ]](0,x,I,0)<1-Cϵ1/2.Proof.
We consider a convex function decreasing from[0,∞[ into [0,1] equal to 1 in 0 and tending to 0 at infinity. Let us introduce
(6.5)αs=exp⁡⁡[sÂ][g(|Vξ|ϵ)I[0,ϵ]](0,x,I,0).
In order to consider the derivative in s of αs, we study the expression
(6.6)βϵ=Â[g(|Vξ|ϵ)I[0,ϵ]](s′,x′,U′,V′).
We have only to consider by (6.3) the case where s′ is small enough, |x′-x| is small enough, |U-I| is small enough, and the positive matrix V′ is small enough. For that we have to estimate
(6.7)γu=∑j(Puj,2[g(|Vξ|ϵ)](s′,x′,U′,V′)-g(|V′ξ|ϵ))
for u between 0 and ϵ. The first derivative of γu has an equivalent -Cϵ-1 when ϵ→0, and its second derivative has a bound Cϵ-2 when ϵ→0. Therefore,
(6.8)0≥γu≥-Cuϵ
on [0,ϵ] and
(6.9)βϵ≥-Cϵ∫0ϵs-1/2ds=-Cϵ-1/2.
We deduce from that that(6.10)αϵ≤1-Cϵ1/2.Remark 6.2.
We could improve (6.4) by showing that
(6.11)exp⁡⁡[ϵÂ][I|Vξ|<C0ϵI[0,ϵ]](s′,x′,U′,V′)<1-Cϵ1/2
if s′ is small enough, |x′-x| is small enough, |U′-I| is small enough, and the positive matrix V′ is small enough.
We consider a very smallα. We slice the time interval [0,ϵα] in ϵα-1 intervals of length ϵ. We have(6.12)exp⁡⁡[tÂ][I[0,l]IV(ξ)≤ϵ](0,x,I,0)≤exp⁡⁡[ϵαÂ][I[0,l]IV(ξ)≤ϵ](0,x,I,0)≤exp⁡⁡[ϵαÂ][I[0,ϵ]IV(ξ)≤ϵ](0,x,I,0)≤{sup⁡|x′-x|≤C0,|U-I|≤C0exp⁡⁡[ϵÂ][I[0,ϵ]IV(ξ)≤ϵ](0,x′,U′,0)}ϵα-1+Cϵp
for a small C0. This last quantity is smaller than Cϵp for all p by the previous lemma if α is small enough. The proof of Theorem 2.2 follows from
(6.13)exp⁡⁡[tÂ][VpI[0,l]](0,x,I,0)≤∞
for all p by using the result of the appendix. The result follows by standard methods (see [15, Equations (4.8) and (4.9)].It remains to show the following.Theorem 6.3.
For allp>0,
(6.14)Pt3,tot[u-p](0,x,0,0)≤∞.Proof.
We remark that if we consider only functions ofu, then
(6.15)Pt3,tot[f](0,x,u,0)=Pt4[f](u),
where Pt4 is a Lévy semi-group with generator
(6.16)A4f(u)=C∫0∞dss3/2(f(s5g(s)+u)-f(u)),
where g(s)=1 on a neighborhood of 0, is with compact support and is positive. The result follows from the adaptation in [17, 18] of the proof of [7] in semi-group theory. We remark that
(6.17)Pt4[u-p](0)=C∫0∞βp-1Pt4[exp⁡[-βu]](0)dβ.
By using the adaptation in semi-group theory of the exponential martingales of Levy process of [17, 18], we have
(6.18)Pt4[exp⁡[-βu]](0)=exp⁡[t∫0∞[exp⁡[-βs5g(s)]-1)dss3/2].
The result holds from the Tauberian theorem of [7, 17, 18].



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