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

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.

1. Introduction

Let 𝑋¹₀,𝑋¹₁,…,𝑋^𝑚₁,𝑋²₀,𝑋²₁,…,𝑋^𝑚₂ be 2𝑚+2 vector fields on ℝ^𝑑 with bounded derivatives at each order. Let𝕃¹=𝜕𝜕𝑠+𝑋¹₀+12_𝑖>0𝑋¹_𝑖² (1.1) be an Hoermander's type operator on ℝ^1+𝑑. Let𝕃²=𝜕𝜕𝑠+𝑋²₀+12_𝑖>0𝑋²_𝑖² (1.2) be a second Hoermander's operator on ℝ^1+𝑑. Bismut [1] considers the generator1𝔸=−2−2𝕃¹−12−2𝕃² (1.3) and the Markov semi-group exp[𝑡𝔸]. This semi-group has a probabilistic representation. We consider a Brownian motion 𝑡→𝑧_𝑡 independent of the others Brownian motions 𝐵^𝑖_𝑡. Bismut introduced the solution of the stochastic differential equation starting at 𝑥 in Stratonovitch sense:𝑑𝑥_𝑡(𝑥)=𝕀_𝑧𝑡<0𝑋¹₀𝑥_t(𝑥)𝑑𝑡+_𝑖>0𝑋¹_𝑖𝑥_𝑡(𝑥)𝑑𝐵^𝑖_𝑡+𝕀_𝑧𝑡>0𝑋²₀𝑥_𝑡(𝑥)𝑑𝑡+_𝑖>0𝑋²_𝑖𝑥_𝑡(𝑥)𝑑𝐵^𝑖_𝑡, (1.4) where 𝑡→𝐵^𝑖_𝑡 are 𝑚 independent Brownian motions.

Let us introduce the local time 𝑡→𝐿_𝑡 associated with 𝑡→𝑧_𝑡 and its right inverse 𝑡→𝐴_𝑡 (see [2, 3]). Then,exp[𝑡𝔸]𝑓(0,𝑥)=𝐸𝑓𝐴_𝑡,𝑥_𝐴𝑡(𝑥). (1.5) Such operator is classically related to the Dirichlet Problem [3].

Classically [4],exp𝑡𝕃¹𝑓(𝑥)=𝐸𝑓𝑥¹_𝑡(𝑥), (1.6) where 𝑥¹_𝑡(𝑥) is the solution of the Stratonovitch differential equation starting at 𝑥:𝑑𝑥¹_𝑡(𝑥)=𝑋¹₀𝑥¹_𝑡(𝑥)𝑑𝑡+𝑋¹_𝑖𝑥¹_𝑡(𝑥)𝑑𝐵^𝑖_𝑡, (1.7) The question is as following: is there an heat-kernel associated with the semi-group exp[𝑡𝕃¹]? This means thatexp𝑡𝕃¹𝑓(𝑥)=_ℝ𝑑𝑓𝑦𝑝_𝑡𝑥,𝑦𝑑𝑦. (1.8) 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→𝑥¹_𝐴𝑡(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→𝑥¹_𝑡(𝑥)) and the tools of the Malliavin Calculus for Poisson process (on the jump process 𝑡→𝐴_𝑡) in order to show that this isthe problem if𝐸𝑓𝐴_𝑡,𝑥_𝐴𝑡(𝑥)=_ℝ1+𝑑𝑞_𝑡𝑠,𝑦𝑓𝑠,𝑦𝑑𝑠𝑑𝑦. (1.9) 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 𝑃_𝑠. Then,−√−𝕃=𝐶^∞₀s^−3/2𝑃_𝑠−𝕀𝑑𝑠. (2.1) The results of this paper could be extended to generators of the type𝔸=^∞₀𝑔(𝑠)𝑃_𝑠−𝕀𝑑𝑠, (2.2) where ∫^∞₀𝑔(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 𝔼_𝑑=ℝ^1+𝑑×𝔾_𝑑×𝕄_d where 𝔾_𝑑 is the space of invertible matrices on ℝ^𝑑 and 𝕄_𝑑 the space of symmetric matrices on ℝ^𝑑. (𝑠,𝑥,𝑈,𝑉) is the generic element of 𝔼_𝑑. 𝑉 is called the Malliavin matrix.

On 𝔼_𝑑, we consider the vector fields:𝑋¹_𝑖=0,𝑋_𝑖,𝐷𝑋¹_𝑖(𝑥),𝑌𝑈,0₁=0,0,0,^𝑚_𝑖=1𝑈⁻¹𝑋_𝑖,⋅². (2.3) We consider the Malliavin generator 𝕃¹ on 𝔼_𝑑:𝐿¹=𝜕+𝑋𝜕𝑠¹⁰+12_𝑖>0𝑋¹_𝑖²+𝑌¹. (2.4) We consider the Malliavin semi-group 𝑃¹_𝑡 associated and −𝐿¹.

We perform the same algebraic considerations on 𝕃². We get 𝐿², 𝑃²_𝑡, and −𝐿². Let us consider the total generator𝔸=−−𝐿¹−−𝐿² (2.5) and the Malliavin semi-group exp[𝑡𝔸].

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 in 𝑥 is checked: exp𝑡𝔸det𝑉^−𝑝(0,𝑥,𝐼,0)<∞ (2.6) holds for all 𝑝, then exp[𝑡𝔸]𝑓(0,𝑥)=_ℝ1+𝑑𝑓𝑠,𝑦𝑞_𝑡𝑠,𝑦𝑑𝑠𝑑𝑦, (2.7) where 𝑞_𝑡(𝑠,𝑦) is the density of a probability measure on ℝ^1+𝑑.

Theorem 2.2. If the quadratic form _𝑖>0𝑋¹_𝑖(𝑥),⋅²+_𝑖>0𝑋²_𝑖(𝑥),⋅² (2.8) 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 on ℝ^1+𝑑+1,𝑋^𝑗,1_{𝑖,𝑠,𝑡}=0,𝑋^𝑗_𝑖(𝑥),𝑍^𝑗_{𝑖,𝑠,𝑡}, (3.1) where 𝑍^𝑗_{𝑖,𝑠,𝑡}=⟨𝜙(𝑥),ℎ^𝑗_𝑠,𝑡⟩_𝑖 (𝜙(𝑥) is a convenient matrix on ℝ^𝑚 which depends smoothly on 𝑥 and whose derivatives at each order are bounded. (𝑠,𝑡)→ℎ^𝑗_𝑠,𝑡 does not depend on 𝑥, and ℎ^𝑗_𝑠,𝑡 belong to ℝ^𝑚). Let 𝑓 be a smooth function on ℝ^1+𝑑+1, 𝐷𝑓 denotes its gradient, and 𝐷²𝑓 denotes its Hessian.

We consider the generator 𝕃^𝑗,1_𝑠,𝑡 acting on smooth functions on ℝ^1+𝑑+1,𝕃^𝑗,1_𝑠,𝑡𝜕𝑓=𝑋𝜕𝑠𝑓+^𝑗₀(𝑥),𝐷𝑓+12_𝑖>0𝐷𝑋^𝑗_𝑖(𝑥)𝑋^𝑗_𝑖(𝑥),𝐷𝑓+12_𝑖>0𝑋^𝑗,1_{𝑖,𝑠,𝑡},𝐷²𝑓,𝑋^𝑗,1_{𝑖,𝑠,𝑡}. (3.2) In (3.2), the generator is written under Itô’s form. It generates a time inhomogeneous in the parameter 𝑠 semi-group 𝑃^𝑗,1_𝑠,𝑡. We can consider−−𝕃^𝑗,1_⋅,𝑡=𝐶^∞₀𝑠^−3/2𝑃^𝑗,1_𝑠,𝑡−𝕀𝑑𝑠. (3.3) We put𝔸¹_𝑡=−−𝕃^𝑗,1_⋅,𝑡−−𝕃^𝑗,2_⋅,𝑡. (3.4) It generates a semi-group 𝑃¹_𝑡.

Let us consider the Hoermander’s type generator associated with the smooth Lipschitz vector fields on ℝ^1+𝑑+𝑑((𝑠,𝑥,𝑈) on ℝ^1+𝑑+𝑑):𝑋_𝑖^𝑗,2=0,𝑋^𝑗_𝑖,𝐷𝑋^𝑗_𝑖𝑈,𝑌^𝑗,2_0,𝑠,𝑡=0,0,𝑋^𝑗_𝑖(𝑥)𝑍^𝑗_{𝑖,𝑠,𝑡}=0,0,𝑌^𝑗_{𝑖,𝑠,𝑡},𝕃^𝑗,2_𝑠,𝑡=𝑋₀^𝑗,2+12_𝑖>1𝑋_𝑖^𝑗,22+𝑌^𝑗,2_0,𝑠,𝑡. (3.5) We consider the heat semi-group associated with 𝕃^𝑗,2_𝑠,𝑡𝜕𝑃𝜕𝑠^{𝑗,2𝑠,𝑡}𝑓=𝕃^𝑗,2_𝑠,𝑡𝑃^𝑗,2_𝑠,𝑡𝑓. (3.6) Let us recall [15, Theorem 2.2] that𝑃^𝑗,1_𝑠,𝑡𝑢𝑓𝑠₀,𝑥₀,0=𝑃^𝑗,2_𝑠,𝑡𝐷𝑓,𝑈𝑠₀,𝑥₀,0, (3.7) 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 ℝ^𝑑. From this, we deduce the following.

Lemma 3.1. One has the relation −𝕃^𝑗,1_⋅,𝑡𝑢𝑓𝑠₀,𝑥₀,0=−𝕃^𝑗,2_⋅,𝑡𝐷𝑓,𝑈𝑠₀,𝑥₀,0. (3.8)

Let us consider the semi-group 𝑃²_𝑡 associated with𝔸²_𝑡=−−𝕃^1,2_⋅,𝑡−−𝕃^2,2_⋅,𝑡. (3.9) We get, with the same notations for (𝑠,𝑥,𝑢,𝑈) the following.

Theorem 3.2. For 𝑓 bounded continuous with compact support in (𝑠,𝑥), one has the following relation: 𝑃²_𝑡𝐷𝑓,𝑈𝑠₀,𝑥₀,0=𝑃¹_𝑡𝑓𝑢𝑠₀,𝑥₀,0. (3.10)

Proof. For the integrability conditions, we refer to the appendix.
We remark that 𝜕/𝜕𝑢 commute with 𝔸¹_𝑡, therefore with 𝑃¹_𝑡. We deduce that 𝑃¹_𝑡𝑓𝑢𝑠₀,𝑥₀,𝑢₀=𝑢₀exp[𝑡𝔸]𝑓𝑠₀,𝑥₀+𝑃¹_𝑡𝑓𝑢𝑠₀,𝑥₀,0. (3.11) By the method of variation of constants, 𝑃¹_𝑡𝑓𝑢𝑠₀,𝑥₀,0=^𝑡₀exp[(𝑡−𝑠)𝔸]𝔸¹_𝑠𝑢exp[𝑠𝔸]𝑓(⋅,⋅,0)𝑠₀,𝑥₀𝑑𝑠. (3.12) In order to show that, we follow the lines of (2.17) and (2.18) in [15]. We apply 𝔸¹_𝑡 to (3.11).
By Lemma 3.1, 𝔸¹_𝑠𝑢exp[𝑠𝔸]𝑓(⋅,⋅)𝑠₁,𝑥₁,0=𝔸²_𝑠𝐷exp[𝑠𝔸],𝑈𝑠₁,𝑥₁,0. (3.13) Let us consider the vector fields on ℝ^1+𝑑×𝔾_𝑑, 𝑋_𝑖^𝑗,3=0,𝑋^𝑗_𝑖,𝐷𝑋^𝑗_𝑖𝑈. (3.14) We consider the Hoermander’s type operator associated with these vector fields: 𝕃^𝑗,3=𝑋₀^𝑗,3+12_𝑖>0𝑋_𝑖^𝑗,32. (3.15) We consider the generator 𝔸³_𝑡=−−𝕃^1,3_𝑠,𝑡−−𝕃^2,3_𝑠,𝑡. (3.16) It generates a semi-group 𝑃³_𝑡. By lemma 3.2 of [15], we have 𝐷exp[𝑠𝔸]𝑓𝑠₁,𝑥₁=𝑃³_𝑠𝐷𝑓𝑉𝑠₁,𝑥₁,𝐼. (3.17) By [15, Equation (3.18)], 𝑃^𝑗,2_𝑠,𝑡𝑃³_𝑡𝐷𝑓𝑈(⋅,𝐼)𝑉𝑠₁,𝑥₁,0=_𝑖^𝑠₀𝑃^𝑗_{𝑠−𝑣,𝑡𝑖}𝑌^𝑗_{𝑖,𝑣,𝑡},𝑃^𝑗,3_𝑣,𝑡𝑃³_𝑡𝐷𝑓𝑈(⋅,𝐼)𝑠₁,𝑥₁,0. (3.18) In [15, Equation (3.18)], we consider the semi-group 𝑃^′_𝑡 instead of the semi-group 𝑃^𝑗,2_𝑠,𝑡 and the test function 𝐷𝑓 instead as of the test function 𝑃³_𝑡[𝐷𝑓𝑈](⋅,𝐼) here. 𝑌^𝑗_{𝑖,𝑣,𝑡} is considered as an element of ℝ and not as a one-order differential operator: 𝜕𝑃𝜕𝑠^{𝑗,3𝑠,𝑡}𝑓=𝕃^𝑗,3_𝑠,𝑡𝑃^𝑗,3_𝑠,𝑡𝑓. (3.19) Therefore, 𝔸²_𝑡𝑃³_𝑡𝐷𝑓𝑉(⋅,𝐼)𝑉𝑠₁,𝑥₁,0=_𝑖,𝑗𝐶^∞₀𝑠^−3/2𝑠₀𝑃^𝑗_{𝑠−𝑣,𝑡𝑖}𝑌^𝑗_{𝑖,𝑣,𝑡},𝑃^𝑗,3_𝑣,𝑡𝑃³_𝑡𝐷𝑓𝑈(⋅,𝐼)𝑠₁,𝑥₁,0𝑑𝑣𝑑𝑠. (3.20) We write 𝔸²_𝑡=𝔸³_𝑡+𝐴³_𝑡, (3.21) where 𝐴³_𝑡𝑓𝑈𝑠₀,𝑥₀,𝑈₀=_𝑗𝐶^∞₀𝑠^−3/2𝑃^𝑗,2_𝑠,𝑡−𝑃^𝑗,3_𝑠,𝑡𝑓𝑈𝑠₀,𝑥₀,𝑈₀𝑑𝑠. (3.22) The Volterra expansion (see [15, Equation (3.17)]) if it converges gives the following formula: 𝑃^𝑗,2_𝑠,𝑡𝑓𝑈𝑠₀,𝑥₀,𝑈₀=_{0<𝑠1<𝑠2<⋯<𝑠𝑛<𝑡}𝑑𝑠₁⋯𝑑𝑠_𝑛𝑃_𝑠^𝑗,3₁𝑌^𝑗_{𝑖,𝑠1,𝑡}⋯𝑃_𝑠^𝑗,3_{𝑛−𝑠𝑛−1}×𝑌^𝑗_{𝑖,𝑠𝑛,𝑡}⋯𝑃^𝑗,3_{𝑡−𝑠𝑛}𝑓𝑈𝑠₀,𝑥₀,𝑈₀. (3.23) But 𝑢₀→𝑃^𝑗,3_𝑠,𝑡[𝑓𝑈](𝑠₀,𝑥₀,𝑈₀) is linear in 𝑢₀. Therefore: 𝑃^𝑗,2_𝑠,𝑡𝑓𝑈𝑠₀,𝑥₀,𝑈₀=𝑃^𝑗,3_𝑠,𝑡𝑓𝑈𝑠₀,𝑥₀,𝑈₀+^𝑠₀𝑃^𝑗_𝑣𝑖𝑌^𝑗_{𝑖,𝑣,𝑡},𝑃^𝑗,3_{𝑠−𝑣,𝑡}𝑓𝑈𝑠₀,𝑥₀,𝑈₀𝑑𝑣. (3.24) In this last formula, 𝑌^𝑗_{𝑖,𝑠,𝑡} are considered as differential operators.
Therefore, 𝐴³_𝑡[𝑓𝑈](𝑠₀,𝑥₀,𝑈₀) does not depend on 𝑈₀ and is equal to _𝑖,𝑗𝐶^∞₀𝑠^−3/2𝑠₀𝑃^𝑗_𝑣𝑖𝑌^𝑗_{𝑖,𝑣,𝑡},𝑃^𝑗,3_{𝑠−𝑣,𝑡}𝑓𝑈𝑠₀,𝑥₀,𝐼𝑑𝑠𝑑𝑣, (3.25) where 𝑌^𝑗_{𝑖,𝑠,𝑡} are considered as elements of ℝ^𝑑. We deduce as in [15, Equation (3.17)], 𝑃²_𝑡𝑓𝑈𝑠₀,𝑥₀,0=^𝑡₀exp[(𝑡−𝑠)𝔸]𝐴³_𝑠𝑃³_𝑠𝑓𝑈𝑠₀,𝑥₀,0𝑑𝑠. (3.26) But 𝑈₀→𝑃³_𝑠[𝑓𝑈](𝑠₀,𝑥₀,𝑈₀) is linear. Therefore, 𝐴³_𝑡𝑃³_𝑡𝑓𝑈𝑠₀,𝑥₀,0=_𝑖,𝑗𝐶^∞₀𝑠^−3/2𝑠₀𝑃^𝑗_𝑣𝑖𝑌^𝑗_{𝑖,𝑣,𝑡},𝑃^𝑗,3_{𝑠−𝑣,𝑡}𝑃³_𝑡𝑓𝑈𝑠₀,𝑥₀,𝐼𝑑𝑠𝑑𝑣. (3.27) 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 generator 𝔸. We can perform the same algebraic construction as in Theorem 3.2. We get two semi-groups 𝑃²_𝑡 and 𝑃¹_𝑡. 𝑌^𝑗_{𝑖,𝑠,𝑡} and 𝑍^𝑗_{𝑖,𝑠,𝑡} are smooth with bounded derivatives in 𝑥=(𝑥,𝑈,𝑈⁻¹,𝑉). We get by the same procedure the following.

Theorem 3.3. If 𝑓 is bounded with bounded derivatives and with compact support in 𝑠, then one gets 𝑃²_𝑡𝐷𝑓,𝑈𝑠₀,𝑥,0=𝑃¹_𝑡𝑓𝑢𝑠₀,𝑥,0, (3.28) where one take does not derivative in the direction of 𝑠 in 𝐷𝑓.

We can perform the same improvements as in [15, page 512]. We define on ℝ^𝑑×ℝ^𝑑1×⋯×ℝ^𝑑𝑘 some vectors fields:𝑋^𝑗,tot_𝑖=𝑋_𝑖^𝑗,1𝑥₁,…,𝑋_𝑖^𝑗,𝑙𝑥₁,…,𝑥_𝑙,𝑋_𝑖^𝑗,𝑘𝑥₁,…,𝑥_𝑘, (3.29) where𝑋_𝑖^𝑗,𝑙𝑥¹,…,𝑥^𝑙=𝑋^𝑗,𝑙_1,𝑖𝑥¹,…,𝑥^𝑙−1𝑥^𝑙𝜕𝜕𝑥^𝑙+𝑋^𝑙_2,𝑖𝑥¹,…,𝑥^𝑙𝜕𝜕𝑥^𝑙+𝑋^𝑙_3,𝑖𝑥¹,…,𝑥^𝑙−1, (3.30) where 𝑋^𝑗,𝑙_1,𝑖,𝑋^𝑗,𝑙_2,𝑖 have derivatives bounded at each order and 𝑋^𝑗,𝑙_3,𝑖 has derivative with polynomial growth.

We can consider the generator 𝔸^tot associated with these vector fields and perform the same algebraic computations as in Theorem 3.2. We get two semi-groups 𝑃^2,tot_𝑡 and 𝑃^1,tot_𝑡⋅𝑌^𝑗_{𝑖,𝑠,𝑡} and 𝑍^𝑗_{𝑖,𝑠,𝑡} are smooth with bounded derivatives in 𝑥=(𝑥,𝑈,𝑈⁻¹,𝑉). We get by the same procedure the following.

Theorem 3.4. If 𝑓^tot is bounded with bounded derivatives and with compact support in 𝑠, then one gets 𝑃^2,tot_𝑡𝐷𝑓^tot,𝑈𝑠₀,𝑥^tot,0=𝑃^1,tot_𝑡𝑓^tot𝑢𝑠₀,𝑥^tot,0, (3.31) where 𝐷𝑓^tot 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 [𝐴_𝑡−,𝐴_𝑡]. On this random time interval, we do the following translation on the leading Brownian motion 𝐵^𝑖_𝑠:(i) if 𝑧_𝑠>0 on this time interval, then 𝑑𝐵^𝑖_𝑠 is transformed in 𝑑𝐵^𝑖_𝑠+𝜆⟨𝜙(𝑥_𝑠),ℎ²_𝑠,𝑡⟩_𝑖𝑑𝑠 for a small parameter 𝜆,(ii)if 𝑧_𝑠<0 on this time interval, then 𝑑𝐵^𝑖_𝑠 is transformed in 𝑑𝐵^𝑖_𝑠+𝜆⟨𝜙(𝑥_𝑠),ℎ¹_𝑠,𝑡⟩_𝑖𝑑𝑠 for a small parameter 𝜆.

According to the fact that 𝑓 has compact support (this means that we consider bounded values of 𝐴_𝑡), 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 𝑥^𝜆_𝑡(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 𝕃^𝑗,2_𝑠,𝑡 and of the type 𝕃^𝑗,3.

4. Integration by Parts on the Subordinator

Let us consider diffusion type generator of the previous part:𝕃=𝑌₀+12𝑌²_𝑖,𝕃^√𝑡=√𝑡²𝑌₀+12_𝑖>0√𝑡𝑌_𝑖². (4.1) Let us consider the semi-group𝜕𝑃𝜕𝑡^𝑡=𝕃𝑃_𝑡 (4.2) and the semi-group𝜕𝑃𝜕𝑠^√𝑡𝑠=𝕃^√𝑡𝑃^√𝑡_𝑠. (4.3) We have classically𝑃_𝑡=𝑃^√𝑡₁, (4.4) where the smooth vector fields are Lipschitz.

Therefore, we can write−√−𝕃=𝐶^∞₀𝑠^−3/2𝑃^√𝑠₁−𝕀𝑑𝑠. (4.5) We consider a diffeomorphsim 𝑓_𝜆(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−𝔸^𝜆=𝐶^∞₀𝑓_𝜆(𝑠))^−3/2𝑃^𝑓𝜆₁^(𝑠)−𝑠^−3/2𝕀𝑑𝑠. (4.6) We do this operation on the two operators on ℝ^1+𝑑 giving 𝔸. We get a generator 𝔸^𝜆.

According the line of stochastic analysis, we consider a generator 𝔸^𝜆,1 on ℝ^1+𝑑+1. If 𝕃¹ is a generator on ℝ^1+𝑑 with associated semi-group 𝑃_𝑠, then we consider 𝔸^𝜆,1 the generator on ℝ^1+𝑑+1,𝔸^𝜆,1𝑓𝑠₀,𝑥₀,𝑢₀=_𝑗𝐶^∞₀𝑓_𝜆(𝑠)^−3/2𝑃^{𝑗,𝑓𝜆}₁^(𝑠)𝑓𝑠₀,𝑥₀,𝑢₀𝐽_𝜆(𝑠)−𝑠^−3/2𝑓𝑠₀,𝑥₀,𝑢₀𝑑𝑠, (4.7) where 𝐽_𝜆(𝑠) is the Jacobian of the transformation 𝑠→𝑓_𝜆(𝑠). By doing this procedure in (1.3), we deduce a global generator 𝔸^𝜆,1 and a semi-group 𝑃_𝑡^𝜆,1 associated with it.

It is not clear that 𝑃_𝑡^𝜆,1 is a Markovian semi-group. We decompose𝔸^𝜆,1=𝔸^𝜆,1,𝜖+𝔸^{𝜆,1,𝜖𝑐}, (4.8) where𝔸^𝜆,1,𝜖𝑓𝑠₀,𝑥₀,𝑢₀=_𝑗𝐶^𝜖₀𝑠^−3/2𝑃^√𝑗,𝑠₁𝑓𝑠₀,𝑥₀,𝑢₀−𝑠^−3/2𝑓𝑠₀,𝑥₀,𝑢₀𝑑𝑠. (4.9)𝔸^𝜆,1,𝜖 generates a Markovian semi-group 𝑃_𝑡^𝜆,1,𝜖. But 𝔸^{𝜆,1,𝜖𝑐} is a bounded operator on the set of bounded continuous functions on ℝ^1+𝑑+1 endowed with the uniform norm. The Volterra expansion converges on this set:𝑃_𝑡^𝜆,1𝑓𝑠₀,𝑥₀,𝑢₀=𝑃_𝑡^𝜆,1,𝜖𝑓𝑠₀,𝑥₀,𝑢₀+_{𝑛0<𝑠1<⋯<𝑠𝑛<𝑡}𝑃_𝑠^𝜆,1,𝜖₁𝔸^{𝜆,1,𝜖𝑐}𝑃_𝑠^𝜆,1,𝜖_2−𝑠1⋯𝔸^{𝜆,1,𝜖𝑐}𝑃^𝜆,1,𝜖_{𝑡−𝑠𝑛}𝑑𝑠₁⋯𝑑𝑠_𝑛. (4.10)

Theorem 4.1 (Girsanov). For 𝑓 with compact support in (𝑠,𝑥), one has 𝑃_𝑡^𝜆,1𝑢𝑓𝑠₀,𝑥₀,1=exp[𝑡𝔸]𝑓𝑠₀,𝑥₀. (4.11)

Proof. By linearity, 𝑃_𝑡^𝜆,1𝑢𝑓𝑠₀,𝑥₀,𝑢₀=𝑢₀𝑃_𝑡^𝜆,1𝑢𝑓𝑠₀,𝑥₀,1. (4.12) But by an elementary change of variable, 𝔸^𝜆,1𝑢𝑃_𝑡^𝜆,1𝑢𝑓(⋅,⋅,1)=𝔸𝑃_𝑡^𝜆,1𝑢𝑓(⋅,⋅,1). (4.13) 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 motion 𝐵^𝑖_𝑡. Here, we do a perturbation of Δ𝐴_𝑠 into 𝑓_𝜆(Δ𝐴_𝑠)=Δ𝐴^𝜆_𝑠. By standard result on Levy processes, the law of the Levy process 𝐴^𝜆_𝑡 is equivalent to the law of 𝐴_𝑡. Moreover, 𝐴^𝜆_𝑡 and 𝐵^𝑖_𝑡 are independents. Therefore, the result.
Bismut’s idea to state hypoellipticity result is to take the derivative in 𝜆 of𝑃_𝑡^𝜆,1𝑢𝑓𝑠₀,𝑥₀,1=exp[𝑡𝔸]𝑓𝑠₀,𝑥₀ (4.14) in order to get an integration by parts.
First of all, let us compute (𝜕/𝜕𝜆)𝑃^𝑓𝜆_𝑡^(𝑠)𝑓 in 𝜆=0. It is fulfilled by the next considerations. Let us consider a generator written under Hoermander’s form:𝕃^𝜆=𝑔_𝜆𝑌₀+12𝑔²_𝜆𝑖>0𝑌²_𝑖, (4.15) where 𝑔_𝜆 are smooth and where the vector fields 𝑌_𝑖 are smooth Lipschitz on ℝ𝑑. We consider the semi-group 𝑃_𝑡^𝜆,⋅ associated with it. Let us introduce the vector fields on ℝ𝑑₊𝑑: 𝑌_𝑖^𝜆,1=𝑔_𝜆𝑌_𝑖,𝑔_𝜆𝐷𝑌_𝑖𝑑𝑈+𝑔𝑑𝜆^𝜆𝑌_𝑖. (4.16) Let us consider the diffusion generator 𝕃^𝜆,1=𝑌₀^𝜆,1+12_𝑖>0𝑌_𝑖^𝜆,12. (4.17) Associated with it there is a semi-group 𝑃_𝑡^𝜆,1,⋅.

Proposition 4.3. For 𝑓 smooth with compact support, one has 𝜕𝑃𝜕𝜆^𝑡𝜆,⋅𝑓(𝑥)=𝑃_𝑡^𝜆,1,⋅𝑑𝑓,𝑈(𝑥,0). (4.18)

Proof. Let us introduce the vector fields on ℝ𝑑₊𝑑: 𝑌_𝑖^𝜆,2=𝑔_𝜆𝑌_𝑖,𝑔_𝜆𝐷𝑌_𝑖𝑈 (4.19) and the generator 𝕃^𝜆,2=𝑌₀^𝜆,2+12_𝑖>0𝑌_𝑖^𝜆,22. (4.20) Associated with it there is a semi-group 𝑃_𝑡^𝜆,2,⋅.
If the Volterra expansion converges, then𝑃_𝑡^𝜆,1,⋅𝑑𝑓,𝑈(𝑥,0)=_{0<𝑠1<⋯<𝑠𝑛<𝑡}𝑑𝑠₁⋯𝑑𝑠_𝑛𝑃_𝑠^𝜆,2,⋅₁𝕃^𝜆,1−𝕃^𝜆,2×𝑃_𝑠^𝜆,2,⋅_2−𝑠1𝕃^𝜆,1−𝕃^𝜆,2⋯𝕃^𝜆,1−𝕃^𝜆,2𝑃^𝜆,2,⋅_{𝑡−𝑠𝑛}𝑑𝑓,𝑈(𝑥,0). (4.21) But 𝑈₀→𝑃_𝑡^𝜆,2,⋅[⟨𝑑𝑓,𝑈⟩](𝑥,𝑈₀) is linear in 𝑈₀ and therefore the quantity (𝕃^𝜆,1−𝕃^𝜆,2)𝑃^𝜆,2,⋅_{𝑡−𝑠𝑛}[⟨𝑑𝑓,𝑈⟩](𝑥,𝑈₀) does not depend on 𝑈₀. Therefore the Volterra expansion reads 𝑃_𝑡^𝜆,1,⋅𝑑𝑓,𝑈(𝑥,0)=^𝑡₀𝑃^𝜆_𝑠𝕃^𝜆,1−𝕃^𝜆,2𝑃^𝜆,2,⋅_𝑡−𝑠𝑑𝑓,𝑈(𝑥,0). (4.22) Let us compute 𝕃^𝜆,1−𝕃^𝜆,2. It is equal to _𝑖>0𝑔_𝜆𝑔^′_𝜆𝐷𝑌_𝑖𝑌_𝑖,𝐷_𝑈+_𝑖>0𝑔_𝜆𝑔^′_𝜆𝑌_𝑖,𝐷_𝑈𝐷_𝑋,𝑌_𝑖+𝑔^′_𝜆𝑌₀,𝐷_𝑈. (4.23) We use the relation (see [15, Lemma 3.2]) 𝐷_𝑋𝑃_𝑡^𝜆,⋅𝑓(𝑥)=𝑃_𝑡^𝜆,2,⋅𝐷𝑓,𝑈(𝑥,𝐼),⋅ (4.24) and the relation 𝐷_𝑈𝑃_𝑡^𝜆,2,⋅𝑑𝑓,𝑈𝑋,0=𝑃_𝑡^𝜆,2,⋅𝑑𝑓,𝑈𝑋,𝕀̃. (4.25) Therefore, 𝕃^𝜆,1−𝕃^𝜆,2𝑃_𝑡^𝜆,2,⋅𝐷𝑓,𝑈𝑥₀,𝑈₀=𝑔^′_𝜆𝑌₀,𝐷𝑃_𝑡^𝜆,⋅+_𝑖>0𝑔_𝜆𝑔^′_𝜆𝐷𝑌_𝑖𝑌_𝑖,𝐷𝑃_𝑡^𝜆,⋅+_𝑖>0𝑔_𝜆𝑔^′_𝜆𝑌_𝑖,𝐷²𝑃_𝑡^𝜆,⋅,𝑌_𝑖. (4.26) We insert this formula in the right-hand side of (4.23) and we see that 𝑃_𝑡^𝜆,1,⋅(⟨𝐷𝑓,𝑈⟩)(𝑥,0) satisfies the same parabolic equation as (𝜕/𝜕𝜆)𝑃_𝑡^𝜆,⋅𝑓(𝑥).

Remark 4.4. Let us show from where this formula comes. Classically, 𝑃_𝑡^𝜆,⋅𝑓(𝑥)=𝐸𝑓𝑥^𝜆_𝑡(𝑥), (4.27) where 𝑥^𝜆_𝑡 is the solution of the Stratonovitch equation starting at 𝑥: 𝑑𝑥^𝜆_𝑠(𝑥)=𝑔_𝜆𝑌₀𝑥^𝜆_𝑠(𝑠)𝑑𝑠+_𝑖>0𝑔_𝜆𝑌_𝑖𝑥^𝜆_𝑠(𝑠)𝑑𝐵^𝑖_𝑠. (4.28) Therefore, 𝑈_𝑠=(𝜕/𝜕𝑠)𝑥^𝜆_𝑠(𝑥) is solution starting at 0 of the Stratonovitch differential equation: 𝑑𝑈_𝑠=𝑔^′_𝜆𝑌₀𝑥^𝜆_𝑠(𝑠)𝑑𝑠+_𝑖>0𝑔^′_𝜆𝑌_𝑖𝑥^𝜆_𝑠(𝑠)𝑑𝐵^𝑖_𝑠+𝑔_𝜆𝐷𝑌₀𝑥_𝑠𝜆(𝑥),𝑈^𝜆_𝑠𝑑𝑠+_𝑖>0𝑔_𝜆𝐷𝑌_𝑖𝑥^𝜆_𝑠(𝑥),𝑈^𝜆_𝑠𝑑𝐵^𝑖_𝑠 (4.29) which can be solved classically by using the method of variation of constant [4].
Let us introduce the generator on ℝ^{1+𝑑+1+1+𝑑}𝔸^𝜆,2:𝔸^𝜆,2𝑓𝑠₀,𝑥₀,𝑢₀,𝑣₀,𝑈₀=_𝑗𝐶^∞₀𝑓_𝜆(𝑠)^−3/2𝑃^{𝑗,𝑓𝜆}₁^(𝑠),2,⋅𝑓𝑠₀,𝑢₀,𝑢₀𝐽_𝜆(𝑠),𝑣₀,𝑈₀−𝑠^−3/2𝑓𝑠₀,𝑥₀,𝑢₀,𝑣₀,𝑈₀𝑑𝑠. (4.30) It generates a semi-group 𝑃_𝑡^𝜆,2. In order to see that, we split the generator by keeping the values od 𝑠⟨𝜖or𝑠⟩𝜖 and we proceed as for 𝔸^𝜆,1 (see (4.10)).
We get the following.

Theorem 4.5. For 𝑓 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: 𝐷𝑃_𝑡^𝜆,1𝑓𝑠₀,𝑥₀,𝑢₀=𝑃_𝑡^𝜆,2𝐷𝑓,𝑣,𝑈𝑠₀,𝑥₀,𝑢₀,1,𝕀. (4.31)

Proof. We have 𝜕𝜕𝑡𝐷𝑃_𝑡^𝜆,1=_𝑗𝐶^∞₀𝑓_𝜆(𝑠)^−3/2𝐷𝑃^{𝑗,𝑓𝜆}₁^(𝑠)𝑃_𝑡^𝜆,1𝑓𝑠₀,𝑢₀,𝑢₀𝐽_𝜆(𝑠)−𝑠^−3/2𝐷𝑃_𝑡^𝜆,1𝑓𝑠₀,𝑥₀,𝑢₀𝑑𝑠. (4.32) But by [15, Lemma 3.2.]: 𝐷𝑃^{𝑗,𝑓𝜆}₁^(𝑠)𝑓𝑠₀,𝑢₀,𝑢₀𝐽_𝜆(𝑠)=𝑃^{𝑗,𝑓𝜆}₁^(𝑠),2,⋅𝐷𝑓,𝑣,𝑈𝑠₀,𝑥₀,𝑢₀𝐽_𝜆(𝑠),1,𝕀 (4.33) Therefore 𝐷𝑃_𝑡^𝜆,1 satisfies the parabolic equation associated with 𝑃_𝑡^𝜆,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. For 𝑓 with compact support in 𝑥 in ℝ𝑑. 𝑃_𝑡^𝜆,1,⋅𝑑𝑓,𝑈𝑥₀,𝑈₀=𝑃_𝑡^𝜆,2,⋅𝑑𝑓,𝑈𝑥₀,𝑈₀+𝑃_𝑡^𝜆,1,⋅𝑑𝑓,𝑈𝑥₀,0 (4.34) if 𝑈, 𝑈₀ belong to ℝ𝑑.

Proof. If the Volterra expansion converges, then 𝑃_𝑡^𝜆,1,⋅𝑑𝑓,𝑈𝑥,𝑈₀=𝑃_𝑡^𝜆,2,⋅𝑑𝑓,𝑈𝑥,𝑈₀+_{0<𝑠1<⋯<𝑠𝑛<𝑡}𝑑𝑠₁⋯𝑑𝑠_𝑛𝑃_𝑠^𝜆,2,⋅₁𝕃^𝜆,1−𝕃^𝜆,2×𝑃_𝑠^𝜆,2,⋅_2−𝑠1𝕃^𝜆,1−𝕃^𝜆,2⋯𝕃^𝜆,1−𝕃^𝜆,2𝑃^𝜆,2,⋅_{𝑡−𝑠𝑛}𝑑𝑓,𝑈𝑥,𝑈₀. (4.35) But 𝑈₀→𝑃_𝑡^𝜆,2,⋅[⟨𝑑𝑓,𝑈⟩](𝑥,𝑈₀) is linear in 𝑈₀ and therefore the quantity (𝕃^𝜆,1−𝕃^𝜆,2)𝑃^𝜆,2,⋅_{𝑡−𝑠𝑛}[⟨𝑑𝑓,𝑈⟩](𝑥,𝑈₀) do not depend of 𝑈₀. Therefore the Volterra expansion reads 𝑃_𝑡^𝜆,1,⋅𝑑𝑓,𝑈𝑥,𝑈₀=𝑃_𝑡^𝜆,2,⋅𝑑𝑓,𝑈𝑥,𝑈₀+^𝑡₀𝑃^𝜆_𝑠𝕃^𝜆,1−𝕃^𝜆,2𝑃^𝜆,2,⋅_𝑡−𝑠𝑑𝑓,𝑈(𝑥,0)𝑑𝑠 (4.36) But the last term in the right-hand side of (4.26) is equal to 𝑃_𝑡^𝜆,1,⋅[⟨𝑑𝑓,𝑈⟩](𝑥,0) by the end of the proof of the Proposition 4.3.

Remark 4.7. Analogous formula works for 𝐷exp[𝑡𝔸]𝑓.
Let us compute 𝛼_𝑡=(𝜕/𝜕𝜆)𝑃_𝑡^0,1[𝑢𝑓](𝑠₀,𝑥₀,1). We remark that𝑃^{𝑗,𝑓𝜆}₁^(𝑠)𝑢𝑓𝑠₀,𝑥₀,𝑢₀𝐽_𝜆(𝑠)=𝑃^{𝑗,𝑓𝜆}₁^(𝑠)𝑓𝑠₀,𝑥₀𝑢₀𝐽_𝜆(𝑠). (4.37) Namely, the generator of 𝑃^{𝑗,𝑓𝜆}_𝑡^(𝑠) does not act on the 𝑢₀ component such that the two sides of (4.37) satisfy the same parabolic equality.
Therefore,𝑑_𝑡𝛼_𝑡=𝔸𝛼_𝑡+_𝑗𝐶^∞₀𝑓^′₀(𝑠)𝑠^−5/2𝑃^√𝑗,𝑠₁exp[𝑡𝔸]𝑓𝑑𝑠+_𝑗𝐶^∞₀𝑠^−3/2𝐽^′₀(𝑠)𝑃^√𝑗,𝑠₁exp[𝑡𝔸]𝑓𝑑𝑠+_𝑗𝐶^∞₀𝑠^−3/2𝜕𝑃𝜕𝜆^{√𝑗,𝑠1}exp[𝑡𝔸]𝑓𝑑𝑠=𝔸𝛼_𝑡+𝑎₁(𝑡)+𝑎₂(𝑡)+𝑎₃(𝑡), (4.38) where 𝐽^′₀(𝑠)=(𝜕/𝜕𝜆)𝐽₀(𝑠), Therefore, 𝛼_𝑡=^𝑡₀exp[(𝑡−𝑠)𝔸]𝑎₁(𝑠)+𝑎₂(𝑠)+𝑎₃(𝑠)𝑑𝑠. (4.39)𝑎₃(𝑡) in the previous expression is the only term which contains a derivative of 𝑓, because by Proposition 4.3, 𝜕𝑃𝜕𝜆^{√𝑗,𝑠1}exp𝑡𝔸𝑓𝑠₀,𝑥₀=𝑃^√𝑗,₁^𝑠,1,⋅𝐷exp[𝑡𝔸]𝑓,𝑢,𝑈𝑠₀,𝑥₀,0,0. (4.40)
Let 𝔸³ be the generator on ℝ^{1+𝑑+1+𝑑}:𝔸³𝑓𝑠₀,𝑥₀,𝑢₀,𝑈₀=𝐶_𝑗^∞₀𝑠^−3/2𝑃^√𝑗,₁^𝑠,1,⋅𝑓𝑠₀,𝑥₀,𝑢₀,𝑈₀−𝑓𝑠₀,𝑥₀,𝑢₀,𝑈₀𝑑𝑠. (4.41) It generates a semi-group, 𝑃³_𝑡. We get the following.

Theorem 4.8. For 𝑓 with compact support in s and with bounded derivatives at each order, we have 𝑃³_𝑡𝑑𝑓,𝑢,𝑈𝑠₀,𝑥₀=,0,0_𝑡0exp[(𝑡−𝑣)𝔸]_𝑗𝐶^∞₀𝑠^−3/2𝑃^√𝑗,₁^𝑠,1,⋅𝐷exp[𝑣𝔸]𝑓,𝑢,𝑈𝑠₀,𝑥₀,0,0𝑑𝑠𝑑𝑣. (4.42)

Proof. If the Volterra expansion converges, then 𝑃³_𝑡𝑑𝑓,𝑈𝑠₀,𝑥₀,0,0=_{𝑛0<𝑠1<⋯<𝑠𝑛<𝑡}𝑑𝑠₁⋯𝑑𝑠_𝑛𝑃²_𝑠1𝔸³−𝔸²⋯𝑃²_{𝑠𝑛−𝑠𝑛−1}𝔸³−𝔸²×𝑃²_{𝑡−𝑠𝑛}𝑑𝑓,𝑢,𝑈𝑠₀,𝑥₀,0,0. (4.43) But 𝑃²_{𝑡−𝑠𝑛}[⟨𝑑𝑓,𝑢,𝑈⟩](𝑠₀,𝑥₀,𝑢₀,𝑈₀) is linear in (𝑢₀,𝑈₀). Let us explain the details of that. We have to compute 𝑃^√𝑗,₁^𝑠,1,⋅−𝑃^√𝑗,₁^𝑠,2,⋅𝑃²_{𝑡−𝑠𝑛}𝑑𝑓,𝑢,𝑈𝑠₀,𝑥₀,𝑢₀,𝑈₀. (4.44) 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: 𝑃³_𝑡𝑑𝑓,𝑈𝑠₀,𝑥₀,0,0=_0<𝑣<𝑡𝑃²_𝑣𝔸³−𝔸²⋅𝑃²_𝑡−𝑣𝑑𝑓,𝑢,𝑈𝑠₀,𝑥₀,0,0𝑑𝑣. (4.45) But 𝔸³−𝔸²⋅𝑃²_𝑡−𝑣𝑑𝑓,𝑢,𝑈𝑠₀,𝑥₀,𝑢₀,𝑈₀. (4.46) does not depend on (𝑢₀,𝑈₀). Therefore, the right-hand side of formula (4.45) is equal to exp[_0<𝑣<𝑡𝑣𝔸]𝔸³−𝔸²⋅𝑃²_𝑡−𝑣𝑑𝑓,𝑢,𝑈𝑠₀,𝑥₀,0,0𝑑𝑣. (4.47) But 𝔸²⋅𝑃²_𝑡−𝑣𝑑𝑓,𝑢,𝑈𝑠₀,𝑥₀,0,0=_𝑗𝐶^∞₀s^−3/2𝑃^√𝑗,_𝑠^2,2,0𝑃²_𝑡−𝑣𝑑𝑓,𝑢,𝑈𝑠₀,𝑥₀,0,0=𝑂 (4.48) because (𝑢₀,𝑈₀)→𝑃^𝑗_𝑡−𝑣[⟨𝑑𝑓,𝑢,𝑈⟩](𝑠₀,𝑥₀,𝑢₀,𝑈₀) is linear in (𝑢₀,𝑈₀) and because the vector fields which give the generator of 𝑃^√𝑗,_𝑠^𝑠,2,⋅ are linear in 𝑢₀,𝑈₀. Therefore, exp[_0<𝑣<𝑡𝑣𝔸]𝔸³−𝔸²⋅𝑃²_𝑡−𝑣𝑑𝑓,𝑢,𝑈𝑠₀,𝑥₀=,0,0𝑑𝑣_0<𝑣<𝑡exp[𝑣𝔸]𝑑𝑣_𝑗𝐶^∞₀𝑠^−3/2𝑃^√𝑗,₁^𝑠,1,⋅𝑃²_𝑡−𝑣𝑑𝑓,𝑢,𝑈𝑠₀,𝑥₀,0,0𝑑𝑠. (4.49) But by an analog of Theorem 4.5, 𝑃²_𝑡−𝑣𝑑𝑓,𝑢,𝑈𝑠₀,𝑥₀,𝑢₀,𝑈₀=𝐷exp[𝑡𝔸],𝑢₀,𝑈₀. (4.50)

We can summarize the previous considerations in the next theorem.

Theorem 4.9. If 𝑓_𝜆(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: 0=_𝑗𝐶^𝑡₀𝑑𝑢exp[(𝑡−𝑢)𝔸]^∞₀𝑓^′₀(𝑠)𝑠^−5/2𝑃^√𝑗,𝑠₁exp[𝑡𝔸]𝑓𝑠₀,𝑥₀+_𝑗𝐶^𝑡₀𝑑𝑢exp[(𝑡−𝑢)𝔸]^∞₀𝐽^′₀(𝑠)𝑠^−3/2𝑃^√𝑗,𝑠₁exp[𝑡𝔸]𝑓𝑠₀,𝑥₀+𝑃³_𝑡𝑑𝑓,𝑢,𝑈𝑠₀,𝑥₀,0,0, (4.51) where 𝐽^′₀(𝑠)=(𝜕/𝜕𝜆)𝐽₀(𝑠).

Theorem 4.10. Let one suppose that 𝑓_𝜆(𝑠)=𝑠+𝜆𝑠⁵ near 0. Then, (4.51) is still true.

Proof. It is enough to show that wecan approximate 𝑓_𝜆(𝑠) by a function 𝑓^𝜖_𝜆(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 𝑓_𝜆(𝑠)𝑠=𝑠+𝑔𝜖𝜆𝑠⁵,𝜕𝑓𝜕𝜆^𝜖0(𝑠)𝑠=𝑔𝜖𝑠⁵,𝜕𝐽𝜕𝜆^𝜖0(𝑠)=𝑔^′𝑠𝜖𝑠⁵𝜖𝑠+5𝑔𝜖𝑠⁴. (4.52) We remark that(i)if 𝑠≤𝜖/2, then 𝑔^′(𝑠/𝜖)𝑠⁵/𝜖=0,(ii)if 𝑠>𝜖, then 𝑔^′(𝑠/𝜖)𝑠⁵/𝜖=0,(iii)if 𝑠∈[𝜖/2,𝜖], then |𝑔^′(𝑠/𝜖)𝑠⁵/𝜖|≤𝐶𝑠⁴.
𝑃_𝑡^3,𝜖 is the semi-group associated with 𝔸^3,𝜖 where we replace in the construction of (4.41) 𝑓_𝜆(𝑠) by 𝑓^𝜖_𝜆(𝑠):𝑃_𝑡^3,𝜖𝑑𝑓,𝑢,𝑈𝑠₀,𝑥₀,0,0⟶𝑃³_𝑡𝑑𝑓,𝑢,𝑈𝑠₀,𝑥₀,0,0. (4.53) By the appendix, 𝑃_𝑠^3,𝜖|𝑢|^𝑝+|𝑈|^𝑝ℎ𝑠,𝑥,𝑢₀,𝑈₀<∞ (4.54) if ℎ is compact support in 𝑠. Let us consider the generator 𝐴^3,𝜖 associated with 𝑓^𝜖_𝜆. If 𝑔=⟨𝑑𝑓,𝑢,𝑈⟩, then we have by Duhamel principle 𝑃³₁𝑔𝑠₀,𝑥₀,0,0=𝑃₁^3,𝜖𝑔𝑠₀,𝑥₀,0,0+¹₀𝑃_𝑠^3,𝜖𝔸−𝔸^3,𝜖𝑃³_1−𝑠𝑔𝑠₀,𝑥₀,0,0. (4.55) By the proof of Theorem 4.8, 𝑃³_1−𝑠[𝑔](𝑠₀,𝑥₀,𝑢₀,𝑈₀) is affine in (𝑢₀,𝑈₀). Namely, in the proof of this theorem, we have removed the 𝑃²_1−𝑠[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, (𝔸^3,𝜖−𝔸)𝑃³_1−𝑠[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, sup_𝑠0,𝑥0𝑃^√𝑗,₁^𝑠,1,0−𝑃^{√𝜖,𝑗,}₁^𝑠,1,⋅𝑔¹𝑠₀,𝑥₀,𝑢₀,𝑈₀≤𝐶(𝜖)𝑠𝑢₀+𝑈₀, (4.56) 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→𝑃^√𝑗,_𝑡^𝑠,1,⋅[𝑔¹] and 𝑡→𝑃^{√𝜖,𝑗,}_𝑡^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 𝑓_𝜆(𝑠)=𝑠+𝜆𝑠⁵ in a neighborhood of 0. We get a generator 𝔸^tot and semi-groups 𝑃^{√𝑗,𝑠,tot}_𝑠 and 𝑃^3,tot_𝑡. We have with the extension of Theorem 4.10 the following.

Theorem 4.11 (Bismut). If 𝑓_𝜆(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 𝑓^tot be a function with compact support in 𝑠, bounded with bounded derivatives at each order. Then, 0=_𝑗𝐶^𝑡₀𝑑𝑢exp(𝑡−𝑢)𝔸^tot∞₀𝑓^′₀(𝑠)𝑠^−5/2𝑃^{√𝑗,𝑠,tot}₁exp𝑡𝔸^tot𝑓^tot𝑠₀,𝑥^tot₀+_𝑗𝐶^𝑡₀𝑑𝑢exp(𝑡−𝑢)𝔸^tot∞₀𝐽^′₀(𝑠)𝑠^−3/2𝑃^{√𝑗,𝑠,tot}₁exp𝑡𝔸^tot𝑓^tot𝑠₀,𝑥^tot₀+𝑃^3,tot_𝑡𝑑𝑓^tot,𝑢,𝑈𝑠₀,𝑥^tot₀,0,0. (4.57)

5. The Abstract Theorem

The proof of Theorem 2.1 follows the idea of Malliavin [5]. If there exist 𝐶_𝑙 such that, for function 𝑓 with compact support in [0,1]×[0,𝑙]^𝑑,exp[𝑡𝔸]𝐷𝑓(0,𝑥)≤𝐶_𝑙𝑓_∞ (5.1) then the heat kernel 𝑞_𝑡(𝑠,𝑦) exists.

There are two partial derivatives to treat:(i)the partial derivative in the time of the subordinator 𝑠,(ii)the partial derivatives in the space of the underlying diffusion 𝑥.

Let us begin by the most original part of Bismut's Calculus on boundary process, that is, the integration by parts in the time 𝑠.

We look at (4.42). We remark (see the next part) that𝑃^3,tot_𝑡𝑢^−𝑝(0,𝑥,0,0)<∞ (5.2) for all 𝑝. So, we take 𝑓^tot(𝑠,𝑥,𝑢)=𝑓(𝑠,𝑥)1/𝑢 and we apply (4.42) for this convenient semi-group. We get[𝜕exp𝑡𝔸]𝑓)𝜕𝑠(0,𝑥=−𝑃^3,tot_𝑡𝐷_𝑥1𝑓,𝑈𝑢(0,𝑥,0,0)+𝑅 (5.3)𝑅 can be estimated by using the appendix by 𝐶_𝑙‖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 Theorem 3.4, one has for 𝑓 with compact support in 𝑠𝑃^2,tot_𝑡𝐷𝑓^tot,𝑈𝑠₀,𝑥^tot₀,0=exp𝑡𝔸^tot𝐷𝑓^tot,𝑈𝑉𝑠₀,𝑥^tot₀,𝐼,0. (5.4)

Proof. If 𝑓 is a function with compact support depending only of 𝑠, 𝑥^tot and 𝑉, we have 𝑃^2,tot_𝑡𝑓𝑠₀,𝑥^tot₀,0=exp𝑡𝔸^tot𝑓(⋅,𝑈𝑉)𝑠₀,𝑥^tot₀,𝐼,0. (5.5) We do the change of variable 𝑈→𝑈 and 𝑉→𝑈𝑉 on the Malliavin generator 𝔸^tot. By using Lemma 3.7 of [15], it is transformed in 𝔸^2,tot𝑓(𝑠,𝑥^tot,𝑈,𝑉) where for 𝔸^2,tot we consider the same type of operator as 𝔸² but with the modified vector fields: 𝑋_𝑖^𝑗,2=0,𝑋^𝑗,tot_𝑖,𝐷𝑋^𝑗_𝑖𝑈,𝐷𝑋^𝑗_𝑖𝑉,𝑌₀^𝑗,2=0,0,0,𝑋^𝑗_𝑖^𝑡𝑈⁻¹𝑋^𝑗_𝑖. (5.6) It remains to use the appendix to show the Lemma.

We consider 𝑍^𝑗_𝑖=^𝑡(𝑈⁻¹𝑋^𝑗_𝑖). By the previous Lemma and Malliavin hypothesis,𝑃^2,tot_𝑡det𝑉^−𝑝𝑔0,𝑥^tot₀,𝐼,0<∞ (5.7) 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 𝑢_𝑖 of 𝑈 in (5.3). We consider the Bismut function 𝑓𝑉⁻¹(𝑢_𝑖/𝑢). We integrate by parts as in Theorem 3.4. We deduce under Malliavin assumption that𝑃^3,tot_𝑡𝐷_𝑥1𝑓,𝑈𝑢(0,𝑥,0,0)≤𝐶_𝑙𝑓_∞ (5.8) if 𝑓 has compact support in [0,𝑙]×[0,𝑙]^𝑑.

By the same way, we deduce that if 𝑓 has compact support in [0,𝑙]×[0,𝑙]^𝑑thenexp[𝑡𝔸]𝐷_𝑥𝑓(0,𝑥,0,0)≤𝐶_𝑙𝑓_∞. (5.9) Therefore, the result is obtained.

Remark 5.2. We could do integration by parts to each order in order to show that the semi-group exp[𝑡𝔸] has a smooth heat-kernel under Malliavin assumption.

6. Inversion of the Malliavin Matrix

Proof of Theorem 2.2. Let 𝑠₁<𝑠₂ and let 𝜉 be of modulus 1. Then, exp𝑡𝔸𝕀_[0,𝑠1]𝕀_{𝑉(𝜉)≤𝜖}(0,𝑥,𝐼,0)≥exp𝑡𝔸𝕀_[0,𝑠1]𝕀_{𝑉(𝜉)≤𝜖}(0,𝑥,𝐼,0). (6.1) These two quantities are equal in 𝑡=0 when we consider the semi-group exp[𝑡𝔸]. 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 𝔸𝕀_[0,𝑠1]𝕀_{𝑉(𝜉)≤𝜖}(𝑠,𝑥,𝑈,𝑉)≥𝔸𝕀_[0,𝑠2]𝕀_{𝑉(𝜉)≤𝜖}(𝑠,𝑥,𝑈,𝑉). (6.2) (These two quantities are negative.)
By the result of the appendix,exp𝑡𝔸𝕀_[0,𝑡]𝕀_|𝑈⁻¹_{−𝐼|>𝐶}+𝕀_{|𝑈−𝐼|>𝐶}+𝕀_{|⋅−𝑥|>𝐶}+𝕀_𝑉>𝐶(0,𝑥,𝐼,0)≤𝐶𝑝𝑡^𝑝 (6.3) for all 𝑝.

Lemma 6.1. If |𝜉|=1, then there exist 𝐶 and 𝐶₀ independent of 𝜉 such that exp𝜖𝔸𝕀_{|𝑉𝜉|<𝐶0𝜖}𝕀_[0,𝜖](0,𝑥,𝐼,0)<1−𝐶𝜖^1/2. (6.4)

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 𝛼_𝑠=exp𝑠𝔸g𝑉𝜉𝜖𝕀_[0,𝜖](0,𝑥,𝐼,0). (6.5) In order to consider the derivative in 𝑠 of 𝛼_𝑠, we study the expression 𝛽_𝜖=𝔸𝑔𝑉𝜉𝜖𝕀_[0,𝜖]𝑠^′,𝑥^′,𝑈^′,𝑉^′. (6.6) 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 𝛾_𝑢=_𝑗𝑃_𝑢^𝑗,2𝑔𝑉𝜉𝜖𝑠^′,𝑥^′,𝑈^′,𝑉^′𝑉−𝑔^′𝜉𝜖 (6.7) for 𝑢 between 0 and 𝜖. The first derivative of 𝛾_𝑢 has an equivalent −𝐶𝜖⁻¹ when 𝜖→0, and its second derivative has a bound 𝐶𝜖⁻² when 𝜖→0. Therefore, 0≥𝛾_𝑢≥−𝐶𝑢𝜖 (6.8) on [0,𝜖] and 𝛽_𝜖𝐶≥−𝜖^𝜖₀𝑠^−1/2𝑑𝑠=−𝐶𝜖^−1/2. (6.9)
We deduce from that that𝛼_𝜖≤1−𝐶𝜖^1/2. (6.10)

Remark 6.2. We could improve (6.4) by showing that exp𝜖𝔸𝕀_{|𝑉𝜉|<𝐶0𝜖}𝕀_[0,𝜖]𝑠^′,𝑥^′,𝑈^′,𝑉^′<1−𝐶𝜖^1/2 (6.11) 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 small 𝛼. We slice the time interval [0,𝜖^𝛼] in 𝜖^𝛼−1 intervals of length 𝜖. We haveexp𝑡𝔸𝕀_[0,𝑙]𝕀_{𝑉(𝜉)≤𝜖}(0,𝑥,𝐼,0)≤exp𝜖^𝛼𝔸𝕀_[0,𝑙]𝕀_{𝑉(𝜉)≤𝜖})𝜖(0,𝑥,𝐼,0≤exp^𝛼𝔸𝕀_[0,𝜖]𝕀_{𝑉(𝜉)≤𝜖})≤(0,𝑥,𝐼,0sup_{𝑥′−𝑥≤𝐶0,||𝑈−𝐼≤𝐶0}exp𝜖𝔸𝕀_[0,𝜖]𝕀_{𝑉(𝜉)≤𝜖}0,𝑥^′,𝑈^′,0^𝜖𝛼−1+𝐶𝜖^𝑝 (6.12) for a small 𝐶₀. This last quantity is smaller than 𝐶𝜖^𝑝 for all 𝑝 by the previous lemma if 𝛼 is small enough. The proof of Theorem 2.2 follows from exp𝑡𝔸𝑉^𝑝𝕀_[0,𝑙](0,𝑥,𝐼,0)≤∞ (6.13) 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 all 𝑝>0, 𝑃^3,tot_𝑡𝑢^−𝑝(0,𝑥,0,0)≤∞. (6.14)

Proof. We remark that if we consider only functions of 𝑢, then 𝑃^3,tot_𝑡𝑓(0,𝑥,𝑢,0)=𝑃⁴_𝑡𝑓(𝑢), (6.15) where 𝑃⁴_𝑡 is a Lévy semi-group with generator 𝔸⁴𝑓(𝑢)=𝐶^∞₀𝑑𝑠𝑠^3/2𝑓𝑠⁵𝑔(𝑠)+𝑢−𝑓(𝑢), (6.16) 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 𝑃⁴_𝑡𝑢^−𝑝(0)=𝐶^∞₀𝛽^𝑝−1𝑃⁴_𝑡exp−𝛽𝑢(0)𝑑𝛽. (6.17) By using the adaptation in semi-group theory of the exponential martingales of Levy process of [17, 18], we have 𝑃⁴_𝑡exp−𝛽𝑢(0)=exp𝑡^∞₀exp−𝛽𝑠⁵𝑔(𝑠)−1𝑑𝑠𝑠^3/2. (6.18) The result holds from the Tauberian theorem of [7, 17, 18].

Appendix

Burkholder-Davies-Gundy Inequality

Theorem A.4. Let 𝑠₀>0 and 𝑝∈ℕ. Then, 𝑃^2,tot_𝑡𝕀_[0,𝑠0]𝑥^tot2𝑝0,𝑥^tot₀,0<∞. (A.1)

Proof. Following the idea of [17, Appendix], we consider the auxiliary function 𝐹_𝐶𝑥^tot=𝑥^tot2𝑝+11+|𝑥^tot|^2𝑘/𝐶. (A.2) We get 𝑑𝑃𝑑𝑡^2,tot𝑡𝕀_[0,𝑠0]𝐹_𝐶𝑥^tot0,𝑥^tot₀=𝑃,0_{2,tot𝑡𝑠00−𝑠}𝑑𝑢𝑢^3/2_𝑗𝑃^𝑗,2,tot_𝑢𝐹_𝐶𝑥^tot−𝐹_𝐶𝑥^tot0,𝑥^tot₀,0. (A.3)
Let us consider an improvement of the Gronwall lemma: if |𝑥_𝑠−𝑥₀|≤∫^𝑠₀|𝑥_𝑢|𝑑𝑢, then |𝑥_𝑡−𝑥₀|≤𝐾𝑡|𝑥₀| if 𝑡∈[0,1].
We remark that𝐿^𝑗,2,tot𝐹_𝐶𝑥^tot≤𝐾𝐹_𝐶𝑥^tot (A.4) for 𝐾 independent of 𝐶. Then, by the modified Gronwall lemma, 𝑃^𝑗,2,tot_𝑢𝐹_𝐶𝑥^tot−𝐹_𝐶𝑥^tot≤𝐾𝑢𝐹_𝐶𝑥^tot,𝑑𝑃𝑑𝑡^2,tot𝑡𝕀_[0,𝑠0]𝐹_𝐶𝑥^tot0,𝑥^tot₀,0≤𝐾𝐹_𝐶𝑥^tot+𝐾𝑃^2,tot_𝑡𝕀_[0,𝑠0]𝐹_𝐶𝑥^tot0,𝑥^tot₀,0, (A.5) where 𝐾 does not depend on 𝐶.
By Gronwall lemma,𝑃^2,tot_𝑡𝕀_[0,𝑠0]𝐹_𝐶𝑥^tot0,𝑥^tot₀,0≤𝐾<∞, (A.6) where 𝐾 does not depend on 𝐶. The result arises by doing 𝐶→∞.

By the same procedure, we get the following.

Theorem A.5. Let be 𝑠₀>0 and 𝑝∈ℕ. Then 𝑃^2,tot_𝑡𝕀_[0,𝑠0]|𝑈|^2𝑝0,𝑥^tot₀,𝑈₀<∞ (A.7)

and we get the following.

Theorem A.6. Let 𝑠₀>0 and 𝑝∈ℕ: 𝑃^3,tot_𝑡𝕀_[0,𝑠0]𝑥^tot+|u|+|𝑈|^2𝑝0,𝑥^tot₀,𝑢₀,𝑈₀<∞. (A.8)

Remark A.7. We can show (6.3) by the same way.