Fractional Differential Equations 2011View this Special Issue
Malliavin Calculus of Bismut Type for Fractional Powers of Laplacians in Semi-Group Theory
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.
Let be vector fields on with bounded derivatives at each order. Let be an Hoermander's type operator on . Let be a second Hoermander's operator on . Bismut  considers the generator and the Markov semi-group . This semi-group has a probabilistic representation. We consider a Brownian motion independent of the others Brownian motions . Bismut introduced the solution of the stochastic differential equation starting at in Stratonovitch sense: where are independent Brownian motions.
Classically , where is the solution of the Stratonovitch differential equation starting at : The question is as following: is there an heat-kernel associated with the semi-group ? This means that There are several approaches in analysis to solve this problem, either by using tools of microlocal analysis or tools of harmonic analysis. Malliavin  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  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 . 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  is that if we consider the jump process , then it is a diffeomorphism almost surely in . So, Bismut mixed the tools of the Malliavin Calculus for diffusion (on the process ) and the tools of the Malliavin Calculus for Poisson process (on the jump process ) in order to show that this isthe problem if 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 . 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 . 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 . Let be a generator of a Markovian semi-group . Then, The results of this paper could be extended to generators of the type where and , 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 where is the space of invertible matrices on and the space of symmetric matrices on . is the generic element of . is called the Malliavin matrix.
On , we consider the vector fields: We consider the Malliavin generator on : We consider the Malliavin semi-group associated and .
We perform the same algebraic considerations on . We get , , and . Let us consider the total generator and the Malliavin semi-group .
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 is checked: holds for all , then where is the density of a probability measure on .
Theorem 2.2. If the quadratic form is invertible in , then the Malliavin condition holds in .
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 , where ( is a convenient matrix on which depends smoothly on and whose derivatives at each order are bounded. does not depend on , and belong to ). Let be a smooth function on , denotes its gradient, and denotes its Hessian.
We consider the generator acting on smooth functions on , In (3.2), the generator is written under Itô’s form. It generates a time inhomogeneous in the parameter semi-group . We can consider We put It generates a semi-group .
Let us consider the Hoermander’s type generator associated with the smooth Lipschitz vector fields on on : We consider the heat semi-group associated with Let us recall [15, Theorem 2.2] that where depends only on . In the left-hand side of (3.7), we apply the enlarged semi-group to the test function and in the right-hand side we apply the semi-group to the test function . belongs to and belongs to . From this, we deduce the following.
Lemma 3.1. One has the relation
Let us consider the semi-group associated with We get, with the same notations for the following.
Theorem 3.2. For bounded continuous with compact support in , one has the following relation:
Proof. For the integrability conditions, we refer to the appendix.
We remark that commute with , therefore with . We deduce that By the method of variation of constants, In order to show that, we follow the lines of (2.17) and (2.18) in . We apply to (3.11).
By Lemma 3.1, Let us consider the vector fields on , We consider the Hoermander’s type operator associated with these vector fields: We consider the generator It generates a semi-group . By lemma 3.2 of , we have By [15, Equation (3.18)], In [15, Equation (3.18)], we consider the semi-group instead of the semi-group and the test function instead as of the test function here. is considered as an element of and not as a one-order differential operator: Therefore, We write where The Volterra expansion (see [15, Equation (3.17)]) if it converges gives the following formula: But is linear in . Therefore: In this last formula, are considered as differential operators.
Therefore, does not depend on and is equal to where are considered as elements of . We deduce as in [15, Equation (3.17)], But is linear. Therefore, It remains to replace by in this last equation and to compare (3.26) with (3.13) and (3.20).
We consider the Malliavin generator . We can perform the same algebraic construction as in Theorem 3.2. We get two semi-groups and . and are smooth with bounded derivatives in . We get by the same procedure the following.
Theorem 3.3. If is bounded with bounded derivatives and with compact support in , then one gets where one take does not derivative in the direction of in .
We can perform the same improvements as in [15, page 512]. We define on some vectors fields: where where have derivatives bounded at each order and has derivative with polynomial growth.
We can consider the generator associated with these vector fields and perform the same algebraic computations as in Theorem 3.2. We get two semi-groups and and are smooth with bounded derivatives in . We get by the same procedure the following.
Theorem 3.4. If is bounded with bounded derivatives and with compact support in , then one gets where does not include derivative in the direction of .
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 . On this random time interval, we do the following translation on the leading Brownian motion :(i) if on this time interval, then is transformed in for a small parameter ,(ii)if on this time interval, then is transformed in for a small parameter .
According to the fact that has compact support (this means that we consider bounded values of ), the transformed Brownian motion has an equivalent law through the Girsanov exponential to the original Brownian motions. The term in in Theorem 3.2 come that from the fact we take the derivative in of the Girsanov exponential. When we do this transformation, we get a random process . Derivation of it in is done classically according to the stochastic flow theorem, which leads to the study of generators of the type and of the type .
4. Integration by Parts on the Subordinator
Let us consider diffusion type generator of the previous part: Let us consider the semi-group and the semi-group We have classically where the smooth vector fields are Lipschitz.
Therefore, we can write We consider a diffeomorphsim of with bounded derivative of first order in equal to if and equals to if (we suppose small). We can write We do this operation on the two operators on giving . We get a generator .
According the line of stochastic analysis, we consider a generator on . If is a generator on with associated semi-group , then we consider the generator on , where is the Jacobian of the transformation . By doing this procedure in (1.3), we deduce a global generator and a semi-group associated with it.
It is not clear that is a Markovian semi-group. We decompose where generates a Markovian semi-group . But is a bounded operator on the set of bounded continuous functions on endowed with the uniform norm. The Volterra expansion converges on this set:
Theorem 4.1 (Girsanov). For with compact support in , one has
Proof. By linearity, But by an elementary change of variable, The result holds by the unicity of the solution of the parabolic equation associated with . To state the integrability of , we refer to .
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 . Here, we do a perturbation of into . By standard result on Levy processes, the law of the Levy process is equivalent to the law of . Moreover, and are independents. Therefore, the result.
Bismut’s idea to state hypoellipticity result is to take the derivative in of in order to get an integration by parts.
First of all, let us compute in . It is fulfilled by the next considerations. Let us consider a generator written under Hoermander’s form: where are smooth and where the vector fields are smooth Lipschitz on . We consider the semi-group associated with it. Let us introduce the vector fields on : Let us consider the diffusion generator Associated with it there is a semi-group .
Proposition 4.3. For smooth with compact support, one has
Proof. Let us introduce the vector fields on :
and the generator
Associated with it there is a semi-group .
If the Volterra expansion converges, then But is linear in and therefore the quantity does not depend on . Therefore the Volterra expansion reads Let us compute . It is equal to We use the relation (see [15, Lemma 3.2]) and the relation Therefore, We insert this formula in the right-hand side of (4.23) and we see that satisfies the same parabolic equation as .
Remark 4.4. Let us show from where this formula comes. Classically,
where is the solution of the Stratonovitch equation starting at :
Therefore, is solution starting at 0 of the Stratonovitch differential equation:
which can be solved classically by using the method of variation of constant .
Let us introduce the generator on : It generates a semi-group . In order to see that, we split the generator by keeping the values od and we proceed as for (see (4.10)).
We get the following.
Theorem 4.5. For smooth with compact support in and with derivatives of each order bounded, one has the relation if one takes only derivatives in of the considered expressions:
Proof. We have But by [15, Lemma 3.2.]: Therefore satisfies the parabolic equation associated with . Only the integrability of puts any problem. It is solved by the appendix since has compact support in .
Theorem 4.6. For with compact support in in . if , belong to .
Proof. If the Volterra expansion converges, then But is linear in and therefore the quantity do not depend of . Therefore the Volterra expansion reads But the last term in the right-hand side of (4.26) is equal to by the end of the proof of the Proposition 4.3.
Remark 4.7. Analogous formula works for .
Let us compute . We remark that Namely, the generator of does not act on the component such that the two sides of (4.37) satisfy the same parabolic equality.
Therefore, where , Therefore, in the previous expression is the only term which contains a derivative of , because by Proposition 4.3,
Let be the generator on : It generates a semi-group, . We get the following.
Theorem 4.8. For with compact support in s and with bounded derivatives at each order, we have
Proof. If the Volterra expansion converges, then But is linear in . Let us explain the details of that. We have to compute By the technique of the beginning of the proof of Proposition 4.3, it does not depend on . Therefore, the Volterra expansion reads: But does not depend on . Therefore, the right-hand side of formula (4.45) is equal to But because is linear in and because the vector fields which give the generator of are linear in . Therefore, But by an analog of Theorem 4.5,
We can summarize the previous considerations in the next theorem.
Theorem 4.9. If is a diffeomorphism of equal to if and if , then one has the following integration by part formula if is with compact support in , bounded with bounded derivatives at each order: where .
Theorem 4.10. Let one suppose that near 0. Then, (4.51) is still true.
Proof. It is enough to show that wecan approximate by a function equal to s if . Let us give some details on this approximation. We consider a smooth function from into equal to zero if and equal to 1 if . We put
We remark that(i)if , then ,(ii)if , then ,(iii)if , then .
is the semi-group associated with where we replace in the construction of (4.41) by : By the appendix, if is compact support in . Let us consider the generator associated with . If , then we have by Duhamel principle By the proof of Theorem 4.8, is affine in . Namely, in the proof of this theorem, we have removed the which is equal to zero in because this expression is linear in . Its component in is smooth with bounded derivatives at each order. By Theorem 4.6, is still affine in and its components in are smooth with bounded derivatives at each order. Moreover, if is affine in with components in smooth with bounded derivatives at each order, then we get that, for , where when . This can be seen as an appliation of the Duhamel formula applied to the two semi-groups and . Then, the result arises from the Duhamel formula (4.55).
Theorem 4.11 (Bismut). If in a neighborhood of 0 and is equal to 1 if , then one has the following integration by parts: let be a function with compact support in , bounded with bounded derivatives at each order. Then,
5. The Abstract Theorem
There are two partial derivatives to treat:(i)the partial derivative in the time of the subordinator ,(ii)the partial derivatives in the space of the underlying diffusion .
Let us begin by the most original part of Bismut's Calculus on boundary process, that is, the integration by parts in the time .
We look at (4.42). We remark (see the next part) that for all . So, we take and we apply (4.42) for this convenient semi-group. We get can be estimated by using the appendix by for with compact support in and by (5.2).
Lemma 5.1. For a conveniently enlarged semi-group in the manner of Theorem 3.4, one has for with compact support in
Proof. If is a function with compact support depending only of , and , we have We do the change of variable and on the Malliavin generator . By using Lemma 3.7 of , it is transformed in where for we consider the same type of operator as but with the modified vector fields: