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

be 2m + 2 vector fields on ℝ^d with bounded derivatives at each order. Let

(1.1)

be an Hoermander′s type operator on ℝ^1+d. Let

(1.2)

be a second Hoermander′s operator on ℝ^1+d. Bismut [1] considers the generator

(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

. Bismut introduced the solution of the stochastic differential equation starting at x in Stratonovitch sense:

(1.4)

where

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,

(1.5)

Such operator is classically related to the Dirichlet Problem [3].

Classically [4],

(1.6)

where

is the solution of the Stratonovitch differential equation starting at x:

(1.7)

The question is as following: is there an heat-kernel associated with the semi-group exp [t𝕃¹]? This means that

(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

, then it is a diffeomorphism almost surely in x. 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 t → A_t) in order to show that this isthe problem if

(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,

(2.1)

The results of this paper could be extended to generators of the type

(2.2)

where

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:

(2.3)

We consider the Malliavin generator

on 𝔼_d:

(2.4)

We consider the Malliavin semi-group

associated and

We perform the same algebraic considerations on 𝕃². We get

, and

. Let us consider the total generator

(2.5)

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 x is checked:

(2.6)

holds for all p, then

(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

(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,

(3.1)

where

(ϕ(x) is a convenient matrix on ℝ^m which depends smoothly on x and whose derivatives at each order are bounded.

does not depend on x, and

belong to ℝ^m). Let

be a smooth function on ℝ^1+d+1,

denotes its gradient, and

denotes its Hessian.

We consider the generator

acting on smooth functions on ℝ^1+d+1,

(3.2)

In (3.2), the generator is written under Itô’s form. It generates a time inhomogeneous in the parameter s semi-group

. We can consider

(3.3)

We put

(3.4)

It generates a semi-group

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):

(3.5)

We consider the heat semi-group associated with

(3.6)

Let us recall [15, Theorem 2.2] that

(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

(3.8)

Let us consider the semi-group

associated with

(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:

(3.10)

Proof. For the integrability conditions, we refer to the appendix.

We remark that ∂/∂u commute with , therefore with . We deduce that

(3.11)

By the method of variation of constants,

(3.12)

In order to show that, we follow the lines of (2.17) and (2.18) in [15]. We apply

to (3.11).

By Lemma 3.1,

(3.13)

Let us consider the vector fields on ℝ^1+d × 𝔾_d,

(3.14)

We consider the Hoermander’s type operator associated with these vector fields:

(3.15)

We consider the generator

(3.16)

It generates a semi-group

. By lemma 3.2 of [15], we have

(3.17)

By [15, Equation (3.18)],

(3.18)

In [15, Equation (3.18)], we consider the semi-group

instead of the semi-group

and the test function Df instead as of the test function

here.

is considered as an element of ℝ and not as a one-order differential operator:

(3.19)

Therefore,

(3.20)

We write

(3.21)

where

(3.22)

The Volterra expansion (see [15, Equation (3.17)]) if it converges gives the following formula:

(3.23)

But

is linear in u₀. Therefore:

(3.24)

In this last formula,

are considered as differential operators.

Therefore, does not depend on U₀ and is equal to

(3.25)

where

are considered as elements of ℝ^d. We deduce as in [15, Equation (3.17)],

(3.26)

But

is linear. Therefore,

(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 . 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 s, then one gets

(3.28)

where one take does not derivative in the direction of s in

We can perform the same improvements as in [15, page 512]. We define on

some vectors fields:

(3.29)

where

(3.30)

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 s, then one gets

(3.31)

where

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 :

(i)
if z_s > 0 on this time interval, then is transformed in for a small parameter λ,
(ii)
if z_s < 0 on this time interval, then is transformed in 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 . 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 and of the type 𝕃^j,3.

4. Integration by Parts on the Subordinator

Let us consider diffusion type generator of the previous part:

(4.1)

Let us consider the semi-group

(4.2)

and the semi-group

(4.3)

We have classically

(4.4)

where the smooth vector fields are Lipschitz.

Therefore, we can write

(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

(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,

(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

associated with it.

It is not clear that

is a Markovian semi-group. We decompose

(4.8)

where

(4.9)

𝔸^λ,1,ϵ generates a Markovian semi-group

. But

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:

(4.10)

Theorem 4.1 (Girsanov). For f with compact support in (s, x), one has

(4.11)

Proof. By linearity,

(4.12)

But by an elementary change of variable,

(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 . Here, we do a perturbation of ΔA_s into . By standard result on Levy processes, the law of the Levy process is equivalent to the law of A_t. Moreover, and are independents. Therefore, the result.

Bismut’s idea to state hypoellipticity result is to take the derivative in λ of

(4.14)

in order to get an integration by parts.

First of all, let us compute in λ = 0. It is fulfilled by the next considerations. Let us consider a generator written under Hoermander’s form:

(4.15)

where g_λ are smooth and where the vector fields Y_i are smooth Lipschitz on

. We consider the semi-group

associated with it. Let us introduce the vector fields on

(4.16)

Let us consider the diffusion generator

(4.17)

Associated with it there is a semi-group

Proposition 4.3. For f smooth with compact support, one has

(4.18)

Proof. Let us introduce the vector fields on :

(4.19)

and the generator

(4.20)

Associated with it there is a semi-group

If the Volterra expansion converges, then

(4.21)

But

is linear in

and therefore the quantity

does not depend on

. Therefore the Volterra expansion reads

(4.22)

Let us compute 𝕃^λ,1 − 𝕃^λ,2. It is equal to

(4.23)

We use the relation (see [15, Lemma 3.2])

(4.24)

and the relation

(4.25)

Therefore,

(4.26)

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,

(4.27)

where

is the solution of the Stratonovitch equation starting at x:

(4.28)

Therefore,

is solution starting at 0 of the Stratonovitch differential equation:

(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+d𝔸^λ,2:

(4.30)

It generates a semi-group

. In order to see that, we split the generator by keeping the values od s〈ϵ or s〉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:

(4.31)

Proof. We have

(4.32)

But by [15, Lemma 3.2.]:

(4.33)

Therefore

satisfies the parabolic equation associated with

. 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 in .

(4.34)

belong to

Proof. If the Volterra expansion converges, then

(4.35)

But

is linear in

and therefore the quantity

do not depend of

. Therefore the Volterra expansion reads

(4.36)

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 D exp [t𝔸]f.

Let us compute . We remark that

(4.37)

Namely, the generator of

does not act on the u₀ component such that the two sides of (4.37) satisfy the same parabolic equality.

Therefore,

(4.38)

where

, Therefore,

(4.39)

a₃(t) in the previous expression is the only term which contains a derivative of f, because by Proposition 4.3,

(4.40)

Let 𝔸³ be the generator on ℝ^1+d+1+d:

(4.41)

It generates a semi-group,

. We get the following.

Theorem 4.8. For f with compact support in s and with bounded derivatives at each order, we have

(4.42)

Proof. If the Volterra expansion converges, then

(4.43)

But

is linear in (u₀, U₀). Let us explain the details of that. We have to compute

(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:

(4.45)

But

(4.46)

does not depend on (u₀, U₀). Therefore, the right-hand side of formula (4.45) is equal to

(4.47)

But

(4.48)

because

is linear in (u₀, U₀) and because the vector fields which give the generator of

are linear in u₀, U₀. Therefore,

(4.49)

But by an analog of Theorem 4.5,

(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:

(4.51)

where

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 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)

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.

is the semi-group associated with 𝔸^3,ϵ where we replace in the construction of (4.41) f_λ(s) by :

(4.53)

By the appendix,

(4.54)

if h is compact support in s. Let us consider the generator A^3,ϵ associated with

. If g = 〈df, u, U〉, then we have by Duhamel principle

(4.55)

By the proof of Theorem 4.8,

is affine in (u₀, U₀). Namely, in the proof of this theorem, we have removed the

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,

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)

where C(ϵ) → 0 when ϵ → 0. 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).

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 and . 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,

(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,

(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) that

(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

(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 s

(5.4)

Proof. If is a function with compact support depending only of s, x^tot and V, we have

(5.5)

We do the change of variable U → U and V → UV on the Malliavin generator

. By using Lemma 3.7 of [15], it is transformed in

where for 𝔸^2,tot we consider the same type of operator as 𝔸² but with the modified vector fields:

(5.6)

It remains to use the appendix to show the Lemma.

We consider

. By the previous Lemma and Malliavin hypothesis,

(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 that

(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]^dthen

(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,

(6.1)

These two quantities are equal in t = 0 when we consider the semi-group

. 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)

(These two quantities are negative.)

By the result of the appendix,

(6.3)

for all p.

Lemma 6.1. If |ξ| = 1, then there exist C and C₀ independent of ξ such that

(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

(6.5)

In order to consider the derivative in s of α_s, we study the expression

(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

(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,

(6.8)

on [0, ϵ] and

(6.9)

We deduce from that that

(6.10)

Remark 6.2. We could improve (6.4) by showing that

(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 have

(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

(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,

(6.14)

Proof. We remark that if we consider only functions of u, then

(6.15)

where

is a Lévy semi-group with generator

(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

(6.17)

By using the adaptation in semi-group theory of the exponential martingales of Levy process of [17, 18], we have

(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,

(A.1)

Proof. Following the idea of [17, Appendix], we consider the auxiliary function

(A.2)

We get

(A.3)

Let us consider an improvement of the Gronwall lemma: if , then |x_t − x₀| ≤ Kt|x₀| if t ∈ [0,1].

We remark that

(A.4)

for K independent of C. Then, by the modified Gronwall lemma,

(A.5)

where K does not depend on C.

By Gronwall lemma,

(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

(A.7)

and we get the following.

Theorem A.6. Let s₀ > 0 and p ∈ ℕ:

(A.8)

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

References

1 Bismut J. M., The calculus of boundary processes, Annales Scientifiques de l′École Normale Supérieure. (1984) 17, no. 4, 507–622, 776287 (86d:60087), ZBL0561.60081.
10.24033/asens.1482
Web of Science® Google Scholar
2 Revuz D. and Yor M., Continuous Martingales and Brownian Motion, 1991, Springer, Berlin, Germany, 1083357 (92d:60053).
10.1007/978-3-662-21726-9
Google Scholar
3 Rogers L. C. G. and Williams D., Diffusions, Markov Processes, and Martingales, 1987, 2, John Wiley & Sons, New York, NY, USA, 921238 (89k:60117).
Google Scholar
4 Ikeda N. and Watanabe S., Stochastic Differential Equations and Diffusion Processes, 1989, 2nd edition, North Holland, Amsterdam, The Netherlands, 1011252 (90m:60069).
Google Scholar
5 Malliavin P., Stochastic calculus of variation and hypoelliptic operators, 1978, Proceedings of the Stochastic Analysis, Kinokuniya, Kyoto, Japan, 195–263, 536013 (81f:60083), ZBL0411.60060.
Google Scholar
6 Bismut J. M., Martingales, the malliavin calculus and hypoellipticity under general hörmander′s conditions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. (1981) 56, no. 4, 469–505, https://doi.org/10.1007/BF00531428, 621660 (82k:60134), ZBL0445.60049.
10.1007/BF00531428
Google Scholar
7 Bismut J. M., Calcul des variations stochastique et processus de sauts, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. (1983) 63, no. 2, 147–235, https://doi.org/10.1007/BF00538963, 701527 (85a:60077), ZBL0494.60082.
10.1007/BF00538963
Google Scholar
8 Bichteler K., Gravereaux J., and Jacod J., Malliavin Calculus for Processes with Jumps, 1988, Gordon and Breach Science Publishers, New York, NY, USA.
Google Scholar
9 Léandre R., Une généralisation du théoreme de Hoermander a divers processus de sauts, Ph.D. thesis, 1984, Université de Franche-Comte, Besançon, France.
Google Scholar
10 Léandre R., Régularité de processus de sauts dégénérés. II, Annales de l′Institut Henri Poincaré. Probabilités et Statistique. (1988) 24, no. 2, 209–236, 953118 (89j:60103), ZBL0669.60068.
Google Scholar
11 Léandre R., J. Azéma, P. A. Meyer, and M. Yor, Calcul des variations sur un brownien subordonné, Séminaire de Probabilités, XXII, 1988, 1321, Springer, Heidelberg, Germany, 414–433, Lecture Notes in Math., https://doi.org/10.1007/BFb0084147, 960537 (89m:60201), ZBL0663.60062.
10.1007/BFb0084147
Google Scholar
12 Ben Arous G., Kusuoka S., and Stroock D. W., The Poisson kernel for certain degenerate elliptic operators, Journal of Functional Analysis. (1984) 56, no. 2, 171–209, https://doi.org/10.1016/0022-1236(84)90086-7, 738578 (85k:35093), ZBL0556.35036.
10.1016/0022-1236(84)90086-7
Web of Science® Google Scholar
13 Cattiaux P., Hypoellipticité et hypoellipticité partielle pour les diffusions avec une condition frontière, Annales de l′Institut Henri Poincaré. Probabilités et Statistique. (1986) 22, no. 1, 67–112, 838373 (87k:60180), ZBL0595.60059.
Google Scholar
14 Cattiaux P., Regularité au bord pour les densités et les densités conditionnelles d′une diffusion réfléchie hypoelliptique, Stochastics. (1987) 20, no. 4, 309–340, 885877 (88h:60153), ZBL0637.60092.
10.1080/17442508708833447
Google Scholar
15 Léandre R., Malliavin calculus of Bismut type without probability, Festchrift in Honour of K. Sinha, 2006, 116, Indian Academy of Sciences. Proceedings. Mathematical Sciences, 507–518, ZBL1127.60053.
10.1007/BF02829706
Google Scholar
16 Léandre R., T. Simos, Girsanov transformation for Poisson processes in semi-group theory, 936, Proceedings of the International Conference of Numerical Analysis and Applied Mathematics, September 2007 , Corfu, Greece, American Institute of Physics Proceedings, 336–338.
Google Scholar
17 Léandre R., J. Sabatier, Malliavin calculus of Bismut type for Poisson processes without probability, Fractional Order Systems, 2008, 715, Journal Europeen des Systemes Automatises, 715–733.
Google Scholar
18 Léandre R., A. Luo, Regularity of a degenerated convolution semi-group without to use the Poisson process, Proceedings of the Non linear Science and Complexity, 2010, Porto, Portugal, Springer, 311–320.
Google Scholar
19 Léandre R., V. Balan, Wentzel-Freidlin estimates for jump process in semi-group theory: lower bound, 17, Proceedings of the International Conference of Differential Geometry and Dynamical Systems, 2010, Bucuresti, Romania, 107–113, Balkan Society of Geometers Proceedings.
Google Scholar
20 Léandre R., H. Arabnia, Wentzel-Freidlin estimates for jump process in semi-group theory: upper bound, Proceedings of the International Conference on Scientific Computing, 2010, Las Vegas, Nev, USA, CSREA Press, 187–193, To appear Compueter technology and applications.
Google Scholar
21 Léandre R., M. Vinagre, Varadhan estimates for a degenerated convolution semi-group: upper bound. To appear, Proceedings of the Fractional Differentiation and applications, October 2010, Badajoz, Spain.
Google Scholar
22 Léandre R., A criterium for the strict positivity of the density of the law of a Poisson process, Advances in Difference Equations. (2011) 2771487, 803508.
Web of Science® Google Scholar
23 Léandre R., A. M. Madureira, Applications of the Malliavin calculus of Bismut type without probability, Proceedings of the Simulation, Modelling and Optimization, 2006, Lisbon, Portugal, 559–564.
Google Scholar
24 Léandre R., Applications of the malliavin calculus of bismut type without probability, WSEAS Transactions on Mathematics. (2006) 5, 1205–1211.
Google Scholar
25 Léandre R., Malliavin calculus of Bismut type in semi-group theory, Far East Journal of Mathematical Sciences. (2008) 30, no. 1, 1–26, 2483411 (2010a:60195), ZBL1161.60020.
Google Scholar
26 Léandre R., Stochastic analysis without probability: study of some basic tools, Journal of Pseudo-Differential Operators and Applications. (2010) 1, no. 4, 389–400, https://doi.org/10.1007/s11868-010-0020-3, 2747902.
10.1007/s11868-010-0020-3
Web of Science® Google Scholar
27 Léandre R., A path-integral approach to the Cameron-Martin-Maruyama-Girsanov formula associated to a bilaplacian. Preprint.
Google Scholar
28 Léandre R., A generalized Fock space associated to a bilaplacian, Proceedings of the International Conference on Applied and Engineering Mathematics, October, 2011, Shanghai, China.
Google Scholar
29 Yosida K., Functional Analysis, 1977, Fourth edition, Springer, Heidelberg, Germany, 0350358 (50#2851).
Google Scholar

Citing Literature

All articles

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

Abstract

1. Introduction

2. Statements of the Theorems

3. Integration by Parts on the Underlying Diffusion

4. Integration by Parts on the Subordinator

5. The Abstract Theorem

6. Inversion of the Malliavin Matrix

Appendix

Burkholder-Davies-Gundy Inequality

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

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

Abstract

1. Introduction

2. Statements of the Theorems

3. Integration by Parts on the Underlying Diffusion

4. Integration by Parts on the Subordinator

5. The Abstract Theorem

6. Inversion of the Malliavin Matrix

Appendix

Burkholder-Davies-Gundy Inequality

References

Citing Literature

References

Related

Information