Komatsu, Takashi

Takeuchi, Atsushi

Osaka City University Repository

Komatsu, T. and Takeuchi, A.
Osaka J. Math.
38 (2001), 681–691
SIMPLIFIED PROBABILISTIC APPROACH
TO THE H ¨ORMANDER THEOREM
TAKASHI KOMATSU and ATSUSHI TAKEUCHI
(Received November 20, 1999)
1. Introduction
In this paper, we shall discuss the problem to find sufficient conditions under
which the probability law of the solution to the stochastic differential equation has a
smooth transition density. There are many approaches to this problem in the theory of
partial differential equations. It is well known that Ho¨rmander ([2]) showed the rela-
tion between the hypoellipticity of second order elliptic differential operators and the
dimension of the Lie algebra generated by vector fields associated with coefficients of
the differential operator. Malliavin ([6], [7]) introduced the new differential calculus on
the Wiener space, and applied its calculus to the probabilistic proof of the Ho¨rmander
theorem. He introduced the Ornstein-Uhlenbeck operator which is an unbounded self-
adjoint non-negative operator on the 2-space over the Wiener space, and obtained the
integration by parts formula on this space ([3], [9]). On the other hand, Bismut ([1])
gave the different approach from Malliavin’s work. He showed the integration by parts
formula by using the Girsanov transformation. For the integration by parts formula,
the integrability of the inverse of so called Malliavin covariance matrix is essential.
Kusuoka and Stroock ([5]) presented a key lemma for the proof of the integrability.
Norris ([8]) gave a simplified proof of the key lemma. His proof of it is still consid-
erably long and complicated.
Instead of the Kusuoka-Stroock-Norris key lemma, we shall present a new lemma
that plays an important role in the Malliavin calculus for SDE’s. This can be proved
easily and directly only by using simple stochastic calculations, that is, the Ito formula
and the Fubini type theorem for stochastic integrals. In order to show the integrability
of the inverse of the Malliavin covariance matrix, it suffices to prove the exponential
decay of the Laplace transform of the quadratic form of the covariance matrix. It is
possible to prove the exponential decay by an iterative application of the new lemma.
Therefore, by using the new key lemma, we can easily show the Ho¨rmander theorem.
The organization of this paper is as follows: in Section 2, we give some prelimi-
naries that need in our argument, and introduce well-known results on the integrability
of the inverse of the Malliavin covariance matrix. In Section 3, our main results are
stated. In the final section, the Ho¨rmander theorem is proved.
682 T. KOMATSU AND A. TAKEUCHI
2. Preliminaries
Let ( 0 F ; {F }) be the -dimensional Wiener space, that is, 0 =
(R+; R | 0 = 0), F = σ[ | ≤ ], F =
∨ F and is the Wiener measure
on ( 0 F). Let = ( ) · ∂ ( = 0 1 . . . ) be smooth vector fields on R such
that derivatives of all orders of ∂ are bounded.
We consider the -dimensional SDE
(1) = 0 +
∫
0
0( ) +
∫
0
∑
=1
( ) ◦
for 0 ∈ R . This equation is equivalent to
φ( ) = φ( 0) +
∫
0
0φ( ) +
∫
0
∑
φ( ) ◦
for φ ∈ ∞(R ; R). From the assumption for coefficients of (1), there exists
the unique solution = ( 0 ) in the pathwise sense. Moreover the mapping
defines a stochastic flow of diffeomorphisms on R . The Jacobi matrix =
((∂/∂ 0 ) ( 0 ))1≤ ≤ of the diffeomorphism satisfies the linear SDE
(2) = +
∫
0
′
0( ) +
∫
0
∑ ′( ) ◦
where = (δ )1≤ ≤ . The symbol ϕ′ denotes the matrix ((∂/∂ )ϕ ( ))1≤ ≤ for
any ϕ ∈ 1(R ; R ). Let be the solution to the linear SDE
(3) = −
∫
0
′
0( ) −
∫
0
∑ ′( ) ◦
It is easily checked that = = . Since coefficients of (1), (2), and (3) satisfy
the linear growth condition, we have
(4)
[
sup
≤
(| | + ‖ ‖ + ‖ ‖ )
]
<∞
for all > 1. For a matrix , its transposed matrix is denoted by ∗. Define
=
∫
0
∑
( ) ( )∗ ∗ = ∗
The matrix is called the Malliavin covariance matrix. The following result is well
known in the Malliavin calculus (cf. [3], [9]).
SIMPLIFIED APPROACH TO H ¨ORMANDER THEOREM 683
Proposition 1. For > 0, if (det )−1 ∈ ⋂ >1 ( 0 ), then the probability
law of has a smooth density.
Throughout this paper, ’s denote certain positive absolute constants, which may
change every lines. It is sufficient for the assumption of the above proposition that
[{inf| |=1( · )}− ] is bounded for all > 1, because
[(det )− ] = [(det ) (det )− ]
≤
[
{ inf
| |=1
( · )}−2
]1/2
Lemma 1. If the condition
sup
| |=1
[( · )− ] <∞
is satisfied for all > 1, then the following one is also satisfied for all > 1.[
{ inf
| |=1
( · )}−
]
<∞
Proof. For any ′ ∈ −1, we have | · − ′ · ′| ≤ 2 ‖ ‖ | − ′|. Hence
it is possible to choose a sequence { } of points on −1 such that, for any > 0,
if · ≥ 2 (1 ≤ ≤ (‖ ‖/ ) −1), then inf| |=1( · ) ≥ . Therefore we
have [
{ inf
| |=1
( · )}−
]
= ( )−1
∫ ∞
0
λ −1
[
exp{−λ inf
| |=1
( · )}
]
λ
≤
∫ ∞
0
λ −1
{ [
exp{−λ inf
| |=1
( · )} ; ‖ ‖ ≤ λ
]
+ [‖ ‖ > λ]
}
λ
=
∫ ∞
0
λ −1
{∫ ∞
0
λ −λ
[
inf
| |=1
( · ) ≤ ‖ ‖ ≤ λ
]
+ [‖ ‖ > λ]
}
λ
≤
∫ ∞
0
λ −1
{∫ ∞
0
λ −λ
(λ/ ) −1∑
=1
[ · ≤ 2 ]
+
( [‖ ‖ +1]
λ +1
)
∧ 1
}
λ
≤
∫ ∞
0
λ2 + −2
{∫ ∞
0
−λ ∑
≥1
2−2
2 [ · ≤ 2 ]
}
λ +
=
∑
≥1
1
2
∫ ∞
0
3−4 − [ · ≤ ] +
684 T. KOMATSU AND A. TAKEUCHI
=
∑
≥1
1
2 [( · )4−4 − ] +
≤ sup
| |=1
[( · )4−4 − ] +
< ∞
The proof is complete.
From Lemma 1 and the equality
[( · )− ] = ( )−1
∫ ∞
0
λ −1 [exp{−λ( · )}] λ
our main purpose is to find a sufficient condition under which
(5) sup [exp{−λ( · )}] = (λ− ) (λ→∞)
is satisfied for all > 1.
3. Lower bounds of functionals of semimartingales
Let ( ) be a continuous semimartingale. Then∫
0
( )2 ≥
{∫
δ
−
∫ −δ
0
}
( )2(6)
=
∫
δ
∫
−δ
( )2 =
∫
0
δ ( ) ( )2
where δ ( ) = ( + δ) ∧ − ∨ δ for δ > 0. This inequality is very simple, but it will
turn out that the new key lemma is a consequence of this inequality.
For the sake of simlicity, set
M [(ζ )] = exp
{
−
∫
0
∑
=1
ζ ( ) − 1
2
∫
0
∑
=1
ζ ( )2
}
for an R -valued continuous semimartingale ζ( ) = {ζ ( )}
=1. Let the symbol ‖ξ‖ de-
note the value sup |ξ( )| for a given process ξ( ).
Consider semimartingales
( ) = 0( ) +
∑
=1
( ) 0( ) = 00( ) +
∑
=1
0 ( )
SIMPLIFIED APPROACH TO H ¨ORMANDER THEOREM 685
and define
(λ) =
{
∈ 0 |
∑
=0
‖ 2‖ +
∑
=0
‖ 20 ‖ > λ1/4
}
Theorem 1. Let λ > 1. There exist positive constants 0 1 2 independent of
(·) and λ, and a positive random variable (λ) with [ (λ)] ≤ 1 such that the
inequality
(7) 0
∫
0
λ4 ( )2 + λ−1/8 log (λ) ≥ 1
∫
0
λ1/4
∑
=0
( )2 − 2
holds on the complement of the set (λ) for sufficiently large λ.
Proof. STEP 1. By using inequality (6) and the Ito formula, we have∫
0
λ4 2 ≥
∫
0
λ4δ 2
=
∫
0
λ4δ
{(∑
2 + 2 0
)
+ 2
∑ }
= − 1
1
log ˜ (1) + ∫
0
λ4δ
∑
2 +
∫
0
2λ4δ 0
−
∫
0
2 1λ8δ2 2
∑
2
= − 1
1
log ˜ (1) + 1 + 2 + 3
where 3λ1/8 < 1 ≤ 3λ7/4 and ˜ (1) = M [(2 1λ4δ )]. We consider lower esti-
mates of 1 2 3 on (λ) . Set δ = λ−3. Since λ3δ = 1 on [δ − δ],
1 =
∫
0
λ
∑
2 −
∫
0
λ(1− λ3δ )
∑
2 ≥
∫
0
λ
∑
2 − 2λ−7/4
Since 0 ≤ δ ≤ δ = λ−3, we see that
2 ≥ −
∫
0
λ4 2 −
∫
0
λ−2 20 ≥ −
∫
0
λ4 2 − λ−7/4
3 ≥ −2 1λ2
∫
0
2
∑
2 ≥ −2 1λ−7/4
∫
0
λ4 2
686 T. KOMATSU AND A. TAKEUCHI
Therefore we obtain the inequality
(8)
∫
0
λ4 ( )2 + 1
1
log ˜ (1) ≥ ∫
0
λ
∑
=1
( )2 −
STEP 2. For the sake of simplicity, set ε = λ−1 and µ = λ1/4. From the Fubini
type theorem for stochastic integrals, we obtain the equality
µ9
∫
ε
(∫
−ε
0( )
)2
= 2µ9
∫
ε
∫
−ε
(∫
0( )
)
0( )
= 2µ9
∫
ε
∫
−ε
{
( )− ( )−
∫ ∑
( )
}
0( )
= 2µ9
∫
ε
∫
−ε
{ ( )− ( )} 0( ) − µ4
∫
0
λ( )
∑
( )
=
1
2
log ˜ (2) + 2µ9∫
ε
∫
−ε
{ ( )− ( )} 0( )
+
2
2
∫
0
µ8 2λ
∑
2
=
1
2
log ˜ (2) + 4 + 5
where 3µ1/2 < 2 ≤ 3µ,
λ( ) = 2µ5
∫ ( +ε)∧
∨ε
∫
−ε
0( ) ˜ (2) = M [( 2µ4 λ )]
We see that
| λ( )| ≤ 2µ5‖ 0‖
∫ +ε
( − + ε) ≤ µ−5/2
on (λ) . To estimate of 4, 5 on (λ) are routine works.
4 ≤
∫
ε
∫
−ε
2µ‖ 0‖µ8| ( )− ( )|
≤ µ2‖ 20 ‖ε( − ε) +
∫
ε
∫
−ε
µ16| ( )− ( )|2
≤ µ−2‖ 20 ‖ +
∫
ε
∫
−ε
2µ16{ ( )2 + ( )2}
≤ µ−1 + 4
∫
0
µ12 2
SIMPLIFIED APPROACH TO H ¨ORMANDER THEOREM 687
5 ≤ 22
∫
0
∑
µ3 2
Therefore we obtain the estimate
µ9
∫
ε
(∫
−ε
0( )
)2
− 1
2
log ˜ (2)(9)
≤
∫
0
µ16 2 +
∫
0
µ4
∑
2 +
STEP 3. Next we consider the lower estimate of the left-hand side of (9). Define
a new process
ξλ( ; ) = ( − ) 0( ) +
∫
−ε
0( ) = ε 0( − ε) +
∫
−ε
( − ) 0( )
Then |ξλ( ; )| ≤ ε‖ 0‖ for − ε ≤ ≤ . By using the Ito formula, we have
ξλ( ; )2
= ε2 0( − ε)2 +
∫
−ε
2( − )ξλ( ; ) 0( ) +
∫
−ε
( − )2
∑
0 ( )2
≥ ε2 0( − ε)2 +
∫
−ε
2( − )ξλ( ; ) 0( )
Hence
µ9
∫
ε
(∫
−ε
0( )
)2
≥ µ
∫
ε
0( − ε)2 + 2µ9
∫
ε
∫
−ε
( − )ξλ( ; ) 0( )
= µ
∫ −ε
0
2
0 +
∫
0
ηλ 0
where
ηλ( ) = 2µ9
∫ ( +ε)∧
∨ε
( − )ξλ( ; )
Since it holds that
|ηλ( )| ≤ 2µ9
∫ +ε
( − ) |ξλ( ; )| ≤ µ−5/2
688 T. KOMATSU AND A. TAKEUCHI
on (λ) , we have
µ9
∫
ε
(∫
−ε
0( )
)2
≥
∫
0
µ 20 − µ−2 +
∫
0
ηλ 0
=
∫
0
µ 20 − µ−2 +
∫
0
ηλ 00 +
∫
0
ηλ
∑
0
≥
∫
0
µ 20 − µ−2 −
1
3
log ˜ (3) − 3
2
∫
0
|ηλ|2
∑
2
0
≥
∫
0
µ 20 − µ−2 −
1
3
log ˜ (3) − 3µ−4
where 3µ1/2 < 3 ≤ 3µ4 and ˜ (3) = M [( 3ηλ 0 )]. Therefore we obtain the estimate
(10) µ9
∫
ε
(∫
−ε
0( )
)2
≥ − 1
3
log ˜ (3) + ∫
0
µ 20 −
on (λ) . Set (λ) = ( ˜ (1))√µ/ 1 ( ˜ (2))√µ/ 2 ( ˜ (3))√µ/ 3 . Then
[ (λ)] ≤ [ ˜ (1)]√µ/ 1 [ ˜ (2)]√µ/ 2 [ ˜ (3)]√µ/ 3 ≤ 1
From (8), (9) and (10), we obtain inequality (7).
4. Existence of the smooth density
In this section, we prove the Ho¨rmander theorem by using Theorem 1. For =
= φ( ) · ∂ (φ ∈ ∞(R ; R )), let [ ] = − , and set
̺[ 0 ] = [ 0 ] +
1
2
∑
=1
[ [ ]] ̺[ ] = [ ] ( = 1 . . . )
By using the Ito formula,
( ) = ̺[ 0 ] +
∑
̺[ ]
We shall introduce sets of vector fields:
A0 = { 1 . . . }
A = {̺[ ] | ∈ A −1 = 0 1 . . . } ( ≥ 1)
Theorem 2. Assume that Ho¨rmander’s condition is satisfied, that is, the linear
space generated by
⋃ A at 0 is R . Then for > 0, the probability law of has
a smooth density.
SIMPLIFIED APPROACH TO H ¨ORMANDER THEOREM 689
Proof. By the Ho¨rmander condition, there exist an integer ≥ 0 and constants
γ, η > 0 such that
(11)
∑
∈S
=0 A
( · φ( ))2 ≥ η
for all ∈ −1 and with | − 0| < γ. Consider the following stopping time:
= inf
{
> 0 | sup
≤
| − 0| ≥ γ or sup
≤
‖ − ‖ ≥ 1/2
}
∧
From the Chebyshev inequality, the Burkholder inequality and (4), we see that
[ < λ−β] ≤
[
sup
≤λ−β
| − 0| ≥ γ
]
+
[
sup
≤λ−β
‖ − ‖ ≥ 1
2
]
(12)
≤
[
sup
≤λ−β
| − 0|
]
+
[
sup
≤λ−β
‖ − ‖
]
=
(
λ− β/2
)
for all ≥ 4/β, where β = 2−1−4 . Remark that | ∗ | ≥ 1 − ‖ − ‖ ≥ 1/2 under
≤ . Set α = 2−1−4 ( = 0 1 . . . ). Consider a functional Q (λα A ) defined
by
Q (λα A ) =
∫
0
∑
φ·∂ ∈A
(λα · φ( ))2
for ∈ −1. For a given vector field = φ( ) · ∂ , let ‖ ‖ denote the value
sup0≤ ≤ | φ( )|. Define subsets (λ) of 0 by 0(λ) = ∅ and
(λ) =
⋃
∈A −1
{
∈ 0 |
∑
=0
‖̺[ ]‖2 +
∑
=0
‖̺[ ̺[ 0 ]]‖2 > λ2α
}
for ≥ 1. From (4), [ (λ)] = (λ− ) as λ→∞ for > 1. For ≥ 1, by Theorem
1, there exists a positive random variable ( λ) with [ ( λ)] ≤ 1 such that the
inequality
(13) Q (λα −1 A −1) ≥ −λ−α log ( λ) + Q (λα A )−
holds on (λ) . Set A′ = ⋃
=0A . By the iterative application of (13),
Q (λα0 A0) ≥ −λ−α1 log (1 λ) + Q (λα1 A′1)−(14)
≥ · · · · · ·
690 T. KOMATSU AND A. TAKEUCHI
≥ −
∑
=1
λ−α log ( λ) + Q (λα A′ )−
on
⋂
=0 (λ) . By the Jensen inequality, we see[∏
=1
( ( λ))2λ−α ] ≤∏
=1
[( ( λ))2 λ−α ]1/ ≤ 1
for sufficiently large λ. From (11), (12) and (14), we have[
exp{−λ( · )} ]
≤ [ exp{−λ( · )} { ≥λ−β} ] + [ < λ−β]
≤
[∏
=1
( ( λ))λ−α exp{− Q (λα A′ )} { ≥λ−β}]
+
[⋃
=0
(λ)
]
+
[
< λ−β
]
≤
[
exp
{
− Qλ−β
(
λα A′ )} { ≥λ−β}]1/2 + (λ− )
≤ exp
{
− λ2α −β inf
| |=1
inf
| − 0|<γ
∑
∈A′
( · φ( ))2
}
+
(
λ−
)
=
(
λ−
)
Since property (5) is satisfied, the transition density of is smooth.
REMARK. The new key lemma can be extended to the general semimartingale
with the jump term. Hence we can discuss the regularity on the transition density asso-
ciated with the jump type SDE. This will be discussed in the forthcoming paper ([4]).
References
[1] J.M. Bismut: Martingales, the Malliavin calculus and hypoellipticity under general Ho¨r-
mander’s conditions, Zeit. fu¨r Wahr., 56 (1981), 469–505.
[2] L. Ho¨rmander: Hypoelliptic second order differential equations, Acta Math., 117 (1967), 147–
171.
[3] N. Ikeda and S. Watanabe: Stochastic differential equations and diffusion processes, 2nd edi-
tion, North-Holland/Kodansha, Tokyo, 1989.
[4] T. Komatsu and A. Takeuchi: On the smoothness of pdf of solutions to SDE of jump type,
Preprint (1999).
[5] S. Kusuoka and D.W. Stroock: Applications of the Malliavin calculus, Part I, Proce. Taniguchi
Symp. SA Kyoto 1982, North-Holland/Kinokuniya, Tokyo, 1984, 271–306.
SIMPLIFIED APPROACH TO H ¨ORMANDER THEOREM 691
[6] P. Malliavin: Stochastic calculus of variation and hypoelliptic operators, Proce. Intern. Symp.
SDE Kyoto 1976, Kinokuniya, Tokyo, 1978, 195–263.
[7] P. Malliavin: -hypoellipticity with degeneracy, Stochastic Analysis, Academic Press, New
York, 1978, 199–214, 327–340.
[8] J. Norris: Simplified Malliavin calculus, Lect. Notes in Math. 1204, 101–130, Springer-Verlag,
Berlin, 1986.
[9] D. Nualart: The Malliavin calculus and related topics, Springer-Verlag, 1995.
Department of Mathematics
Osaka City University
Sugimoto-3, Sumiyoshi-ku
Osaka 558-8585, Japan


Simplified probabilistic approach to the Hormander theorem

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Simplified probabilistic approach to the Hormander theorem

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