Download Hypoelliptic non homogeneous diffusions

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Patrick Cattiaux12 · Laurent Mesnager2
Hypoelliptic non homogeneous diffusions
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Abstract. Let Lt =
d
d
m
X
1 X
1X 2
aij (t, x)∂i ∂j +
bi (t, x)∂i =
Xj (t, x) + X0 (t, x) be a time dependent
2 i,j=1
2 j=1
i=1
second order operator, written in usual or Hörmander form. We study the regularity of the law of the associated non homogeneous (time dependent) diffusion process, under Hörmander’s like conditions. Coefficients
are only Hölder continuous in time. The main tool is Malliavin calculus. Our results extend and correct
previous ones ([17] and related works, [15]). Related topics like filtering theory, killed or reflected processes,
parabolic hypoellipticity are also discussed.
1. Introduction and Summary
For j = 0, . . . , m, consider measurable flows t 7→ Xj (t, •) of C ∞ vector fields on Rd , such that for
any multi-index α, ∂xα Xj (t, •) are bounded on any time-space compact subset [s, t] × K.
One can thus find a pathwise unique solution of the stochastic differential equation

m
X

 dx =
Xj (t, xt ) dwtj + X0 (t, xt ) ,
t
j=1


xs = x
(1.1.a)
t>s
written in Stratonovich form, where (w1 , . . . , wm ) is a standard Brownian motion, or equivalently
of the Ito stochastic differential equation
(1.1.b)
m
where X̄0 = X0 +

m
X

 dx =
Xj (t, xt ) δwtj + X̄0 (t, xt ) ,
t
j=1


xs = x
t>s
1 X ∂Xj
Xj .
2 j=1 ∂x
We will assume that the solution is non explosive, for instance assuming when necessary that
∂xα Xj (t, x) are bounded on [s, t] × Rd . If we denote by xst (x) the solution of (1.1) , then
Tst : Cb −
7 → Cb
f −
7 → E (f (xst (x)))
Ecole Polytechnique, CMAP, F-91128 Palaiseau Cedex
Modal’X, Université de Paris X, UFR SEGMI, 200 avenue de la République, F-92001 Nanterre Cedex
e-mail: [email protected]
e-mail: [email protected]
2
Patrick Cattiaux, Laurent Mesnager
is a non-homogeneous (strong) Markov semigroup with (local) generator
m
Lt =
1X 2
X (t, •) + X0 (t, •)
2 j=1 j
s
written in Hörmander form. Ifwe denote
? by µst (x, dy) the law of xt (x), then µst (x, dy) is a solution
∂
of the weak forward equation
+ Lt µst = 0, µss = δx . We are first interested in the existence
∂t
of (smooth) densities for the laws, i.e.
(1.2)
µst (x, dy) = pst (x, y) dy
where pst is a (smooth) function on Rd ( or a smooth kernel on Rd ⊗Rd ). Since the homogeneous case
(when Lt does not depend on t) is now very well known, we shall really focus on the non-homogeneous
case. However, we will extensively use the time-space process (which is homogeneous), namely the
solution of
(1.3)









dyt =
m
X
Xj (ut , yt ) dwtj + X0 (ut , yt ) ,
j=1
dut = dt,
(u0 , y0 ) = (u, x)
which is given by a flow ψt (w, u, x) such that x 7→ φt (w, u, x) = Πx ψt (w, u, x) is , for each u, a
C ∞ diffeomorphism of Rd . Here of course Πx denotes the projection operator from R ⊗ Rd onto
Rd . Note that the standard proof, lying on Kolmogorov continuity theorem, allows to choose such
a version for each u, but non necessarily for all u simultaneously. In particular if one wants to use
the pathwise composition of flows formula, one has to make an additional regularity assumption on
the Xj . For instance one may assume that ∂xα Xj (•, •) are β-Hölder continuous, for some β > 0. In
this case one can build continuous versions of ∂xα φt (w, •, •).
Of course the key point is that
(1.4)
xst (x) = φt−s (w, s, x) in Law t > s.
Without loss of generality, we may and will assume that s = 0, and write φt (w, x) instead of
φt (w, 0, x).
In order to prove (1.2) one can call upon celebrated Hörmander’s sum of squares theorem ([18]) i.e.
∂
+ X0
(1.5) If the Xj ’s are Cb∞ on [s, T ] × Rd and the Lie algebra generated by Xj , 1 ≤ j ≤ m and
∂t
d
d
is full at each (t, y) ∈]s, T ] × R , then (1.2) holds for all t ∈]s, T ], all x ∈ R and y 7→ pst (x, y)
is smooth (i.e. C ∞ ).
Actually Hörmander’s result is stronger since it deals with parabolic-hypoellipticity. Note that the
statement of (1.5) includes the time component i.e.
(1.6)
dim span Lie
∂
+ X0 , Xj , 1 ≤ j ≤ m,
∂t
(t, y) = d + 1.
Extending Kohn’s proof of Hörmander’s result, Chaleyat-Maurel and Michel [14] have shown a weak
form of parabolic-hypoellipticity, for coefficients which are only measurable in time. This yields
Hypoelliptic non homogeneous diffusions
3
(1.7) If all ∂xα Xj , 0 ≤ j ≤ m are measurable and bounded on [s, T ] × Rd and
dim span Lie (Xj , 1 ≤ j ≤ m)(t, y) = d
for all (t, y) ∈ [s, T ] × Rd , then the same conclusion as in (1.5) holds.
Note that the time component (as vector fields) disappears, but that Lie brackets with X0 are no
∂
more allowed (this is sometimes called the restricted Hörmander’s hypothesis). The role of
will
∂t
be enlightened by a very simple example in a moment.
Notice that when the vector fields are real analytic, the Hörmander’s condition is almost necessary
(see [16]).
Malliavin stochastic calculus of variations initiated in the 80th’s ([25][24]) , proposed a new approach
based on a variational method for s.d.e. (1.3) which leads to the following principle
(1.8) Let Ct (x) =
m Z
X
j=1
t
φs?−1 Xj >< φ?−1
Xj ds be (part of) the Malliavin covariance matrix. If
s
0
Ct−1 (x) belongs to all the Lp ’s (p ∈ [1, +∞[), then µ0t (x, dy) = pt (x, y)dy with y 7→ pt (x, y) ∈
C ∞ Rd .
In (1.8), we are using Bismut notations
−1
∂φs
(w, x)
Xj (s, φs (w, x))
∂x

Y >< Y = Y t Y,


φ?−1
Xj =
s
so that Ct is a nonnegative symmetric matrix. In order to study Ct the standard way is to use
Ito-Kunita-Bismut formula
(1.9) for all ξ ∈ Rd , j = 1, . . . , m
hξ, φt?−1 Xj i(x) = hξ, Xj (0, x)i +
m Z
X
k=1
Z t*
+
ξ, φ?−1
s
0
t
hξ, φ?−1
[Xk , Xj ]iδwsk
s
0
!+
m
∂
1X
+ X0 , X j +
[Xk , [Xk , Xj ]]
ds
∂t
2
k=1
where [Y, Z] is the Lie bracket between vector fields Y and Z. It follows from a now well known
procedure (see e.g. [31] or [5]) that
(1.10) When the Xj ’s, j = 0, . . . , m are Cb∞ on [0, T ] × Rd , a sufficient condition for Ct−1 (x) to be in
all the Lp ’s for all t ∈]0, T ] is that
∂
dim span Lie Xj , 1 ≤ j ≤ m,
+ X0 (0, x) = d + 1
∂t
(for another approach of this result also see Bally [4])
4
Patrick Cattiaux, Laurent Mesnager
The main difference with (1.5) is that the assumption on the Lie brackets is only made at the
starting point (0, x). This could be considered as a slight improvement, but Malliavin calculus has
now proved its strength in many areas where analytic methods are very difficult or impossible to
use (see e.g. the recent book of Malliavin himself [26]). Let us also mention that one can recover
exactly Hörmander’s result using the Markov property. This is partly explained by the following
elementary remark, which has already been made by numerous authors
Remark 1.11 Assume that for s0 > s, µss0 (x, dy) = pss0 (x, y)dy. Then for t > s0 ,
E (f (xst (x))) = E (f (φt−s (w, s, x)))
= E [E (f (φt−s0 (w0 , s0 , φs0 −s (w, s, x))))]
Z
= pss0 (x, y) E (f (φt−s0 (w0 , s0 , y))) dy
Z
−1
0
0
dz
= f (z) E pss0 (x, φ−1
t−s0 (w , s0 , z)) Jacobian φt−s0 (w , s0 , z)
provided all manipulations are allowed. In particular, if pss0 is smooth with a good behaviour at
infinity, pst is also smooth thanks to the regularity of the flow φ−1 and µst (x, dy) = pst (x, y)dy with
−1
pst (x, y) = E pss0 x, φ−1
t−s0 (w, s0 , y) Jacobian φt−s0 (w, s0 , y)
.
This allows to improve (1.7) assuming the restricted Hörmander’s hypothesis only near the time
origin (but everywhere in space).
∂
At this point we shall make an additional important remark about the role of the vector field ∂t
.
In the homogeneous case an equivalent formulation for (1.6) is
dim span Xj , 1 ≤ j ≤ m, Lie brackets of order ≥ 2 of the Xj0 s, 0 ≤ j ≤ m (t, y) = d.
In the non homogeneous case, cancellations may occur because of
∂
∂t .
Here is an example
(1.12) example [35]. Take d = 2, m = 1 and consider the process
2
xt (x) = x1 + wt , x2 + t (x1 + wt )
which law, supported by the curve y2 = x2 + ty12 , is singular. The associated vector fields are
X0 (t, x) = x21
So X1 and [X1 , [X1 , X0 ]] = 2
∂
,
∂x2
X1 (t, x) =
∂
∂
+ 2tx1
∂x1
∂x2
∂
∂
span R2 at each (t, x). Of course (1.6) is not satisfied because
+ X0 , X 1 =
∂x2
∂t
0 at each (t, x).
This fact is not related to the unboundedness of the coefficients. One can easily modify this example
(y2 − x2 = f (t, y1 ) = exp − 12 y12 + (1 + t)2
without changing the situation. In particular, the statement of Theorem 1.1.3 in [17] is wrong.
Let us now turn to the case when the vector fields are no more regular in the time
From
∂component.
Malliavin calculus point of view, the only difficulty is that (1.9) (which involves ∂t
, Xj ) is no more
valid or cannot be iterate. Nevertheless three cases are almost immediate
Hypoelliptic non homogeneous diffusions
5
(1.13.1) vector fields are measurable in time, and Lt is uniformly elliptic in a neighborhood of (0, x),i.e.
m
X
2
hξ, Xj (t, x)i2 ≥ c |ξ| for some c > 0 and all (t, y) ∈ [0, s0 ] × B(x, R);
j=1
(1.13.2) ∂xα Xj belong to Cb1 (R × Rd ) for all α and dim span Lie (Xj , 1 ≤ j ≤ m)(0, x) = d;
1
(1.13.3) Xj (t, x) = fj (t)Wj (x) where 0 < c ≤ |fj | ≤ and usual Hörmander’s hypothesis holds for
c
the Wj ’s.
In these three cases (1.2) holds (with s = 0). (1.13.1) is really immediate while (1.13.2) and (1.13.3)
are mainly contained in Taniguchi [35]. As we already said the tentative of Florchinger [17] to relax
ellipticity or regularity is incorrect. It seems that the derivations of pages 211-212 are incorrect.
Some related works by some authors which use the same discretization method also contain wrong
results.
Another approach was proposed by Chen and Zhou [15]. Since one cannot directly use the stochastic
calculus for Xj (s, φs (w, x)) once s 7→ Xj (s, y) is not C 1 , one can freeze the time, writing
(1.14) for some ad-hoc X̃0 ,
hξ, φt?−1 Xj i(x)
= hξ, Xj (t, x)i +
m Z
X
k=1
+
Z tD
0
t
hξ, φ?−1
[Xk (s, •), Xj (t, •)]i δwsk
s
0
h
iE
ξ, φ?−1
X̃0 (s, •), Xj (t, •) ds.
s
The authors use Kusuoka-Stroock method [20] for obtaining some expansion of Ct in terms of
repeated integrals. They then state (see [15] Thm 1.1)
(1.15) if the ∂xα Xj0 s are uniformly β-Hölder continuous in time, for some β >
(Xj , 1 ≤ j ≤ m) = d, then (1.2) holds.
1
2,
and dim span Lie
Condition β > 21 is used in the derivation of lemma 4.3 of [15], but is bad if one wants to apply such
a result to conditional laws.
Actually the proof of (1.15) given in [15] is still somewhat obscure for us because it lies on controls
on repeated integrals without checking possible cancellations. Let us explain this point on example
(1.12).
(1.16) example (continuation) Of course vector fields being smooth in time, one can on one hand apply
(1.9). For ξ = (0, 1) one gets
hξ, φ?−1
X1 i(x) = hξ, X1 (0, x)i = 0
t
On the other hand if we use (1.14) we have to calculate
∂
∂x2
∂
[X0 (s, •), X1 (t, •)] = −2x1
∂x2
[X1 (s, •), [X1 (s, •), X1 (s, •)]] = 0
−1 ∂φs
1
0
(w, x)
=
−2s(ws + x1 ) 1
∂x
[X1 (s, •), X1 (t, •)] = 2(t − s)
6
Patrick Cattiaux, Laurent Mesnager
Thus
hξ, φt?−1 X1 i(x)
Z
t
Z
t
= hξ, X1 (t, x)i +
2 ξ2 (t − s)δws −
2 ξ2 (x1 + ws )ds
0
0
Z t
Z t
= 2tx1 + 2
(t − s)δws − 2
(x1 + ws )ds.
0
0
Of course this is equal to 0, as the terminal value at time t of hξ, φ?−1
X1 i(x) = 2(t − θ)(x1 + wθ ).
θ
But this is due to cancellations at time t of terms which do not play the same role: initial
condition, bounded variation term, stochastic integral. In other words repeated integrals with
deterministic (but non constant) time dependent integrands may cancel at time t.
Remark and Apologies: Let us add to the previous remark that our first tentative, based on the
same idea (freezing the time) in connection with Norris lemma, contains a similar mistake. All our
apologies for those who had in hands the first version of the paper. We warmly thank an anonymous
referee for pointing out this mistake.
In order to bypass this difficulty, one can use a time discretization procedure and use time regularity
of the process. This is what the authors do in [15], but in a quite intricate way.
In section 2 we shall give an alternate proof of (1.15). It is essentially based on the same ideas, but
in the spirit of Norris lemma instead of Kusuoka-Stroock. We hope it will help to clarify Chen and
Zhou paper. Condition β > 12 will clearly appear as a limitation due to the use of time regularity
of the process. To overcome this limitation other ideas are necessary. These ideas are introduced
in section 3. In addition to all that was used before (time discretization, time freezing, stochastic
calculus), the new points are: first replace L2 norms by L∞ norms (thanks to the regularity of the
coefficients), next use the continuity of the argsup of some one dimensional semi-martingales. The
strategy is quite intricate, so a brief account of it is given at the beginning of section 3. In section 4
we collect some useful estimates on the density. We underline that these estimates are certainly not
sharp. But they are sufficient to derive a parabolic-hypoelliptic theorem which is an improvement
of Chaleyat-Maurel and Michel. In section 5 various applications are quickly discussed: filtering
theory, killed and reflected processes, infinite degeneracy. In all these cases we shall only mention
new difficulties arising from non homogeneity. Precise statements are mainly left to the reader.
Once again we warmly thank an anonymous referee for a careful reading of a first version of this
work. We also wish to thank D. Nualart who mentioned to us the paper of Chen and Zhou [15].
2. Estimates on Malliavin covariance matrix : first approach
For a Rl valued function t 7→ h(t) and β > 0, we define the β-Hölder norm of h as
khkβ = sup (t − s)−β |h(t) − h(s)|
(2.1)
0≤s≤T
Throughout this section we will assume that : for all j = 0, ..., m, all multi-index α, there exists
β(α) > 0 such that
sup ∂yα Xj (•, y)β(α) < +∞
(2.2)
y∈Rd
In particular sup ∂yα Xj (t, •)∞ < ∞, i.e. the Xj ’s belong to Cb∞ uniformly in time on [0, T ]. In
0≤t≤T
addition we assume that the restricted Hörmander’s hypothesis holds i.e.
Hypoelliptic non homogeneous diffusions
7
dim span Lie(Xj , 1 ≤ j ≤ m)(0, x) = d
(2.3)
or equivalently, there exist N (x) = N ∈ N, cN (x) = cN > 0 such that for all ξ ∈ S d−1
N
X
X
0
(2.3 )
2
hξ, Zi (0, x) ≥ cN
k=0 Z∈Σk
where Σ0 = {Xj , 1 ≤ j ≤ m}, Σk+1 = {[Xj , Z] , 1 ≤ j ≤ m, Z ∈ Σk }. N being given by (2.30 ) we
can define some constants
(2.4.a)
(2.4.b)
(2.4.c)
K = sup
max ∂yα Xj (s, •)∞
max
0≤s≤T 0≤|α|≤N +2 0≤j≤m
β a common Hölder coefficient for all α s.t. |α| ≤ N + 2
Kβ = sup
max
max ∂yα Xj (•, y)β
y∈Rd 0≤|α|≤N +2 0≤j≤m
Using time-space continuity, one can then find (t0 , R) ∈ R+ ×R+ s.t. for all s ∈ [0, t0 ], all y ∈ B(x, R),
all ξ ∈ S d−1
N
X
X
(2.5)
2
hξ, Zi (s, y) ≥
k=0 Z∈Σk
cN
2
We shall prove the following version of classical estimates (see [30])
1
Theorem 2.6 Let (2.2), (2.3) and (2.4) hold, for some β > . Then for all p ∈ [1, +∞[, there
2
exists cp which only depends on p, t, x, N , cN , K, β and Kβ such that for all ε > 0


Z tX
m
2
sup P 
ξ, φ?−1
Xj (x) ds ≤ ε ≤ cp εp .
s
ξ∈S d−1
0 j=1
Note that for β = 1, this result essentially follows from the estimates of [30, p. 119], thanks to (1.9).
This is what can be done for obtaining (1.13.2).
Proof: The proof will follow the standard scheme of [30], but requires additional work. For k = 0,
. . . , N , let m(k) be a positive constant we will choose later. Then for a given ξ ∈ S d−1 we introduce
the sets
Ek =
(
X Z t
Z∈Σk
0
2
ξ, φ?−1
Z
s
(x) ds ≤ ε
Choosing m(0) = 1, what we need is a bound for P (E0 ). But
m(k)
)
.
8
Patrick Cattiaux, Laurent Mesnager
P (E0 ) ≤
N
−1
X
c
P(Ek ∩ Ek+1
) + P (E) with E =
N
\
Ek .
k=0
k=1
Step 1 : Upper bound for P(E).
It can be obtained as in [30, p. 121] which yields P(E) ≤ cp εp , where cp depends only on t, x, p, K,
N , cN , R, t0 (defined in (2.5)) and min m(k).
k=0,...,N
c
Step 2 : Upper bound for P(Ek ∩ Ek+1
).
We will more generally get a bound for
(2.7)


Z t
m Z t
X
2
2
P(B) = P 
ξ, φ?−1
Z (x)ds ≤ εq ,
ξ, φ?−1
[Xj , Z] (x)ds ≥ ε
s
s
0
0
j=1
for a vector field Z satisfying (2.2) with bounds (2.4). The ad-hoc value of q will determine the ratio
m(k)/m(k + 1). As usual it is useful to bound the process introducing for θ > 0
τ = inf z ≥ 0,
(2.8)
?−1
φs (ω, x) ≥ ε−θ ∧ t
which satisfies P (τ < t) ≤ cp εp where cp only depends on p, t, x, K and θ. So we only have to bound
P (B, τ = t). To overcome the difficulty explained in (1.16) we introduce a time discretization
(2.9)
si =
it
,
n
i = 0, . . . , n,
ui =
si + si+1
2
and define
Bi =

Z

si+1
si
ξ, φ?−1
Z
s
2
(x)ds ≤ εq ,
m Z
X
j=1
si+1
si

ε
2
ξ, φ?−1
[Xj , Z] (x)ds ≥
s
n
so that
P(B, τ = t) ≤
n−1
X
P(Bi , τ = t).
i=0
Again
P(Bi , τ = t) ≤ P(Bi1 , τ = t) + P(Bi2 , τ = t)
where
Hypoelliptic non homogeneous diffusions
Bi1 = Bi ∩

m Z
X

(2.10)
Bi2 = Bi ∩
9
j=1
j=1
si

m Z
X

ui


ε
2
ξ, φ?−1
[Xj , Z] (x)ds ≥
s
2n 
si+1
ui


ε
2
ξ, φ?−1
[Xj , Z] (x)ds ≥
s
2n 
We shall get an estimate for P(Bi1 , τ = t). The same estimate holds for P(Bi2 , τ = t) just changing si
into si+1 in the derivation below. To get this estimate we use the Ito formula as in (1.14), freezing
the time for Z(•, y). Hence for s ∈ [si , si+1 ], P a.s.
(2.11)
2
ξ, φ?−1
Z
s
(x) =
2
ξ, φ?−1
si Z
+Ais +
m
X
(x) +
m Z
X
j=1
s
si
2
ξ, φ?−1
[Xj (u, •), Z(s, •)] (x)du
u
Msij
j=1
with
Msij =2
Z
Ais =2
Z
s
si
?−1
ξ, φ?−1
[Xj (u, •), Z(s, •)] (x)δwuj
u Z(s, •) (x) ξ, φu
?−1
ξ, φ?−1
[X0 (u, •), Z(s, •)] (x)du
u Z(s, •) (x) ξ, φu
s
si
+
Z
s
si
m
X
ξ, φ?−1
Z(s,
•
)
(x)
ξ, φ?−1
[Xj (u, •), [Xj (u, •), Z(s, •)]] (x)du
u
u
j=1
Now using Kolmogorov continuity criterion, it is not hard to see that both hand sides in (2.11) have
a continuous version, so that (2.11) holds for P a.s. for all s ∈ [si , si+1 ]. So we can integrate (2.11)
with respect to ds (up to a fixed negligible set). Applying Fubini and a stochastic Fubini theorem
(see e.g. [19, p. 116], replacing boundedness by integrability of any order), we get
(2.12)
Z
si+1
si
with
ξ, φ?−1
Z
s
2
(x) ds ≥ Ci + Ai +
m
X
j=1
Mij
10
Patrick Cattiaux, Laurent Mesnager
Z
Ai =
si+1
Ais ds
si
Z
Ci =
si+1
Z
s
si
Z
Mij =
m
si+1 X
j=1
2
ξ, φ?−1
[Xj (s, •), Z(u, •)] (x) du ds
s
si+1
αsij δws
si
Z
αsij =
si+1
s
ξ, φ?−1
Z(u, •) (x) ξ, φ?−1
[Xj (s, •), Z(u, •)] (x) du
s
s
But on {τ = t}, for u and s less than t, the following holds
β
ξ, φ?−1
Z (x) − ξ, φ?−1
Z(u, •) (x) ≤ ε−θ Kβ |s − u|
s
s
(2.13)
and
(2.130 )
β
ξ, φ?−1
[Xj , Z] (x) − ξ, φ?−1
[Xj (s, •), Z(u, •)] (x) ≤ 2ε−θ KKβ |s − u|
s
s
Using (2.13) and (2.130 ), we shall successively get a lower bound of Ci , and upper bounds for |Ai |
and |Mi,j |, up to subsets of small Probability.
Lower bound for Ci
On Bi1 ∩ {τ = t} one has, according to (2.130 ) and (a + b)2 ≥ a2 − 2|ab|,
Ci ≥
Z
si+1
(si+1 − s)
si
−4ε−θ KKβ
m
X
j=1
Z
si+1
si
2
ξ, φ?−1
[Xj , Z] (x) ds
s
m
(si+1 − s)1+β X ξ, φ?−1
[Xj , Z] (x) ds
s
1+β
j=1
ε
(si+1 − si )2+β
− 8mε−2θ K 3 Kβ
2n
(1 + β)(2 + β)
ε
1
≥
− 4mK 3 Kβ ε−2θ 2+β
4n2
n
≥ (si+1 − ui )
(we explicitly used Bi1 for obtaining the second inequality)
We will choose n = [ε−µ ] (integer part of ε−µ ). Hence, if βµ − 2θ > 1, one can find ε0 which only
depends on m, K, Kβ , µ, θ such that for ε < ε0 ,
(2.14)
Ci ≥
1 1+2µ
ε
on Bi1 ∩ {τ = t} .
8
(βµ − 2θ > 1 ensures that −2θ + (2 + β)µ > 1 + 2µ, so that the principal of Ci is
Upper bound for |Ai |
ε
∼ ε1+2µ )
n2
Hypoelliptic non homogeneous diffusions
11
On {τ = t},
i
As ≤ c ε−θ
Z
≤ c ε−θ
Z
s
ξ, φ?−1
u Z(s, •) (x) du
si
s
(s − si )1+β
−2θ
ξ, φ?−1
Kβ
u Z (x) du + cε
1+β
si
where c only depends on K and m according to (2.13). It follows that on {τ = t}
Z
|Ai | ≤ cε−θ
si+1
si
(si+1 − si )2+β
(si+1 − s) ξ, φ?−1
Z (x)|ds + cε−2θ Kβ
s
(1 + β)(2 + β)
and on Bi1 ∩ {τ = t}, by Cauchy-Schwarz inequality,
q
|Ai | ≤ cε−θ ε 2
1
n
3
2
+ cε−2θ Kβ
1
n2+β
.
q
3
+ µ > 1 + 2µ, on can find ε00 which only depends on m, K, Kβ ,
2 2
β, µ, θ and q such that on Bi1 ∩ {τ = t}
Thus if βµ − 2θ > 1 and −θ +
|Ai | ≤
(2.15)
1 1+2µ
ε
for ε < ε00 .
32
Upper bound for |Mi,j |
As usual, we shall use the classical martingale inequality
(2.16)
b
Z
P(
αs2
a
Z
b
1
ds ≤ η , αs δws ≥ η l ) ≤ 2 exp − η 2(l−k) .
a
2
2k
As before on {τ = t}, αsij ≤ cε−θ
si+1
Z
s
cording to (2.13), on {τ = t},
ξ, φ?−1
Z(u, •) (s) du where c only depends on K. Acs
ij 2
2
αs ≤ 2c2 ε−2θ (si+1 − s)2 ξ, φ?−1
Z (x) + 2c2 ε−4θ Kβ2 (si+1 − s)2+2β .
s
Thus on {τ = t} ∩ Bi1 ,
Z
si+1
si
ij 2
αs ds ≤ 2c2 ε−2θ εq 1 + 2c2 ε−4θ Kβ2 1 .
n2
n3+2β
Hence, if q + 2µ − 2θ > 2 + 4µ + 2r and −4θ + (3 + 2β)µ > 2 + 4µ + 2r, on can find ε000 which only
depends on K, Kβ , β, µ, θ and r such that for ε ≤ ε000 , on {τ = t} ∩ Bi1 ,
Z
si+1
si
2+4µ+2r
ij 2
αs ds ≤ ε
.
1024m2
12
Patrick Cattiaux, Laurent Mesnager
Applying (2.16), we get
(2.17)
P(τ = t, Bi1 , |Mij | ≥
1
1 1+2µ
ε
) ≤ 2 exp − ε−2r .
32m
2
To conclude, assume that all conditions (2.14), (2.15) and (2.17) are satisfied, and recall (2.12).
Then for ε small enough (ε ≤ min(ε0 , ε00 , ε000 )) the set
{τ = t} ∩
Bi1
m \
∩
|Mij | <
j=1
1 1+2µ
ε
32m
will be empty provided q > 1 + 2µ. Hence, in this case,
1
P(τ = t, Bi1 ) ≤ 2m exp − ε−2r
2
and if r > 0,
1
P(B, τ = t) ≤ 4m ε−µ exp − ε−2r ≤ cp εp
2
for ε small enough (universal). Thanks to (2.8), we finally get P(B) ≤ cp εp hence a bound for
m(k)
c
P(Ek+1
∩ Ek ) (recall that q =
). This achieves step 2 and the proof of Theorem 2.6,
m(k + 1)
provided we can find θ, q, µ, r which satisfies all conditions cited above i.e.
i)
ii)
iii)
iv)
r > 0, θ > 0, µ > 0, q > 0.
βµ − 2θ > 1.
−4θ + (3 + 2β)µ > 2 + 4µ + 2r.
q > 2µ + 1, q + 2µ − 2θ > 2 + 4µ + 2r, −2θ + q + 3µ > 2 + 4µ.
1 + 2θ
and q > max(2µ + 1, 2µ + 2θ + 2r + 2, µ + 2 + 2θ) so that
β
i), ii) and iv) are satisfied. The delicate point is iii) which yields
Fix r > 0, θ > 0. Then choose µ >
(2.18)
(2β − 1)µ > 2 + 4θ + 2r
and can be satisfied only if 2β − 1 > 0.
2
1
Remark 2.19 As the preceding proof clearly shows, condition β > appears in the control of the
2
martingale term.
precisely,
Ito formula leads to bound ξ, φ?−1
u Z(s, •) (x), while the initial
More
control is on ξ, φ?−1
Z(s, •) (x). The error (s − u)β is not important for the bounded variation
s
term (it only imposes βµ − 2θ > 1), and becomes important for the martingale term (imposing
condition iii) above). Since the martingale term appears to be the leading one, one should try to get
a better control on it. In a sense, it is what we are going to do in the next section, but the strategy
will require more refined tools.
Hypoelliptic non homogeneous diffusions
13
3. Estimates on Malliavin covariance matrix : second approach
To help the reader we first explain the flavor of the strategy on an example
example 3.1: Consider the 2-dimensional process

 xt = w
Zt
t
 yt =
h(u, wu ) δBu
0
∂h
where (w, B) is a Brownian motion. Suppose that h(0, 0) = 0 but
(0, 0) 6= 0. Then X1 (0, 0) and
∂x
1
0
0
2
[X1 , X2 ](0, 0) span R (X1 = 0 , X2 = h(x) , X0 = 0 ), so that Hörmander’s condition is satisfied.
Take ξ = (0, 1). It is easy to check
ξ, φ?−1
X1 (0, 0) = 0,
ξ, φ?−1
X2 (0, 0) = h(t, wt ).
t
t
Z t
Hence we have to control P
h2 (s, ws )ds ≤ ε at least for small t. If h satisfies the β-Hölder
0
∂h
1 ∂h
condition (2.2), we may assume (λ, 0) ≥ (0, 0) > 0. We may thus apply Norris Lemma
∂x
2 ∂x
[30, lemma 2.3.2] to the process
h(λ, ws ) = h(λ, 0) +
s
Z
0
∂h
1
(λ, wu ) δwu +
∂x
2
Z
s
0
∂2h
(λ, wu ) du
∂x2
and get that
P
Z
t
h2 (λ, ws ) ds ≤ ε
≤ cp εp
0
where cp does not depend on λ. We claim first that a continuity argument, or a modification of
Norris Lemma, shows that actually
for all λ ∈ [0, t],
P
2
sup h (λ, ws ) ≤ ε
≤ cp εp .
0≤s≤t
Here again cp does not depend on λ.
ε
The second claim is the following : for almost all w, sup h2 (λ, ws ) will be greater than for all λ,
2
0≤s≤t
1
−β
if it is greater than
ε for a finite number of them
λ1 , ..., λK , K being roughly ε .
Accordingly, P ∀λ ∈ [0, t], sup h2 (λ, ws ) ≥ ε ≥ 1 − cp (β)εp .
0≤s≤t
Now assume that for almost all w, for all λ, the supremum is attained at some single time sλ , i.e.
sλ = argsup h2 (λ, ws ). Of course sλ is random, but for a fixed w, λ 7→ sλ (w) will then be continuous
0≤s≤t
from [0, t] into itself, hence admits at least one fixed point s? . On the above set of big Probability,
we will then have h2 (s? , ws? ) ≥ ε, thus sup h2 (s, ws ) ≥ ε. Using β-Hölder continuity, this yields
0≤s≤t
14
Patrick Cattiaux, Laurent Mesnager
P
Z
t
2
h (s, ws ) ds ≤ ε
1
1+ β
≤ c0p (β) εp
0
i.e. the desired result up to an obvious change of notation. Unfortunately our assumption
P-a.s.,
∀λ ∈ [0, t], the supremum is attained at a single point sλ
has very few chances to be true, because of the exchange of P-a.s. and for all λ. Here again one
has to use continuity to modify this assumption, without modifying
the conclusion.
Moreover how
∂h
to prove that for a fixed λ, sλ exists (is unique) ? Because of (λ, x) ≥ α > 0 for λ ∈ [0, t],
∂x
x ∈ R (which is a stronger assumption than the one we did, but can be used if we modify h and
introduce stopping times), the process h(λ, w• ) is up to a Girsanov transform and a time change,
just a Brownian motion. For Brownian motion the uniqueness of the supremum is a well known
consequence of the existence of a density for the joint law of (Bt , sup Bs ).
0≤s≤t
So we may expect that a careful refinement of the argument above will furnish the result.
From now on, we assume that hypothesis (2.2) and (2.3) are satisfied and we use the notations of
section 2. Our aim is to prove the following
Proposition 3.2 There exists q > 0 such that for all Z satisfying (2.2) with bounds (2.4)


Z t
m Z t
X
2
2
P(B) = P 
ξ, φ?−1
Z (x)ds ≤ εq ,
ξ, φ?−1
[Xj , Z] (x)ds ≥ ε ≤ cp εp
s
s
0
j=1
0
where cp only depends on β, K, Kβ , p, x, m.
1
from now.
This statement is exactly (2.7) in the previous section, but we will not assume β >
2
To get rid of this assumption, we shall deeply use β-Hölder continuity. To this end, we state a
straightforward application of Kolmogorov continuity criterion
Lemma 3.3 Let Y be a flow of vector fields satisfying (2.2) with the bounds (2.4). Then for all
1
β 0 < β ∧ , there exists a version of
2
* +
−1
∂φ
?−1
ξ, φs Y (x) = ξ,
(w, x)
Y (φs (w, x))
∂x s
which is β 0 -Hölder continuous on [0, T ]. Furthermore if we set
then
Kβ 0 (w, Y ) = ξ, φ?−1
Y (x)β 0
•
E Kβp0 (w, Y ) ≤ cp < +∞
where cp only depends on p, x, β 0 , K and Kβ .
Hypoelliptic non homogeneous diffusions
15
The proof is left to the reader. Since the proof of Proposition 3.2 is quite long, we split it into several
steps.
Step 1 : From L2 estimates to L∞ estimates.
Thanks to Lemma 3.3, we first replace integral terms by supremums. In all what follows, we fix once
for all β 0 given by Lemma 3.3. As a first consequence of β-Hölder continuity, we get
1
1
Lemma 3.4 Let Y be as in Lemma 3.3 and ε <
∧ t. Then if a > 2 1 + 0 + 1, for all
4
β
p ∈ [1, +∞[,
P
t
Z
0
ξ, φ?−1
Y
s
2
(x) < εa , sup ξ, φ?−1
Y (x) ≥ ε
s
≤ cp εp
0≤s≤t
where cp only depends on t, p, x, β 0 , K and Kβ (we shall call it a “good” constant).
0
Proof: Let zs = ξ, φs?−1 Y (x). According to 3.3, |zs (w) − zu (w)| ≤ Kβ 0 (w, Y )|s − u|β and
1
P Kβ 0 (w, Y ) ≥
2ε
≤ cp εp
1
where cp is a “good” constant. So we may assume that w ∈ Kβ 0 (w, Y ) <
from now on.
2ε
1 0
For such an w and |s? − s| ≤ εη , one has |zs (w)| ≥ |zs? (w)| − εβ η−1 . Moreover assume that
2
sup |zs (w)| ≥ ε. Then one has for ε < t
0≤s≤t
2
1 β 0 η−1
≥
|zs? (w)| − ε
ds
2
0
interval of size εη
Z t
2
1 2+ 2
where |zs? (w)| = sup |zs (w)|. So choosing η = 0 , we get
zs2 (w)ds ≥ ε β0 . Since a >
β
4
0≤s≤t
0
2
1
1
1
2+
2 1 + 0 + 1 and ε ≤ , it holds εa < ε β0 and accordingly the set
β
4
4
Z
Z
0
t
t
Z
zs2 (w) ds
2
ξ, φ?−1
Y
s
a
(x) < ε , sup ξ, φ?−1
Y (x) ≥ ε
s
0≤s≤t
1
∩ Kβ 0 (w, Y ) <
2ε
is empty.
2
Thanks to Lemma 3.4, if we put q = ra with r > 0 and a = 2 1 +
1
β0
+ 1, the proof of Proposition
3.2 reduces to show that for some r,
(3.5)


m Z t
X
2
2
P (C) = P  sup ξ, φ?−1
Z (x) ≤ εr ,
ξ, φ?−1
[Xj , Z] (x)ds ≥ ε ≤ cp εp
s
s
0≤s≤t
for some “good” constant cp .
Step 2 : Time discretization.
j=1
0
16
Patrick Cattiaux, Laurent Mesnager
We introduce a dicretization of the time interval [0, t]
(3.6)
si =
h 4i
it
−
with n = ε β0 .
n
Introduce the sets
Ci =

m
Z t X

0 j=1
We then show

m

X
3
2
2
?−1
2
ξ, φ?−1
[X
,
Z]
(x)
≥
ε
and
∃s
∈
[s
,
s
],
ξ,
φ
[X
,
Z]
(x)
<
ε
j
i
i+1
j
s
s

1
Lemma 3.7 If ε 2 <
j=1
1
then
2t
P
n−1
\
Ci
!
≤ cp εp
i=1
for some “good” cp .
Z t
ε
Proof: If
is greater than ε, then sup is greater than . So as before, up to a set of probability
t
0≤s≤t
0
− 21
p
less than cp ε , we may assume that Kβ 0 (w, Y ) ≤ ε
for all Y of the form [Xj , Z]. Thus there exists
2
0
β
at least one interval I of length ε such that for s ∈ I,
m
X
j=1
2
1
3
ε
ε
ξ, φ?−1
[Xj , Z] (x) ≥ − ε2 ε− 2 ≥
> ε2 .
s
t
2t
Finally to conclude, it suffices that I contains at least one interval [si , si+1 ] which ensures that the
n−1
n
o
\
1
set
Ci ∩ Kβ 0 (w, Y ) ≤ ε− 2 is empty.
2
i=0
Introduce the family of sets

m

X
2
ξ, φ?−1
[Xj , Z] (x) ≥ ε for all s ∈ [si , si+1 ]
Di =
sup ξ, φ?−1
Z (x) ≤ εr ,
s
s
si ≤s≤si+1



j=1
and the set D =
n−1
[
Di . As in section 2, we can introduce the stopping time τ defined by (2.8)
i=0
and remark that P (τ < t) ≤ cp εp . Hence we are reduced to estimate P (C ∩ {τ = t}). According to
3
3
Lemma 3.7 (just changing ε into ε 2 and replacing r par r), one has P (C ∩ Dc ) ≤ cp εp . So we only
2
have to estimate P (C ∩ D ∩ {τ = t}). Note now that
P (C ∩ D ∩ {τ = t}) ≤ P (D ∩ {τ = t}) ≤
n−1
X
i=0
h 4i
−
P (Di ∩ {τ = t}) ≤ ε β0 max P (Di ∩ {τ = t}) .
i
Hypoelliptic non homogeneous diffusions
17
So we are reduced to show that
P (Di ∩ {τ = t}) ≤ cp εp
(3.8)
for some “good” constant independent of i.
Step 3 : Freezing the time.
On {τ = t}, one has
(3.9)
sup
(s,λ)∈[si ,si+1 ]2
4β
ξ, φ?−1
[Xj , Z] (x) − ξ, φ?−1
[Xj (s, •), Z(λ, •)] (x) ≤ 2KKβ ε−θ ε β0
s
s
Set
Ei =


inf
(s,λ)∈[si ,si+1 ]2
X
j=1

ε
2
.
ξ, φ?−1
[Xj (s, •), Z(λ, •)] (x) ≥
s
2
1
Thanks to (3.9), one can find ε0 which only depends on m, K, Kβ such that, provided θ < , for
2
all ε < ε0
Di ∩ {τ = t} ⊂ Di ∩ {τ = t} ∩ Ei = Fi ∩ {τ = t}
where
Fi =


sup
si ≤s≤si+1
ξ, φs?−1 Z (x) ≤ εr ,
inf
(s,λ)∈[si ,si+1 ]2
m
X
j=1

ε
2
.
ξ, φ?−1
[X
(s,
•
),
Z(λ,
•
)]
(x)
≥
j
s
2
So we are reduced to show
P (Fi , τ = t) ≤ cp εp .
(3.10)
Without loss of generality, we shall only deal with the case i = 0. This will avoid notational
intricacies, since the only important point in our future arguments will be the length si+1 − si .We
now use Ito formula which yields
(3.11)
ξ, φs?−1 Z(λ, •)
(x) = ys (λ) = y0 (λ) +
Z
0
s
βu (λ)du +
m Z
X
j=1
s
αuj (λ)δwuj
0
with
αuj (λ) = ξ, φu?−1 [Xj (u, •), Z(λ, •)] (x)

+
*
m


X
1
βu (λ) = ξ, φu?−1 [X0 (u, •), Z(λ, •)] +
[Xj (u, •), [Xj (u, •), Z(λ, •)]]
(x).


2 j=1
18
Patrick Cattiaux, Laurent Mesnager
As we explained in example 3.1, we want to study the supremum of ys (λ), but be sure that this
supremum is attained at a single point. So we will have to restrict ourselves to a random time
interval rather than working on [0, s1 ] and then evaluate the supremum.
Step 4 : Existence of the argsup.
Since we will use a Girsanov transform, we have to work on the full Probability space Ω. To this
end, we have first to modify y• (λ). Taking if necessary an extension of Ω, we may find another
Brownian motion wm+1 which is independent of (w1 , ..., wm ) (when we work on [si , si+1 ], we have
to consider wsj − wsji rather than wsj and choose wm+1 which is independent of ysi (λ) too). Next we
define
(3.12)
ȳs (λ) = y0 +
Z
s
β̄u (λ)du +
0
m+1
XZ s
j=1
ᾱuj (λ)δwuj
0
where β̄, ᾱj are defined as follows. We choose two smooth functions Mθ,ε and χε defined on R such
that
Mθ,ε (z) = z if |z| < ε−θ ,
±2ε−θ if |z| > 2ε−θ and |Mθ,ε | is bounded by 2ε−θ
and
χε (z) = 1 if |z| ≥
ε
,
2
0 if |z| ≤
ε
,
4
χε is even and non decreasing on R+ .
These functions may be viewed as smooth versions of the cut-off by ±2ε−θ and the indicator 1I|z|≥ 2ε .
We then set
β̄u (λ) = Mθ,ε (βu (λ)) (roughly βu (λ) ∧
ε−θ ∨ −ε−θ )
!
m
X
2
ᾱuj (λ) = Mθ,ε αuj (λ) χε
αuk (λ)
for j = 1, ..., m
k=1
!!

r
m

X

2
ε

m+1

ᾱ
(λ)
=
1
−
χ
αuk (λ)

ε
 u
2








(3.13)
k=1
Notice that
if ω ∈ F0 ,
(3.14)
for s ≤ s1 ,
ȳs (λ, ω) = ys (λ, ω)
i.e. ȳ is actually a modification of y. The second and main fact is
m+1
X
(3.15)
j=1
Indeed if
m
X
j=1
2
ε
ᾱuj (λ) = ᾱu2 (λ) ≥ for all u.
8
2
ε
ε
ε
ᾱuj (λ) is greater than or less that then ᾱu2 (λ) ≥ . Otherwise
2
4
2
Hypoelliptic non homogeneous diffusions
ᾱu2 (λ) ≥
m
X
j=1
19
ε 1
2 2
2
ᾱuj (λ)
χε + (1 − χε ) ≥ × .
4 2
˜ given by the Girsanov density
Thanks to (3.15), we can introduce a new Probability measure P


Z t j
Z t 2
m+1
X
˜
dP
ᾱu (λ)β̄u (λ) j
1
β̄u (λ) 
|F = exp −
δwu −
du
2 (λ)
2
dP t
ᾱ
2
ᾱ
0
u
u (λ)
j=1 0
˜
which is equivalent to P on each Ft . Furthermore, P-almost
surely,
ȳs (λ) = y0 (λ) +
m+1
XZ s
j=1
ᾱuj (λ)δ w̃uj
0
˜
where (w̃1 , ..., w̃m+1 ) is a P-Brownian
motion. Now if we define
As (λ) =
Z
s
ᾱu2 (λ)du and Vs (λ) = inf a ≥ 0,
0
Z
a
Au (λ)du = s
0
˜
there exists a P-Brownian
motion B such that for all s
ȳs (λ) = y0 (λ) + BAs (λ) i.e. ȳVs (λ) (λ) = y0 (λ) + Bs
˜
P-a.s.
But as we recall in example 3.1, it is well known that the supremum of a linear Brownian motion
on a deterministic time interval is a.s. attained at a single time. The same holds with is absolute
ε
value. Take T1 = s1 . Since
8
ε
s ≤ As (λ) ≤ 4ε−2θ (m + 1)s
8
(3.16)
then
ε2θ
8
s ≤ Vs (λ) ≤ s.
4(m + 1)
ε
Hence
(3.17)
4
t
1+2θ+ β20
ε
≤ VT1 (λ) = S1 (λ) ≤ tε β0 = s1 .
32(m + 1)
We get from what precedes
˜
(3.18) ∀λ ∈ [0, s1 ], P-a.s.,there
exists an unique (random) time u1 (ω, λ) = argsup |ȳu (λ)| .
0≤u≤S1 (λ)
20
Patrick Cattiaux, Laurent Mesnager
˜ are equivalent on Fs and S1 (λ) ≤ s1 , (3.18) holds P-a.s. and we can thus define
Since P and P
1
u1 (ω, λ) = argsup |ȳu (λ)|
(3.19)
P-a.s. for all λ ∈ [0, s1 ].
0≤u≤S1 (λ)
Step 5 : Estimates for the supremum.
t
1+2θ+ β40
Define t1 =
ε
thanks to (3.17). Note now that sup |ȳu (λ)| ≤ sup |ȳu (λ)|. We
32(m + 1)
0≤u≤t1
0≤u≤s1
will estimate sup |ȳu (λ)| similarly as in the classical Norris lemma. Ito’s formula yields
0≤u≤t1
ȳt21 (λ) = ȳ02 (λ) + 2
(3.20)
t1
Z
ȳu (λ)β̄u (λ)du + 2
0
m+1
X Z t1
j=1
ȳu (λ)ᾱuj (λ)δwuj +
0
Z
t1
ᾱy2 (λ)du.
0
We then show
3
2
+ 3θ + 0 , there exists ε0 (m, t) s.t. if ε < ε0
2
β
0
0 −2θ
P
sup |ȳu (λ)| ≤ εr ≤ ce−c ε
Lemma 3.21 If r0 >
0≤u≤t1
where c and c0 depend on m and t.
0
Proof: on
sup |ȳu (λ)| ≤ εr , one has
0≤u≤t1
Z
2
(i)
t1
0
Z
(ii)
t1
0
0
ȳu (λ)β̄u (λ)du ≤ 4ε−θ εr t1 =
t
1+ 4 +r 0 +θ
ε β0
8(m + 1)
2
0
ȳu (λ)ᾱuj (λ) du ≤ 4ε−2θ εr t1 =
t
1+2r 0 + β40
ε
8(m + 1)
so that

m+1
Z t1
X
0
0
r + 2 + 1 −θ
P  sup |ȳu (λ)| ≤ εr , 2
ȳy (λ)ᾱuj δwuj ≥ ε β0 2 
0≤u≤t1
j=1 0

0 −θ
is less than ce−c ε
thanks to the martingale inequality we recall in (2.16).
Z
(iii)
t1
0
ᾱu2 (λ)du ≥
ε
t
2+2θ+ β40
t1 =
ε
8
256(m + 1)
so if we choose r0 such that
2r0 > 2 + 2θ +
4
,
β0
1+
4
4
+ r0 + θ > 2 + 2θ + 0 ,
0
β
β
1
2
4
+ r0 + 0 − θ > 2 + 2θ + 0
2
β
β
Hypoelliptic non homogeneous diffusions
21
i.e.
r0 >
3
2
+ 3θ + 0 ,
2
β
0
sup |ȳu (λ)| ≤ εr implies that the
then there is an ε0 depending on m and t s.t. for all ε < ε0 ,
0≤u≤t1
martingale term in (3.20) is bigger than ε
r 0 + β20 + 12 −θ
and we may apply (ii).
2
We will now get an estimate for all λ simultaneously i.e. we prove
Lemma 3.22 Let r0 be as in lemma 3.21. Then there exists ε00 s.t. for ε < ε00
P ∃λ ∈ [0, s1 ], sup |ȳu (λ)| ≤ ε
r0
≤ cp εp
0≤u≤t1
for some “good” cp .
Proof: First of all remark that we can again apply Kolmogorov continuity criterion in order to
build a β 0 -Hölder continuous version of (s, λ) 7→ ȳs (λ) thanks to the mollifying (smooth) functions
Mθ,ε and χε . Denote by Γβ 0 ,ε (ω) the β 0 -Hölder norm kȳ• (λ)kβ 0 . The only point is that moments of
Γβ 0 ,ε (ω) depend on ε. Indeed if ε < 1 then the first derivative of Mθ,ε is bounded by 1 but the first
4
derivative of χε is bounded by . Standard applications of BDG inequalities and easy calculations
ε
then yield
E Γβp0 ,ε (ω) ≤ cp ε−p
(3.23)
r0
−2 β
0
for some “good” cp independent of ε. Divide [0, s1 ] into K = s1 ε
+ 1 intervals of length at
2r 0
most ε β0 . Denote by (λi )i=0,...,K the extremities of these intervals. Then
P ∃λ ∈ [0, s1 ], sup |ȳu (λ)| ≤ ε
0≤u≤t1
2r
≤
K−1
X
P ∃λ ∈ [λi , λi+1 ], sup |ȳu (λ)| ≤ ε
r0
.
0≤u≤t1
i=0
0
But if |λ − λ0 | ≤ ε β0 then
n
o
0
Γβ 0 ,ε ≤ ε−r . In particular
r0
0
0
sup |ȳu (λ) − ȳu (λ0 )| ≤ Γβ 0 ,ε ε2r will be less than εr on the set
0≤u≤t1
r0
−r 0
r0
∃λ ∈ [λi , λi+1 ], sup |ȳu (λ)| ≤ ε , Γβ 0 ,ε ≤ ε
⊂
sup |ȳu (λi )| ≤ 2ε
= Gi
0≤u≤t1
0≤u≤t1
using
0
0
|ȳu (λi )| ≤ |ȳu (λ) − ȳu (λi )| + |ȳu (λi )| ≤ Γβ 0 ,ε ε2r + εr .
Moreover one has
22
Patrick Cattiaux, Laurent Mesnager
n
o
0
0
0
∃λ ∈ [λi , λi+1 ], sup |ȳu (λ)| ≤ εr , Γβ 0 ,ε > ε−r ⊂ Γβ 0 ,ε > ε−r = H.
0≤u≤t1
0
But P(Gi ) ≤ c exp −c0 ε−2θ by lemma 3.21, and P(H) ≤ cγ ε(r −1)γ thanks to (3.23) and Markov
inequality. Accordingly
(3.24)
0
0
0
−2θ
0
−2 r
−2 r
P ∃λ ∈ [0, s1 ], sup |ȳu (λ)| ≤ εr ≤ s1 ε β0 ce−cε
+ cγ s1 ε β0 ε(r −1)γ .
0≤u≤t1
3
2
γ − r0
+ 0 . Hence choosing
= p, the right hand side of (3.24) will
2 β
β0
be less than cp εp for ε < ε00 universal.
2
4
Recall that s1 = ε β0 and r0 >
Step 6 : Conclusion.
Let we return to y• (λ). We may of course choose a continuous version of (s, λ) 7→ ys (λ), and
accordingly of (s, λ) 7→ (As (λ), Vs (λ)). So (3.14) may be rewritten
if ω ∈ F0 , for all λ ∈ [0, s1 ], all s ≤ s1 , ȳs (λ, ω) = ys (λ, ω)
and Lemma 3.22 yields
where F̃0 = F0 ∩
(
∀λ ∈ [0, s1 ],
P(F0 ) ≤ P(F̃0 ) + cp εp
)
sup
|yu (λ)| > εr
0
. We also know that for each fixed λ, the
0≤u≤S1 (λ)
supremum is a.s. attained at a single point. But here again we have to be careful, because the
negligible set depends on λ. So first define u1 (ω, λ) = argsup0≤u≤S1 (λ) ȳu (λ) for all λ ∈ Q ∩ [0, s1 ]
which is well defined P-a.s. since Q is countable. We claim that for P-almost all ω, λ 7→ u1 (ω, λ)
is continuous from [0, s1 ] ∩ Q into [0, s1 ]. This claim is almost standard (continuity of the argsup),
but let we prove it. If λn goes to λ, S1 (λn ) goes to S1 (λ) and u1 (ω, λn ) is bounded. If ū(ω, λ) is a
cluster point, we have (using index n, instead of subsequence)
∀η > 0, ∀s ≤ S1 (λ) − η, ∃N s.t if n ≥ N, S1 (λn ) ≥ S1 (λ) − η and ys (λn ) ≤ yu1 (ω,λn ) (λn ).
Taking limits in both hand sides we get ∀s ≤ S1 (λ) − η, ys (λ) ≤ yū(ω,λ) (λ) , and by letting η go to
0, we get ys (λ) ≤ yū(ω,λ) (λ), ∀s ≤ S1 (λ) which implies ū(ω, λ) = u1 (ω, λ) by uniqueness.
So we can extend (λ 7→ u1 (ω, λ), λ ∈ Q) into a continuous function ū1 (ω, λ), λ ∈ [0, s1 ]. Since ū1 is
continuous from [0, s1 ] into itself, it admits at least one fixed point s̄ = ū1 (ω, s̄) ∈ [0, s1 ].
0
0
For P-almost all ω ∈ F̃0 , yu1 (ω,λ) (λ) > εr for all λ ∈ Q. By continuity yū1 (ω,λ) (λ) ≥ εr for all
0
λ ∈ [0, s1 ], hence yū1 (ω,λ) (s̄) = ys̄ (s̄) ≥ εr . But ys̄ (s̄) = ξ, φ?−1
Z(s̄, •) (x) = ξ, φ?−1
Z (x). If
s̄
s̄
r0 < r, it follows that F̃0 is empty.
Hence P(F0 ) ≤ cp εp , as well as all P(Fi ) ≤ cp εp , i.e. (3.10) is satisfied and the proof of proposition
3.2 is complete.
As a consequence we get exactly as in section 2
Hypoelliptic non homogeneous diffusions
23
Theorem 3.25 Let (2.2), (2.3) and (2.4) holds for some β > 0. Then for all p ∈ [1, +∞[, there
exists cp which only depends on p, t, x, N , cN , K, β and Kβ such that for all ε > 0


Z tX
m
2
sup P 
ξ, φ?−1
Xj (x)ds ≤ ε ≤ cp εp .
s
ξ∈S d−1
0 j=1
4. Existence of regular densities, estimates, parabolic-hypoellipticity
Theorem 3.25 (or Theorem 2.6) implies that Malliavin covariance matrix Ct (x) (see (1.8)) is P
almost surely invertible ad that Ct−1 (x) belongs to all the Lp . Furthermore one can estimate these
moments, namely
Proposition 4.1 If (2.2), (2.3) and (2.4) hold for some β > 0, then one can find constants cp ,
Kp , lp depending on p, x, cN , β, K and Kβ , such that
h
p i
E Ct−1 (x)
≤ cp t−Kp (N lp +1) .
The proof is the same as in the homogeneous case (see [31]). Notice that in the proof of Theorem
m(k)
is rather big (it contains some part in ( β1 )2 ) when β is small. The
3.25, the size of q =
m(k + 1)
constants cp , Kp , lp will then really be big, and certainly not sharp. Also notice that the methods
developed in both previous sections cannot be extended to the case of (uniformly) continuous flows
of vector fields, rather than β-Hölder flows. Indeed for each time discretization we need a polynomial
− 4
− 1
control on the cardinality of the discretization (i.e. ε β0 or ε β0 at some points). However the very
strong global assumption (2.2) can be weakened into a local one, thanks to the following argument:
Lemma 4.2 If the hypotheses of Proposition 4.1 are fulfilled, then for all stopping time τ satisfying
\
\
1
∈
Lp , Cτ (x) is P almost surely invertible and Cτ−1 (x) ∈
Lp .
τ
1≤p<∞
1≤p<∞
This is an easy consequence of the estimates 4.1 (see e.g. [7] or [9] for an extensive use). In particular
if (2.2) is replaced by a local assumption, one can use for τ the exit time of some ball centered at x.
All these remarks, combined with the usual methodology of Malliavin calculus, yield the following
Theorem
Theorem 4.3 Let (xt )t≥s be the solution of the stochastic differential equation (1.1). Assume that
(i) t 7→ Xj (t, •) is a measurable flow of C ∞ vector fields such that ∂xα Xj , 0 ≤ j ≤ m, are bounded
on [s, T ] × Rd for any multiindex α of length at least 1;
(ii) there exists s0 > s and R > 0 such that for all j = 0, . . . , m and all multiindex α, there
exists β(α) > 0 s.t. ∂xα Xj (•, y) is β(α)-Hölder continuous for all y with |y − x| ≤ R and
sup k∂xα Xj (•, y)kβ(α) < +∞ on the time interval [s, s0 ];
|y−x|≤R
(iii) dim span Lie (Xj , 1 ≤ j ≤ m) (s, x) = d.
Then for all t ∈]s, T ], the law µst (x, dy) of xt has a density pst (x, y) and y 7→ pst (x, y) ∈ Cb∞ (Rd ).
Furthermore there exists a constant K which only depends on the assumptions i) − iii) such that for
all multiindex α
α
∂y pst (x, •) ≤ c(|α|)(1 ∧ (t − s))−K(|α|+ d2 ) exp (c(|α|, x)(t − s)).
(4.4)
∞
24
Patrick Cattiaux, Laurent Mesnager
Actually one can expect more, i.e. that pst (x, •) belongs to the Schwartz space of rapidly decreasing
functions. Recall that Malliavin integration by parts formula yields the existence of a random variable
θt (α) with moments of any order such that
E [∂ α f (xt )] = E [f (xt )θt (α)]
In particular if f is compactly supported by Γ and x ∈
/ Γ , one can introduce the hitting time of Γ
in the right hand side. As in [11] this furnishes the more precise estimate
d
1
(4.5) |∂yα pst (x, y)| ≤ c1 (1 ∧ (t − s))−K(|α|+ 2 ) |x − y|− 2 exp c2 (t − s) exp − c3
|x − y|2 1 ∧ (t − s)
where c1 , c2 , c3 depend on the assumptions i) − iii), c1 and c2 also depend on α.
As we already said these estimates are not sharp, and when β > 21 , the proof of Theorem 2.6 furnishes
better estimates than the ones one get thanks to Theorem 3.25. As in the homogeneous case one can
use the diffeomorphism property of the flow in order to study the regularity of the kernels pst (•, •).
One can also use the Markov property of the time-space process (we called ψt (w, s, x)) to localize
regularity in an open domain (see e.g. [11] section 1). In the latter case however, Hörmander’s
hypothesis has to be satisfied for all u ∈ [s, t], since it requires the use of stopping times. Up to this
restriction, results like Proposition 1.12 in [11] are still true in the non homogeneous case.
We shall now study parabolic-hypoellipticity as discussed in [11], [14] or [21], that is the regularity
of solutions of
∂u
+ Lt u + cu = f ; u(T, x) = g(x) ;
∂t
(4.6)
t ∈ [S, T ].
A natural candidate for solving (4.6) is
(4.7)
"
u(t, x) = E g(xtT ) exp
Z
T
#
c(s, xts )ds − E
"Z
t
T
f (s, xts ) exp
t
Z
s
c(v, xtv )dv ds
#
t
where xts is the solution of (1.1) which starts at x at time t.
One can easily show that if all coefficients are C ∞ in space, uniformly in time (as well as f , g, c)
then u is actually a smooth (in space) solution of (4.6) in the sense of Schwartz distributions. It
∂u
easily follows that
is a bounded function. When everything is continuous in time, so is u and
∂t
∂u
then
. Conversely by applying Ito formula, any classical solution of (4.6) is given by (4.7).
∂t
Definition 4.8 Let U be an open subset of Rd . The operator
hypoelliptic in ]S, T [×U if for any u ∈ ℘0 (]S, T [ × U ) ,
(
∂
+ Lt + c is said to be parabolic∂t
∂
+ Lt + c)u ∈ L∞ ((]S 0 , T 0 [ , C ∞ (V ))
∂t
implies u ∈ L∞ ((]S 0 , T 0 [ , C ∞ (V )) for any ]S 0 , T 0 [⊆]S, T [ and any open subset V of U .
Hypoelliptic non homogeneous diffusions
25
∂
Recall that in the homogeneous case, Hörmander’s theorem implies hypoellipticity of ∂t
+ Lt + c.
This result was derived, using Malliavin calculus, by Kusuoka and Stroock ([20], Thm 8.6 and Thm
8.13). Another, slightly less complete, but simpler approach is proposed in [11] section 2. In the non
homogeneous case, Thm 1.1 of Chaleyat-Maurel and Michel is some kind of parabolic-hypoellipticity
in ]S, T [×Rd (actually it is a global statement, but it can be rephrased in local terms).
Namely they show that, if Hörmander’s hypothesis is everywhere satisfied (with some uniform
bounds on [S, T ] × Γ, Γ compact), then any solution of (4.6) which can be written dµt × dt for a
locally bounded flow of Radon measures µt , belongs to L∞ ( [S, T ] , C ∞ (Rd ) .
Remark than in order to prove parabolic-hypoellipticity, it is enough to prove
(4.9) for all h ∈ C0∞ (]S, T [×U ), all multiindex α and all multiindex η of length |η| = 0 or 1,
|h∂tη ∂xα h , ui| ≤ C khk∞ , where C only depends on the data and the support of h.
But if u solves (4.6) in ℘0 , then
h∂t ∂xα h , ui = hL?t ∂xα h + c ∂xα h , ui − h∂xα h , f i
hence it is enough to show (4.9) with η = 0, i.e. integrate by parts in the space direction. So we can
state
Theorem 4.10 Assume that 4.3 i) is fulfilled, 4.3 ii) holds on any compact subset of Ū on the full
∂
+ Lt + c is parabolic-hypoelliptic in
time interval [S, T ], and 4.3 iii) holds at each x ∈ Ū . Then
∂t
]S, T [×U .
The proof can be done exactly as in [11] p.433-446, using smoothness of the densities pcst (x, •) of
the law of the process exponentially killed at rate c, which follows from 4.1, 4.2 and usual tools of
Malliavin calculus. We do not know how to adapt Kusuoka-Stroock method which is based on small
time precise estimates.
In regard with Chaleyat-Maurel and Michel result, Theorem 4.10 holds for any distribution solution
instead of measure solution, but we have to assume β-Hölder continuity in time, while these authors
only need L∞ . However it is quite possible that their proof can be improved, while, as we said, our
approach really needs Hölder continuity.
In view of the intricacies of sections 2 and 3, we hope it is possible to get a simpler proof of existence
and regularity for pst . But Malliavin calculus has shown to be a very efficient tool in studying various
problems, so that Theorem 2.6 and Theorem 3.25 have their own interest. In the next section we
shall briefly indicate some possible applications.
5. Further applications
5.1. Filtering
One of the object of [14] or [17] was to use Malliavin’s calculus for time dependent systems in order
to get smoothness of conditional laws as the ones which appear in filtering theory. Our results give
the correct hypothesis for section 2 of [17]. However the best way to study conditional laws is the
use of the partial Malliavin calculus introduced by Bismut and Michel [8]. The same can be done for
time dependent coefficients, the only difference being that the drift X0 is forbidden in the statement
of partial Hörmander’s hypotheses.
5.2. Infinitely degenerate diffusions
The terminology is due to Bell and Mohammed [6] (also see [29]). The non homogeneous case is
studied in [32], but with wrong hypotheses and proofs. With the material of the present paper, [32]
can be corrected.
26
Patrick Cattiaux, Laurent Mesnager
5.3. Killed and reflected processes
The study of reflected diffusions in a domain M with non characteristic boundary ∂M or of the
killed process when it exits such a domain was done, in the homogeneous case in [12] [13][10][9].
One of the main geometric tools is to perform a change of coordinates in a neighborhood of boundary
points such that the transversal coordinate becomes a drifted (reflected) Brownian motion.
In the non homogeneous case, one can mimic what is done in the above papers, except this change
of coordinates. Indeed it leads to use Ito formula with a time dependent function. If the coefficients
are C 1 in time, there is no problem. For less time-regular coefficients the situation is more delicate.
We will not enter into details here.
5.4. Miscellaneous
We should mention various others applications like systems driven by an infinite number of B.M. as
in [27] (this would correct [33]), manifold values s.d.e and so on.
In the numerous applications of Malliavin calculus, small time behaviour of heat kernels takes an
important place (see [21], [2] [3] [22] [23]). We do not know what can be done for non homogeneous
heat kernels in the same spirit.
Another open question is the study of the set of strict positivity of the density (see [1] or [28] for
the homogeneous case).
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