Hypoelliptic non homogeneous di usions · The main tool is Malliavin calculus. Our results extend and correct previous ones ([17] and related works, [15]). Related topics like ltering

Probability Theory and Related Fields manuscript No.(will be inserted by the editor)

Patrick Cattiaux12 · Laurent Mesnager2

Hypoelliptic non homogeneous diffusions

the date of receipt and acceptance should be inserted later

Abstract. Let Lt =1

2

d∑i,j=1

aij(t, x)∂i∂j +

d∑i=1

bi(t, x)∂i =1

2

m∑j=1

X2j (t, x) + X0(t, x) be a time dependent

second order operator, written in usual or Hormander form. We study the regularity of the law of the associ-ated non homogeneous (time dependent) diffusion process, under Hormander’s like conditions. Coefficientsare only Holder continuous in time. The main tool is Malliavin calculus. Our results extend and correctprevious 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 forany multi-index α, ∂αxXj(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

(1.1.a)

dxt =

m∑j=1

Xj (t, xt) dwjt + X0 (t, xt) dt, t > s

xs = x

written in Stratonovich form, where (w1, . . . , wm) is a standard Brownian motion, or equivalentlyof the Ito stochastic differential equation

(1.1.b)

dxt =

m∑j=1

Xj (t, xt) δwjt + X0 (t, xt) dt, t > s

xs = x

where X0 = X0 +1

2

m∑j=1

∂Xj

∂xXj .

We will assume that the solution is non explosive, for instance assuming when necessary that∂αxXj(t, x) are bounded on [s, T ]× Rd. If we denote by xst (x) the solution of (1.1) , then

Tst : Cb 7−→ Cbf 7−→ E (f(xst (x)))

Ecole Polytechnique, CMAP, F-91128 Palaiseau Cedex

Modal’X, Universite de Paris X, UFR SEGMI, 200 avenue de la Republique, 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

Lt =1

2

m∑j=1

X2j (t, •) + X0(t, •)

written in Hormander form. If we denote by µst(x, dy) the law of xst (x), then µst(x, dy) is a solution

of the weak forward equation

(∂

∂t+ Lt

)?µst = 0, µss = δx. We are first interested in the existence

of (smooth) densities for the laws, i.e.

µst(x, dy) = pst(x, y) dy(1.2)

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-homogeneouscase. However, we will extensively use the time-space process (which is homogeneous), namely thesolution of

(1.3)

dyt =

m∑j=1

Xj (ut, yt) dwjt + X0 (ut, yt) ,

dut = dt,(u0, y0) = (u, x)

which is given by a flow ψt(w, u, x) such that x 7→ φt(w, u, x) = Πxψt(w, u, x) is , for each u, aC∞ diffeomorphism of Rd. Here of course Πx denotes the projection operator from R ⊗ Rd ontoRd. Note that the standard proof, lying on Kolmogorov continuity theorem, allows to choose sucha version for each u, but non necessarily for all u simultaneously. In particular if one wants to usethe pathwise composition of flows formula, one has to make an additional regularity assumption onthe Xj . For instance one may assume that ∂αxXj(•, •) are β-Holder continuous, for some β > 0. Inthis case one can build continuous versions of ∂αxφt(w, •, •).Of course the key point is that

xst (x) = φt−s(w, s, x) in Law t > s.(1.4)

Without loss of generality, we may and will assume that s = 0, and write φt(w, x) instead ofφt(w, 0, x). This will not introduce any confusion and will simplify the notations. However somestatements in sections 4 and 5 and just below, are written with a starting time s.

In order to prove (1.2) one can call upon celebrated Hormander’s sum of squares theorem ([18]) i.e.

(1.5) If the Xj ’s are C∞b on [s, T ]×Rd and the Lie algebra generated by Xj , 1 ≤ j ≤ m and∂

∂t+X0

is full at each (t, y) ∈]s, T ] × Rd, then (1.2) holds for all t ∈]s, T ], all x ∈ Rd and y 7→ pst(x, y)is smooth (i.e. C∞).

Actually Hormander’s result is stronger since it deals with parabolic-hypoellipticity. Note that thestatement of (1.5) includes the time component i.e. for each fixed (t, y),

dim span Lie

(∂

∂t+X0, Xj , 1 ≤ j ≤ m,

)(t, y) = d+ 1.(1.6)

Hypoelliptic non homogeneous diffusions 3

Extending Kohn’s proof of Hormander’s result, Chaleyat-Maurel and Michel [14] have shown a weakform of parabolic-hypoellipticity, for coefficients which are only measurable in time. This yields

(1.7) If all ∂αxXj , 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 Hormander’s hypothesis). The role of∂

∂twill

be enlightened by a very simple example in a moment.Notice that when the vector fields are real analytic, the Hormander’s condition is almost necessary(see [16]).

Malliavin stochastic calculus of variations initiated in the 80th’s ([26][25]) , proposed a new approachbased on a variational method for s.d.e. (1.3) which leads to the following principle

(1.8) Let Ct(x) =

m∑j=1

∫ t

0

φ?−1s Xj(x) >< φ?−1

s Xj(x) ds be (part of) the Malliavin covariance matrix.

If C−1t (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 notationsφ?−1s Xj(x) =

(∂φs∂x

(w, x)

)−1

Xj (s, φs(w, x))

Y >< Z = Y tZ,

so that Ct is a nonnegative symmetric matrix. In order to study Ct the standard way is to useIto-Kunita-Bismut formula

(1.9) for all ξ ∈ Rd, j = 1, . . . ,m

〈ξ, φ?−1t Xj〉(x) = 〈ξ,Xj(0, x)〉+

m∑k=1

∫ t

0

〈ξ, φ?−1s [Xk, Xj ]〉(x)δwks

+

∫ t

0

⟨ξ, φ?−1

s

([∂

∂t+X0, Xj

]+

1

2

m∑k=1

[Xk, [Xk, Xj ]]

)⟩(x)ds

where [Y,Z] is the Lie bracket between vector fields Y and Z. It follows from a now well knownprocedure (see e.g. [32] or [5]) that

(1.10) When the Xj ’s, j = 0, . . . ,m are C∞b on [0, T ] × Rd, a sufficient condition for C−1t (x) to be in

all the Lp’s for all t ∈]0, T ] is that

dim span Lie

(Xj , 1 ≤ j ≤ m,

∂

∂t+X0

)(0, x) = d+ 1

(for another approach of this result also see Bally [4])


The main difference with (1.5) is that the assumption on the Lie brackets is only made at thestarting point (0, x). This could be considered as a slight improvement, but Malliavin calculus hasnow proved its strength in many areas where analytic methods are very difficult or impossible touse (see e.g. the recent book of Malliavin himself [27]). Let us also mention that one can recoverexactly Hormander’s result using the Markov property. This is partly explained by the followingelementary 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(w′, s0, φs0−s(w, s, x))))] (where (w,w′) is the generic pair on Ω ⊗Ω)

=

∫pss0(x, y) E (f(φt−s0(w′, s0, y))) dy

=

∫f(z) E

(pss0(x, φ−1

t−s0(w′, s0, z))∣∣Jacobian φ−1

t−s0(w′, s0, z)∣∣) dz

provided all manipulations are allowed. In particular, if pss0 is smooth with a good behaviour atinfinity, pst is also smooth thanks to the regularity of the flow φ−1 and µst(x, dy) = pst(x, y)dy with

pst(x, y) = E(pss0

(x, φ−1

t−s0(w, s0, y)) ∣∣Jacobian φ−1

t−s0(w, s0, y)∣∣) .

This allows to improve (1.7) assuming the restricted Hormander’s hypothesis only near the timeorigin (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 theX ′j 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 [36]. Take d = 2, m = 1 and consider the process

xt(x) =(x1 + wt, x2 + t (x1 + wt)

2)

which law, supported by the curve y2 = x2 + ty21 , is singular. The associated vector fields are

X0(t, x) = x21

∂

∂x2, X1(t, x) =

∂

∂x1+ 2tx1

∂

∂x2

So X1 and [X1, [X1, X0]] = 2∂

∂x2span R2 at each (t, x).

Of course (1.6) is not satisfied because

[∂

∂t+X0, X1

]= 0 at each (t, x).

This fact is not related to the unboundedness of the coefficients. One can easily modify this example.For instance taking

(y2 − x2 = f(t, y1) = exp(− 1

2

(y2

1 + (1 + t)2))

and looking at the starting point y = 0, the situation is unchanged. In particular, the statement ofTheorem 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 component. FromMalliavin calculus point of view, the only difficulty is that (1.9) (which involves

[∂∂t , Xj

]) is no more

valid or cannot be iterated. Nevertheless three cases are almost immediate


(1.13.1) vector fields are measurable in time, and Lt is uniformly elliptic in a neighborhood of (0, x),i.e.m∑j=1

〈ξ,Xj(t, x)〉2 ≥ c |ξ|2 for some c > 0 and all (t, y) ∈ [0, s0]×B(x,R);

(1.13.2) ∂αxXj belong to C1b(R× Rd) for all α and dim span Lie (Xj , 1 ≤ j ≤ m)(0, x) = d;

(1.13.3) Xj(t, x) = fj(t)Wj(x) where 0 < c ≤ |fj | ≤1

cand usual Hormander’s hypothesis holds for

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 [36]. As we already said the tentative of Florchinger [17] to relaxellipticity 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 wrongresults.Another approach was proposed by Chen and Zhou [15]. Since one cannot directly use the stochasticcalculus for Xj(s, φs(w, x)) once s 7→ Xj(s, y) is not C1, one can freeze the time, writing

(1.14)

〈ξ, φ?−1t Xj〉(x) = 〈ξ,Xj(t, x)〉+

m∑k=1

∫ t

0

〈ξ, φ?−1s [Xk(s, •), Xj(t, •)]〉(x) δwks

+

∫ t

0

⟨ξ, φ?−1

s

([X0(s, •), Xj(t, •)] +

1

2

m∑k=1

[Xk(s, •), [Xk(s, •), Xj(t, •)]]

)⟩(x)ds.

The authors use Kusuoka-Stroock method [21] for obtaining some expansion of Ct in terms ofrepeated integrals. They then state (see [15] Thm 1.1)

(1.15) if the ∂αxX′j s are uniformly β-Holder continuous in time, for some β > 1

2 , and dim span Lie(Xj , 1 ≤ j ≤ m) = d, then (1.2) holds.

Condition β > 12 is used in the derivation of lemma 4.3 of [15]. But it is too restrictive in many

applications (for example in filtering) where dependence in time is given by some extra stochasticprocess, like another diffusion process, for which we have to consider β < 1

2 .Actually the proof of (1.15) given in [15] is still somewhat obscure for us because it lies on controlson 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

〈ξ, φ?−1t X1〉(x) = 〈ξ,X1(0, x)〉 = 0

On the other hand if we use (1.14) we have to calculate

[X1(s, •), X1(t, •)] = 2(t− s) ∂

∂x2

[X0(s, •), X1(t, •)] = −2x1∂

∂x2

[X1(s, •), [X1(s, •), X1(t, •)]] = 0(∂φs∂x

(w, x)

)−1

=

(1 0

−2s(ws + x1) 1

)


Thus

〈ξ, φ?−1t X1(t, •)〉(x) = 〈ξ,X1(t, x)〉+

∫ t

0

2 ξ2(t− s)δws −∫ t

0

2 ξ2(x1 + ws)ds

= 2tx1 + 2

∫ t

0

(t− s)δws − 2

∫ t

0

(x1 + ws)ds.

Of course this is equal to 0, as the terminal value at time t of 〈ξ, φ?−1θ X1〉(x) = 2(t−θ)(x1 +wθ).

But this is due to cancellations at time t of terms which do not play the same role: initialcondition, bounded variation term, stochastic integral. In other words repeated integrals withdeterministic (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 thesame idea (freezing the time) in connection with Norris lemma, contains a similar mistake. All ourapologies for those who had in hands the first version of the paper. We warmly thank an anonymousreferee for pointing out this mistake.

In order to bypass this difficulty, one can use a time discretization procedure and use time regularityof 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, butin the spirit of Norris lemma instead of Kusuoka-Stroock. We hope it will help to clarify Chen andZhou paper. Condition β > 1

2 will clearly appear as a limitation due to the use of time regularityof the process. To overcome this limitation other ideas are necessary. These ideas are introducedin section 3. In addition to all that was used before (time discretization, time freezing, stochasticcalculus), the new points are: first replace L2 norms by L∞ norms (thanks to the regularity of thecoefficients), next use the continuity of the argsup of some one dimensional semi-martingales. Thestrategy is quite intricate, so a brief account of it is given at the beginning of section 3. In section 4we collect some useful estimates on the density. We underline that these estimates are certainly notsharp. But they are sufficient to derive a parabolic-hypoelliptic theorem which is an improvementof Chaleyat-Maurel and Michel. In section 5 various applications are quickly discussed: filteringtheory, killed and reflected processes, infinite degeneracy. In all these cases we shall only mentionnew 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 thiswork, and a second referee for the same careful and courageous reading of this version. We also wishto 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-Holder norm of h as

‖h‖β = sup0≤s,t≤T

(t− s)−β |h(t)− h(s)|(2.1)

Throughout this section we will assume that : for all j = 0, ..., m, all multi-index α, there existsβ(α) > 0 such that

supy∈Rd

∥∥∂αyXj(•, y)∥∥β(α)

< +∞(2.2)


In particular sup0≤t≤T

∥∥∂αyXj(t, •)∥∥∞ < ∞, i.e. the Xj ’s belong to C∞b uniformly in time on [0, T ]. In

addition we assume that the restricted Hormander’s hypothesis holds i.e.

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 ∈ Sd−1

N∑k=0

∑Z∈Σk

〈ξ, Z〉2 (0, x) ≥ cN(2.3′)

where Σ0 = Xj , 1 ≤ j ≤ m, Σk+1 = [Xj , Z] , 1 ≤ j ≤ m, Z ∈ Σk. N being given by (2.3′) wecan define some constants

K = sup0≤s≤T

max0≤|α|≤N+2

max0≤j≤m

∥∥∂αyXj(s, •)∥∥∞(2.4.a)

β a common Holder coefficient for all α s.t. |α| ≤ N + 2(2.4.b)

Kβ = supy∈Rd

max0≤|α|≤N+2

max0≤j≤m

∥∥∂αyXj(•, y)∥∥β

(2.4.c)

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 ∈ Sd−1

N∑k=0

∑Z∈Σk

〈ξ, Z〉2 (s, y) ≥ cN2

(2.5)

We shall prove the following version of classical estimates (see [31])

Theorem 2.6 Let (2.2), (2.3) and (2.4) hold, for some β >1

2. 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

supξ∈Sd−1

P

∫ t

0

m∑j=1

⟨ξ, φ?−1

s Xj

⟩2(x) ds ≤ ε

≤ cp εp.Note that for β = 1, this result essentially follows from the estimates of [31, 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 [31], but requires additional work. For k = 0,. . . , N , let m(k) be a positive constant we will choose later. Then for a given ξ ∈ Sd−1 we introducethe sets

Ek =

∑Z∈Σk

∫ t

0

⟨ξ, φ?−1

s Z⟩2

(x) ds ≤ εm(k)

.


Choosing m(0) = 1, what we need is a bound for P (E0). But

P (E0) ≤N−1∑k=0

P(Ek ∩ Eck+1) + P (E) with E =

N⋂k=0

Ek.

Step 1 : Upper bound for P(E).

It can be obtained as in [31, p. 121] which yields P(E) ≤ cpεp, where cp depends only on t, x, p, K,N , cN , R, t0 (defined in (2.5)) and min

k=0,...,Nm(k).

Step 2 : Upper bound for P(Ek ∩ Eck+1).

We will more generally get a bound for

P(B) = P

∫ t

0

⟨ξ, φ?−1

s Z⟩2

(x)ds ≤ εq,m∑j=1

∫ t

0

⟨ξ, φ?−1

s [Xj , Z]⟩2

(x)ds ≥ ε

(2.7)

for a vector field Z satisfying (2.2) with bounds (2.4). The ad-hoc value of q will determine the ratiom(k)/m(k + 1). As usual it is useful to bound the process introducing for θ > 0

τ = infz ≥ 0,

∣∣φ?−1s (ω, x)

∣∣ ≥ ε−θ ∧ t(2.8)

which satisfies P (τ < t) ≤ cpεp where cp only depends on p, t, x, K and θ. So we only have to boundP (B, τ = t). To overcome the difficulty explained in (1.16) we introduce a time discretization

si =it

n, i = 0, . . . , n, ui =

si + si+1

2(2.9)

and define

Bi =

∫ si+1

si

⟨ξ, φ?−1

s Z⟩2


∫ si+1

si

⟨ξ, φ?−1

s [Xj , Z]⟩2

(x)ds ≥ ε

n

(2.10)

so that

P(B, τ = t) ≤n−1∑i=0

P(Bi, τ = t).

Again

P(Bi, τ = t) ≤ P(B1i , τ = t) + P(B2

i , τ = t)

where


B1i = Bi ∩

m∑j=1

∫ ui

si

⟨ξ, φ?−1

s [Xj , Z]⟩2

(x)ds ≥ ε

2n

B2i = Bi ∩

m∑j=1

∫ si+1

ui

⟨ξ, φ?−1

s [Xj , Z]⟩2

(x)ds ≥ ε

2n

(2.11)

We shall get an estimate for P(B1i , τ = t). The same estimate holds for P(B2

i , τ = t) just changingsi into si+1 in the derivation below.

It should be surprising to the reader that we separate the quadratic variation part in two terms.The reason will become apparent when studying Ci.

To get this estimate we use the Ito formula as in (1.14), freezing the time for Z(•, y). Hence fors ∈ [si, si+1], P a.s.

⟨ξ, φ?−1

s Z⟩2

(x) =⟨ξ, φ?−1

si Z(s, •)⟩2

(x) +

m∑j=1

∫ s

si

⟨ξ, φ?−1

u [Xj(u, •), Z(s, •)]⟩2

(x)du(2.12)

+Ais +

m∑j=1

M ijs

with

M ijs =2

∫ s

si

⟨ξ, φ?−1

u Z(s, •)⟩

(x)⟨ξ, φ?−1

u [Xj(u, •), Z(s, •)]⟩

(x)δwju

Ais=2

∫ s

si

⟨ξ, φ?−1

u Z(s, •)⟩

(x)⟨ξ, φ?−1

u [X0(u, •), Z(s, •)]⟩

(x)du

+

∫ s

si

⟨ξ, φ?−1

u Z(s, •)⟩

(x)

m∑j=1

⟨ξ, φ?−1

u [Xj(u, •), [Xj(u, •), Z(s, •)]]⟩

(x)du

Now using Kolmogorov continuity criterion, it is not hard to see that both hand sides in (2.12) havea continuous version, so that (2.12) holds for P a.s. for all s ∈ [si, si+1]. So we can integrate (2.12)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

∫ si+1

si

⟨ξ, φ?−1

s Z⟩2

(x) ds ≥ Ci +Ai +

m∑j=1

Mij(2.13)

with


Ai=

∫ si+1

si

Ais ds

Ci=

∫ si+1

si

∫ si+1

s

m∑j=1

⟨ξ, φ?−1

s [Xj(s, •), Z(u, •)]⟩2

(x) du ds

Mij=

∫ si+1

si

αijs δws

αijs =

∫ si+1

s

⟨ξ, φ?−1

s Z(u, •)⟩

(x)⟨ξ, φ?−1

s [Xj(s, •), Z(u, •)]⟩

(x) du

But on τ = t, for u and s less than t, the following holds

∣∣⟨ξ, φ?−1s Z

⟩(x)−

⟨ξ, φ?−1

s Z(u, •)⟩

(x)∣∣ ≤ ε−θKβ |s− u|β(2.14)

and

∣∣⟨ξ, φ?−1s [Xj , Z]

⟩(x)−

⟨ξ, φ?−1

s [Xj(s, •), Z(u, •)]⟩

(x)∣∣ ≤ 2ε−θKKβ |s− u|β(2.14′)

Using (2.14) and (2.14′), 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

According to (2.14′) and (a+ b)2 ≥ a2 − 2|ab|, one has

Ci ≥∫ si+1

si

(si+1 − s)m∑j=1

⟨ξ, φ?−1

s [Xj , Z]⟩2

(x) ds

−4ε−θKKβ

∫ si+1

si

(si+1 − s)1+β

1 + β

m∑j=1

∣∣⟨ξ, φ?−1s [Xj , Z]

⟩(x)∣∣ ds.

In order to get a lower bound for the first term, we have to bound si+1 − si from below in theintegral. This explains why we have to divide the time interval in two subintervals.Hence, on B1

i ∩ τ = t one has,

Ci ≥ (si+1 − ui)ε

2n− 8mε−2θK3Kβ

(si+1 − si)2+β

(1 + β)(2 + β)

≥ ε t

4n2− 4mK3Kβε

−2θ t2+β

n2+β

We will choose n = [ε−µ] (integer part of ε−µ). Hence, if βµ − 2θ > 1, one can find ε0 which onlydepends on m, K, Kβ , µ, θ and t such that for ε < ε0,


Ci ≥1

8ε1+2µ on B1

i ∩ τ = t .(2.15)

(βµ− 2θ > 1 ensures that −2θ+ (2 + β)µ > 1 + 2µ, so that the principal part of Ci isε

n2∼ ε1+2µ)

Upper bound for |Ai|

On τ = t,

∣∣Ais∣∣ ≤ c ε−θ∫ s

si

∣∣⟨ξ, φ?−1u Z(s, •)

⟩(x)∣∣ du

≤ c ε−θ∫ s

si

∣∣⟨ξ, φ?−1u Z

⟩(x)∣∣ du+ cε−2θKβ

(s− si)1+β

1 + β

where c only depends on K and m according to (2.14). It follows that on τ = t

|Ai| ≤ cε−θ∫ si+1

si

(si+1 − s)|⟨ξ, φ?−1

s Z⟩

(x)|ds+ cε−2θKβ(si+1 − si)2+β

(1 + β)(2 + β)

and on B1i ∩ τ = t, by Cauchy-Schwarz inequality,

|Ai| ≤ cε−θεq2t32

n32

+ cε−2θKβt2+β

n2+β.

Thus if βµ− 2θ > 1 and −θ +q

2+

3

2µ > 1 + 2µ, on can find ε′0 which only depends on m, K, Kβ ,

t, β, µ, θ and q such that on B1i ∩ τ = t

|Ai| ≤1

32ε1+2µ for ε < ε′0.(2.16)

Upper bound for |Mi,j |

As usual, we shall use the classical martingale inequality

P(

∫ b

a

α2s ds ≤ η2k,

∣∣∣∣∣∫ b

a

αs δws

∣∣∣∣∣ ≥ ηl) ≤ 2 exp

(−1

2η2(l−k)

).(2.17)

As before on τ = t,∣∣αijs ∣∣ ≤ cε−θ

∫ si+1

s

∣∣⟨ξ, φ?−1s Z(u, •)

⟩(x)∣∣ du where c only depends on K.

According to (2.14), on τ = t,

∣∣αijs ∣∣2 ≤ 2c2ε−2θ(si+1 − s)2⟨ξ, φ?−1

s Z⟩2

(x) + 2c2ε−4θK2β(si+1 − s)2+2β .

Thus on τ = t ∩B1i ,


∫ si+1

si

∣∣αijs ∣∣2 ds ≤ 2c2ε−2θεqt2

n2+ 2c2ε−4θK2

β

t2+2β

n3+2β.

Hence, if q + 2µ− 2θ > 2 + 4µ+ 2r and −4θ+ (3 + 2β)µ > 2 + 4µ+ 2r, one can find ε′′0 which onlydepends on K, Kβ , t, β, µ, θ and r such that for ε ≤ ε′′0 , on τ = t ∩B1

i ,

∫ si+1

si

∣∣αijs ∣∣2 ds ≤ ε2+4µ+2r

1024m2.

Applying (2.17), we get

P(τ = t, B1i , |Mij | ≥

1

32mε1+2µ) ≤ 2 exp

(−1

2ε−2r

).(2.18)

To conclude, assume that all conditions (2.15), (2.16) and (2.18) are satisfied, and recall (2.13).Then for ε small enough (ε ≤ min(ε0, ε

′0, ε′′0)) the set

τ = t ∩B1i ∩

m⋂j=1

|Mij | <

1

32mε1+2µ

will be empty provided q > 1 + 2µ. Hence, in this case,

P(τ = t, B1i ) ≤ 2m exp

(−1

2ε−2r

)and if r > 0,

P(B, τ = t) ≤ 4mε−µ exp

(−1

2ε−2r

)≤ cpεp

for ε small enough (universal). Thanks to (2.8), we finally get P(B) ≤ cpεp hence a bound for

P(Eck+1 ∩ Ek) (recall that q =m(k)

m(k + 1)). This achieves step 2 and the proof of Theorem 2.6,

provided we can find θ, q, µ, r which satisfies all conditions cited above i.e.

i) r > 0, θ > 0, µ > 0, q > 0.ii) βµ− 2θ > 1.

iii) −4θ + (3 + 2β)µ > 2 + 4µ+ 2r.iv) q > 2µ+ 1, q + 2µ− 2θ > 2 + 4µ+ 2r, −2θ + q + 3µ > 2 + 4µ.

Fix r > 0, θ > 0. Then choose µ >1 + 2θ

βand q > max(2µ+ 1, 2µ+ 2θ+ 2r+ 2, µ+ 2 + 2θ) so that

i), ii) and iv) are satisfied. The delicate point is iii) which yields

(2β − 1)µ > 2 + 4θ + 2r(2.19)

and can be satisfied only if 2β − 1 > 0. 2


Remark 2.20 As the preceding proof clearly shows, condition β > 12 appears in the control of the

martingale term. More precisely, Ito formula leads to bound⟨ξ, φ?−1

u Z(s, •)⟩

(x), while the initial

control is on⟨ξ, φ?−1

s Z(s, •)⟩

(x). The error (s − u)β is not important for the bounded variationterm (it only imposes βµ − 2θ > 1), and becomes important for the martingale term (imposingcondition iii) above). Since the martingale term appears to be the leading one, one should try to geta better control on it. In a sense, it is what we are going to do in the next section, but the strategywill require more refined tools.

Remark 2.21 One can relax the (uniform) boundedness conditions on the coefficients and theirmodulus of continuity; i.e. K and Kβ may have polynomial growth in R, where R is the radius ofthe ball in Rd. Indeed, this will only change the constants in front of θ, and will not change thefinal β > 1

2 . But in order to show regularity of the density in section 4 we shall use a localizationprocedure, which enables to only look at the uniformly bounded case.

One can also see that ε0 has some polynomial growth in 1t when t goes to 0. This will be used in

section 4.

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 =x+ wt

yt = y +

∫ t

0

h(u,wu) δBu

where (w,B) is a Brownian motion. Suppose that h(0, 0) = 0 but∂h

∂x(0, 0) 6= 0. Then X1(0, 0)

and [X1, X2](0, 0) span R2 (X1 =(

10

), X2 =

(0

h(0,x)

), X0 =

(00

)), so that Hormander’s condition is

satisfied. Take ξ = (0, 1). It is easy to check

⟨ξ, φ?−1

t X1

⟩(0, 0) = −

∫ t

0

∂h

∂x(u,wu) δBu,

⟨ξ, φ?−1

t X2

⟩(0, 0) = h(t, wt).

Hence it is enough to control P(∫ t

0

h2(s, ws)ds ≤ ε)

at least for small t. If h satisfies the β-Holder

condition (2.2), we may assume

∣∣∣∣∂h∂x (λ, 0)

∣∣∣∣ ≥ 1

2

∣∣∣∣∂h∂x (0, 0)

∣∣∣∣ > 0 for λ small enough. We may first

freeze the time component in the vector field and thus apply Norris Lemma [31, lemma 2.3.2] to theprocess

h(λ,ws) = h(λ, 0) +

∫ s

0

∂h

∂x(λ,wu) δwu +

1

2

∫ s

0

∂2h

∂x2(λ,wu) du

and get that


P(∫ t

0

h2(λ,ws) ds ≤ ε)≤ cpεp

where cp does not depend on λ (step 3 in the general case where we shall also need a time dis-cretization). We claim that a regularity argument, or a modification of Norris Lemma, shows thatactually (step 1 in the general case where we shall replace the L2 norm by the L∞ norm since theBrownian paths are Holder continuous)

for all λ ∈ [0, t], P(

sup0≤s≤t

h2(λ,ws) ≤ ε)≤ cpεp.

Here again cp does not depend on λ.The second claim is the following : thanks to the regularity of h in λ, we can show

P(∀λ ∈ [0, t], sup

0≤s≤th2(λ,ws) ≥ ε

)≥ 1− cp(β)εp.

Now observe that because of

∣∣∣∣∂h∂x (λ, x)

∣∣∣∣ ≥ α > 0 for λ ∈ [0, t], 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 processh(λ,w•) is up to a Girsanov transform and a time change, just a Brownian motion. For Brownianmotion the uniqueness of the supremum is well known [20, Remark 8.16, p.102]. Hence , we shallassume that for almost all w, for all λ, the supremum is attained at some single time sλ, i.e.sλ = argsup

0≤s≤th2(λ,ws). Of course sλ is random, but for a fixed w, λ 7→ sλ(w) will then be continuous

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

0≤s≤th2(s, ws) ≥ ε. Using Holder continuity of s 7→ h(s, ws),

this yields (step 4 in the general case)

P(∫ t

0

h2(s, ws) ds ≤ ε1+ 1β

)≤ c′p(β) εp

for some β 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 hasto use continuity to modify this assumption, without modifying the conclusion.

¿From now on, we assume that hypotheses (2.2), (2.3)and (2.4) are satisfied and we use the notationsof 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)

P(B) = P

∫ t

0

⟨ξ, φ?−1

s Z⟩2


∫ t

0

⟨ξ, φ?−1

s [Xj , Z]⟩2

(x)ds ≥ ε

≤ cpεpwhere cp only depends on β, K, Kβ, p, x, m and t.


This statement is exactly (2.7) in the previous section, but we do no more assume β >1

2from now.

In the proof, we shall use similar arguments as those exposed in the example above, and first toget rid of the assumption on β, we shall deeply use β-Holder continuity. To this end, we state astraightforward 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

β′ < β ∧ 1

2, there exists a version of

⟨ξ, φ?−1

s Y⟩

(x) =

⟨ξ,

(∂φs∂x

(w, x)

)−1

Y (φs(w, x))

⟩

which is β′-Holder continuous on [0, T ]. Furthermore if we set

Kβ′(w, Y ) =∥∥⟨ξ, φ?−1

• Y⟩

(x)∥∥β′

then

E(Kpβ′(w, Y )

)≤ cp < +∞

where cp only depends on p, x, β′, K and Kβ.

The proof is left to the reader. Since the proof of Proposition 3.2 is quite long, we split it into severalsteps.

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 oncefor all β′ given by Lemma 3.3. As a first consequence of β-Holder continuity, we get

Lemma 3.4 Let Y be as in Lemma 3.3 and ε <1

4∧ t. Then if a ≥ 2

(1 +

1

β′

)+ 1, for all

p ∈ [1,+∞[,

P(∫ t

0

⟨ξ, φ?−1

s Y⟩2

(x)ds < εa, sup0≤s≤t

∣∣⟨ξ, φ?−1s Y

⟩(x)∣∣ ≥ ε) ≤ cpεp

where cp only depends on t, p, x, β′, K and Kβ (we shall call it a “good” constant).

Proof: Let zs =⟨ξ, φ?−1

s Y⟩

(x). According to 3.3, |zs(w)− zu(w)| ≤ Kβ′(w, Y )|s− u|β′ and

P(Kβ′(w, Y ) ≥ 1

2ε

)≤ cpεp

where cp is a “good” constant. So we may assume that w ∈Kβ′(w, Y ) <

1

2ε

from now on.

For such an w and |s? − s| ≤ εη, one has |zs(w)| ≥ |zs?(w)| − 1

2εβ′η−1. Moreover assume that

sup0≤s≤t

|zs(w)| ≥ ε. Then one has for ε < t and η > 1


∫ t

0

z2s(w) ds ≥

∫interval of size εη

(|zs?(w)| − 1

2εβ′η−1

)2

ds

where |zs?(w)| = sup0≤s≤t

|zs(w)|. So choosing η =2

β′, we get

∫ t

0

z2s(w)ds ≥ 1

4ε

2+ 2β′ . Since a ≥

2

(1 +

1

β′

)+ 1 and ε ≤ 1

4, it holds εa <

1

4ε

2+ 2β′ and accordingly the set

∫ t

0

⟨ξ, φ?−1

s Y⟩2

(x)ds < εa, sup0≤s≤t

∣∣⟨ξ, φ?−1s Y

⟩(x)∣∣ ≥ ε ∩ Kβ′(w, Y ) <

1

2ε

is empty. 2

Thanks to Lemma 3.4, if we put q = ra with r > 0 and a = 2

(1 +

1

β′

)+1, the proof of Proposition

3.2 reduces to show that for some r,

P (C) = P

sup0≤s≤t

∣∣⟨ξ, φ?−1s Z

⟩(x)∣∣ ≤ εr, m∑

j=1

∫ t

0

⟨ξ, φ?−1

s [Xj , Z]⟩2

(x)ds ≥ ε

≤ cpεp(3.5)

for some “good” constant cp.

Step 2 : Time discretization.

We introduce a dicretization of the time interval [0, t]

si =it

nwith n =

[ε− 4β′].(3.6)

Introduce the sets

Ci =

∫ t

0

m∑j=1

⟨ξ, φ?−1

s [Xj , Z]⟩2

(x)ds ≥ ε and ∃s ∈ [si, si+1],

m∑j=1

⟨ξ, φ?−1

s [Xj , Z]⟩2

(x) < ε32

We then show

Lemma 3.7 If ε12 <

1

2tthen

P

(n−1⋂i=1

Ci

)≤ cpεp

for some “good” cp.


Proof: If

∫ t

0

is greater than ε, then sup0≤s≤t

is greater thanε

t. So as before, up to a set of probability

less than cpεp, we may assume that Kβ′(w, Y ) ≤ ε− 1

2 for all Y of the form [Xj , Z]. Thus there exists

at least one interval I of length ε2β′ such that for s ∈ I,

m∑j=1

⟨ξ, φ?−1

s [Xj , Z]⟩2

(x) ≥ ε

t− ε2ε−

12 ≥ ε

2t> ε

32 .

Finally to conclude, it suffices that I contains at least one interval [si, si+1] which ensures that the

set

n−1⋂i=0

Ci ∩Kβ′(w, Y ) ≤ ε− 1

2

is empty. 2

Introduce the family of sets

Di=

supsi≤s≤si+1

∣∣⟨ξ, φ?−1s Z

⟩(x)∣∣ ≤ εr, m∑

j=1

⟨ξ, φ?−1

s [Xj , Z]⟩2

(x) ≥ ε for all s ∈ [si, si+1]

and the set D =

n−1⋃i=0

Di. As in section 2, we can introduce the stopping time τ defined by (2.8)

and remark that P (τ < t) ≤ cpεp. Hence we are reduced to estimate P (C ∩ τ = t). According to

Lemma 3.7 (just changing ε into ε32 and replacing r by

3

2r), one has P (C ∩Dc) ≤ cpεp. So we only

have to estimate P (C ∩D ∩ τ = t). Note now that

P (C ∩D ∩ τ = t) ≤ P (D ∩ τ = t) ≤n−1∑i=0

P (Di ∩ τ = t) ≤[ε− 4β′]

maxi

P (Di ∩ τ = t) .

Since p is arbitrary, we are reduced to show that

P (Di ∩ τ = t) ≤ cpεp(3.8)

for some “good” constant independent of i.

Step 3 : Freezing the time.On τ = t, one has

sup(s,λ)∈[si,si+1]2

∣∣⟨ξ, φ?−1s [Xj , Z]

⟩(x)−

⟨ξ, φ?−1

s [Xj(s, •), Z(λ, •)]⟩

(x)∣∣ ≤ 2KKβε

−θε4ββ′(3.9)

Set

Ei =

inf(s,λ)∈[si,si+1]2

m∑j=1

⟨ξ, φ?−1

s [Xj(s, •), Z(λ, •)]⟩2

(x) ≥ ε

2

.


Thanks to (3.9), one can find ε0 which only depends on m, K, Kβ such that, provided θ <1

2in

order to ensure that −θ + 4ββ′ >

32 , for all ε < ε0

Di ∩ τ = t ⊂ Di ∩ τ = t ∩ Ei = Fi ∩ τ = t

where

Fi =

supsi≤s≤si+1

∣∣⟨ξ, φ?−1s Z

⟩(x)∣∣ ≤ εr, inf

(s,λ)∈[si,si+1]2

m∑j=1

⟨ξ, φ?−1

s [Xj(s, •), Z(λ, •)]⟩2

(x) ≥ ε

2

.

So we are reduced to show

P (Fi, τ = t) ≤ cpεp.(3.10)

Without loss of generality, we shall only deal with the case i = 0. This will avoid notationalintricacies, since the only important point in our future arguments will be the length si+1 − si.Wenow use Ito formula which yields

⟨ξ, φ?−1

s Z(λ, •)⟩

(x) =: ys(λ) = y0(λ) +

∫ s

0

βu(λ)du+

m∑j=1

∫ s

0

αju(λ)δwju(3.11)

with

αju(λ) =⟨ξ, φ?−1

u [Xj(u, •), Z(λ, •)]⟩

(x)

βu(λ) =

⟨ξ, φ?−1

u

[X0(u, •), Z(λ, •)] +1

2

m∑j=1

[Xj(u, •), [Xj(u, •), Z(λ, •)]]

⟩

(x).

Note that there exists a constant C depending only on m, K and |ξ| such that, on τ = t,

|αju(λ)|+ |βu(λ)| ≤ Cε−θ

for all j, u and λ.As we explained in example 3.1, we want to study the supremum of ys(λ), but be sure that thissupremum is attained at a single point. So we will have to restrict ourselves to a random timeinterval rather than working on [0, s1] and then evaluate the supremum. The next step will consistin showing the existence of the argsup and is deeply related to the existence of the argsup forBrownian motion.

Step 4 : Existence of the argsup.Since we will use a Girsanov transform, we have to work on the full Probability space Ω and notonly on F0. To this end, we have first to modify y•(λ). Taking if necessary an extension of Ω, wemay find another Brownian motion wm+1 which is independent of (w1, ..., wm) (when we work on[si, si+1], we have to consider wjs − wjsi rather than wjs and choose wm+1 which is independent ofysi(λ) too). Next we define


ys(λ) = y0(λ) +

∫ s

0

βu(λ)du+

m+1∑j=1

∫ s

0

αju(λ)δwju(3.12)

where β, αj are defined as follows. We choose two smooth functions Mθ,ε and χε defined on R suchthat

Mθ,ε(z) = z if |z| < Cε−θ, ±2Cε−θ if |z| > 3Cε−θ and |Mθ,ε| is bounded by 2Cε−θ

so that we may also assume that its derivative is bounded by 1, 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 ±2Cε−θ and the indicator1I|z|≥ ε2 . We then set

βu(λ) = Mθ,ε (βu(λ)) (roughly βu(λ) ∧ Cε−θ ∨ −Cε−θ)

αju(λ) = Mθ,ε

(αju(λ)

)χε

(m∑k=1

(αku(λ)

)2)for j = 1, ...,m

αm+1u (λ) =

√ε

2

(1− χε

(m∑k=1

(αku(λ)

)2))(3.13)

Notice that

if ω ∈ τ = t , for s ≤ s1, ys(λ, ω) = ys(λ, ω)(3.14)

i.e. y is actually a modification of y. The second and main fact is

m+1∑j=1

(αju(λ)

)2= α2

u(λ) ≥ ε

8for all u.(3.15)

Indeed if

m∑j=1

(αju(λ)

)2is greater than

ε

2or less that

ε

4then α2

u(λ) ≥ ε

2. Otherwise

α2u(λ) ≥

m∑j=1

(αju(λ)

)2 (χ2ε + (1− χε)2

)≥ ε

4× 1

2.

Thanks to (3.15), we can introduce a new Probability measure P given by the Girsanov density

dPdP|Ft = exp

−m+1∑j=1

∫ t

0

αju(λ)βu(λ)

α2u(λ)

δwju −1

2

∫ t

0

β2u(λ)

α2u(λ)

du


which is equivalent to P on each Ft. Furthermore, P-almost surely,

ys(λ) = y0(λ) +

m+1∑j=1

∫ s

0

αju(λ)δwju

where (w1, ..., wm+1) is a P-Brownian motion. Now if we define

As(λ) =

∫ s

0

α2u(λ)du and Vs(λ) = inf u ≥ 0, Au(λ) ≥ s

there exists a P-Brownian motion B such that for all s

ys(λ) = y0(λ) +BAs(λ) i.e. yVs(λ)(λ) = y0(λ) +Bs P-a.s.

But as we recall in example 3.1, it is well known that the supremum of the absolute value of a linear

Brownian motion on a deterministic time interval is a.s. attained at a single time. Take T1 =ε

8s1.

Since

ε

8s ≤ As(λ) ≤ 4C2ε−2θ(m+ 1)s(3.16)

then

C2ε2θ

4(m+ 1)s ≤ Vs(λ) ≤ 8

εs.

Hence

C2t

32(m+ 1)ε

1+2θ+ 4β′ ≤ VT1(λ) ≤ tε

4β′ = s1.(3.17)

¿From now on, to simplify the notation, we set S1(λ) = VT1(λ). We get from what precedes

∀λ ∈ [0, s1], P-a.s.,there exists an unique (random) time u1(ω, λ) = argsup0≤u≤S1(λ)

|yu(λ)| .(3.18)

Since P and P are equivalent on Fs1 and S1(λ) ≤ s1, (3.18) holds P-a.s. and we can thus define

u1(ω, λ) = argsup0≤u≤S1(λ)

|yu(λ)| P-a.s. for all λ ∈ [0, s1].(3.19)

Since we only have the existence of the argsup for the modified process y(λ), we need to control thesupremum of this process.

Step 5 : Estimates for the supremum.


Define t1 =C2t

32(m+ 1)ε

1+2θ+ 4β′ . Thanks to (3.17), sup

0≤u≤t1|yu(λ)| ≤ sup

0≤u≤S1(λ)

|yu(λ)|. We will esti-

mate sup0≤u≤t1

|yu(λ)| similarly as in the classical Norris lemma. Ito’s formula yields

y2t1(λ) = y2

0(λ) + 2

∫ t1

0

yu(λ)βu(λ)du+ 2

m+1∑j=1

∫ t1

0

yu(λ)αju(λ)δwju +

∫ t1

0

α2u(λ)du.(3.20)

We then show

Lemma 3.21 If r′ >3

2+ 3θ +

2

β′, there exists ε0(m, t) s.t. if ε < ε0

P(

sup0≤u≤t1

|yu(λ)| ≤ εr′)≤ ce−c

′ε−2θ

where c and c′ depend on m and t.

Proof: on

sup

0≤u≤t1|yu(λ)| ≤ εr

′

, one has

∣∣∣∣2 ∫ t1

0

yu(λ)βu(λ)du

∣∣∣∣ ≤ 4ε−θεr′t1 =

C2t

8(m+ 1)ε

1+ 4β′+r

′+θ(i)

∫ t1

0

(yu(λ)αju(λ)

)2du ≤ 4ε−2θεr

′t1 =

C2t

8(m+ 1)ε

1+2r′+ 4β′(ii)

so that

P

sup0≤u≤t1

|yu(λ)| ≤ εr′,

∣∣∣∣∣∣2m+1∑j=1

∫ t1

0

yu(λ)αjuδwju

∣∣∣∣∣∣ ≥ εr′+ 2β′+

12−θ

is less than ce−c

′ε−2θ

thanks to the martingale inequality we recall in (2.17).

∫ t1

0

α2u(λ)du ≥ ε

8t1 =

C2t

256(m+ 1)ε

2+2θ+ 4β′(iii)

so if we choose r′ such that

2r′ > 2 + 2θ +4

β′, 1 +

4

β′+ r′ + θ > 2 + 2θ +

4

β′,

1

2+ r′ +

2

β′− θ > 2 + 2θ +

4

β′

i.e.

r′ >3

2+ 3θ +

2

β′,

then there is an ε0 depending on m and t s.t. for all ε < ε0, sup0≤u≤t1

|yu(λ)| ≤ εr′

implies that the

martingale term in (3.20) is bigger than εr′+ 2

β′+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 r′ be as in lemma 3.21. Then there exists ε′0 s.t. for ε < ε′0

P(∃λ ∈ [0, s1], sup

0≤u≤t1|yu(λ)| ≤ εr

′)≤ cpεp

for some “good” cp.

Proof: First of all remark that we can again apply Kolmogorov continuity criterion in order to builda β′-Holder continuous version of (s, λ) 7→ ys(λ) thanks to the mollifying (smooth) functions Mθ,ε

and χε. Denote by Γβ′,ε(ω) the β′-Holder norm ‖y•(•)‖β′ . The only point is that moments of Γβ′,ε(ω)depend on ε. Indeed if ε < 1 then the first derivative of Mθ,ε is bounded but the first derivative of

χε is bounded by4

ε. Standard applications of BDG inequalities and easy calculations then yield

E(Γ pβ′,ε(ω)

)≤ cpε−p(3.23)

for some “good” cp independent of ε. Divide [0, s1] into K =

[s1ε−2 r

′β′

]+ 1 intervals of length at

most ε2r′β′ . Denote by (λi)i=0,...,K the extremities of these intervals. Then

P(∃λ ∈ [0, s1], sup

0≤u≤t1|yu(λ)| ≤ εr

′)≤K−1∑i=0

P(∃λ ∈ [λi, λi+1], sup

0≤u≤t1|yu(λ)| ≤ εr

′).

But if |λ− λ′| ≤ ε2 r′β′ then sup

0≤u≤t1|yu(λ)− yu(λ′)| ≤ Γβ′,εε

2r′ will be less than εr′

on the setΓβ′,ε ≤ ε−r

′

. In particular

∃λ ∈ [λi, λi+1], sup

0≤u≤t1|yu(λ)| ≤ εr

′, Γβ′,ε ≤ ε−r

′⊂

sup0≤u≤t1

|yu(λi)| ≤ 2εr′

= Gi

using

|yu(λi)| ≤ |yu(λ)− yu(λi)|+ |yu(λi)| ≤ Γβ′,εε2r′ + εr′.

Moreover one has

∃λ ∈ [λi, λi+1], sup

0≤u≤t1|yu(λ)| ≤ εr

′, Γβ′,ε > ε−r

′⊂Γβ′,ε > ε−r

′

= H.

But P(Gi) ≤ c exp(−c′ε−2θ

)by lemma 3.21, and P(H) ≤ cγε

(r′−1)γ thanks to (3.23) and Markovinequality. Accordingly

P(∃λ ∈ [0, s1], sup

0≤u≤t1|yu(λ)| ≤ εr

′)≤ s1ε

−2 r′β′ ce−cε

−2θ

+ cγs1ε−2 r

′β′ ε(r′−1)γ .(3.24)

Recall that s1 = ε4β′ and r′ >

3

2+

2

β′. Hence choosing

γ − r′

β′= p, the right hand side of (3.24) will

be less than cpεp for ε < ε′0 universal. 2


Step 6 : Conclusion.

Let us return to y•(λ). We may of course choose a continuous version of (s, λ) 7→ ys(λ), andaccordingly 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, ys(λ, ω) = ys(λ, ω).

Lemma 3.22 (recall that sup0≤u≤t1

|yu(λ)| ≤ sup0≤u≤S1(λ)

|yu(λ)|) yields

P(F0) ≤ P(F0) + cpεp

where F0 = F0 ∩

∀λ ∈ [0, s1], sup

0≤u≤S1(λ)

|yu(λ)| > εr′

.

We also know that for each fixed λ, the supremum is a.s. attained at a single point. But hereagain we have to be careful, because the negligible set depends on λ. So first define u1(ω, λ) =argsup0≤u≤S1(λ) yu(λ) for all λ ∈ Q ∩ [0, s1] which is well defined P-a.s. since Q is countable. Weclaim 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 us prove it. If λn goes to λ, S1(λn)goes to S1(λ) and u1(ω, λn) is bounded. If u(ω, λ) is a cluster point, we have (using index n, insteadof 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) ≤ yu(ω,λ)(λ) , and by letting η go to0, we get ys(λ) ≤ yu(ω,λ)(λ), ∀s ≤ S1(λ) which implies u(ω, λ) = u1(ω, λ) by uniqueness.

So we can extend (λ 7→ u1(ω, λ), λ ∈ Q) into a continuous function u1(ω, λ), λ ∈ [0, s1]. Since u1 iscontinuous from [0, s1] into itself, it admits at least one fixed point s = u1(ω, s) ∈ [0, s1].

For P-almost all ω ∈ F0, yu1(ω,λ)(λ) > εr′

for all λ ∈ Q. By continuity yu1(ω,λ)(λ) ≥ εr′

for all

λ ∈ [0, s1], hence yu1(ω,s)(s) = ys(s) ≥ εr′. But ys(s) =

⟨ξ, φ?−1

s Z(s, •)⟩

(x) =⟨ξ, φ?−1

s Z⟩

(x). If

r′ < r, it follows that F0 is empty.

Hence P(F0) ≤ cpεp, as well as all P(Fi) ≤ cpε

p, i.e. (3.10) is satisfied and the proof of proposition3.2 is complete.

As a consequence we get exactly as in section 2

Theorem 3.25 Let (2.2), (2.3) and (2.4) holds for some β > 0. Then for all p ∈ [1,+∞[, thereexists cp and ε0 which only depends on p, t, x, N , cN , K, β and Kβ such that for all ε ≤ ε0

supξ∈Sd−1

P

∫ t

0

m∑j=1

⟨ξ, φ?−1

s Xj

⟩2(x)ds ≤ ε

≤ cpεp.Remark 3.26 The statements of Remark 2.21 are still available, with a different polynomial growth.


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 Palmost surely invertible ad that C−1

t (x) belongs to all the Lp. Furthermore one can estimate thesemoments, 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

E[(C−1t (x)

)p] ≤ cp t−Kp(Nlp+1).

The proof is not very different as in the homogeneous case (see [32]). Indeed, although one can’tuse the scaling properties since coefficients are time dependent, we can nevertheless use remark 3.26

on the growth in time of ε0. Notice that in the proof of Theorem 3.25, the size of q =m(k)

m(k + 1)is

rather big (it contains some part in ( 1β )2) when β is small. The constants cp, Kp, lp will then really

be big, and certainly not sharp. Also notice that the methods developed in both previous sectionscannot be extended to the case of (uniformly) continuous flows of vector fields, rather than β-Holderflows. Indeed for each time discretization we need a polynomial control on the cardinality of the

discretization (i.e. ε− 1β′ or ε

− 4β′ 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 times τ satisfying1

τ∈

⋂1≤p<∞

Lp, Cτ (x) is P almost surely invertible and C−1τ (x) ∈

⋂1≤p<∞

Lp.

This is an easy consequence of the estimates 4.1 (see e.g. [7] or [9] for an extensive use). In particularif (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 followingTheorem

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 ∂αxXj , 0 ≤ j ≤ m, are boundedon [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, thereexists β(α) > 0 s.t. ∂αxXj(•, y) is β(α)-Holder continuous for all y with |y − x| ≤ R and

sup|y−x|≤R

‖∂αxXj(•, y)‖β(α) < +∞ on the time interval [s, s0];

(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) ∈ C∞b (Rd).Furthermore there exists a constant K which only depends on the assumptions i)− iii) such that forall multiindex α∥∥∂αy pst(x, •)∥∥∞ ≤ c(|α|)(1 ∧ (t− s))−K(|α|+ d

2 ) exp (c(|α|, x)(t− s)).(4.4)

where c(•) stands for constants depending only on their arguments.

Actually one can expect more, i.e. that pst(x, •) belongs to the Schwartz space of rapidly decreasingfunctions. Recall that Malliavin integration by parts formula yields the existence of a random variableθt(α) with moments of any order such that

E [∂αf(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 Γin the right hand side. As in [11] this furnishes the more precise estimate

|∂αy pst(x, y)| ≤ c1(1 ∧ (t− s))−K(|α|+ d2 ) |x− y|− 1

2 exp c2(t− s) exp − c3( |x− y|2

1 ∧ (t− s)

)(4.5)

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 > 12 , 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 canuse 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 localizeregularity in an open domain (see e.g. [11] section 1). In the latter case however, Hormander’shypothesis has to be satisfied for all u ∈ [s, t], since it requires the use of stopping times. Up to thisrestriction, 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 [22], that is the regularityof solutions of

∂u

∂t+ Lt u+ cu = f ; u(T, x) = g(x) ; t ∈ [S, T ].(4.6)

A natural candidate for solving (4.6) is

u(t, x) = E

[g(xtT ) exp

∫ T

t

c(s, xts)ds

]− E

[∫ T

t

f(s, xts) exp

(∫ s

t

c(v, xtv)dv

)ds

](4.7)

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

easily follows that∂u

∂tis a bounded function. When everything is continuous in time, so is u and

then∂u

∂t. Conversely by applying Ito formula, any classical solution of (4.6) is given by (4.7).

Definition 4.8 Let U be an open subset of Rd. The operator∂

∂t+ Lt + c is said to be parabolic-

hypoelliptic in ]S, T [×U if for any u ∈ D′ (]S, T [× U) ,

(∂

∂t+ Lt + c)u ∈ L∞((]S′, T ′[ , C∞(V ))

implies u ∈ L∞((]S′, T ′[ , C∞(V )) for any ]S′, T ′[⊆]S, T [ and any open subset V of U .

Recall that in the homogeneous case, Hormander’s theorem implies hypoellipticity of ∂∂t + Lt + c.

This result was derived, using Malliavin calculus, by Kusuoka and Stroock ([21], Thm 8.6 and Thm8.13). Another, slightly less complete, but simpler approach is proposed in [11] section 2. In the nonhomogeneous case, Thm 1.1 of Chaleyat-Maurel and Michel is some kind of parabolic-hypoellipticityin ]S, T [×Rd (actually it is a global statement, but it can be rephrased in local terms).Namely they show that, if Hormander’s hypothesis is everywhere satisfied (with some uniformbounds on [S, T ] × Γ, Γ compact), then any solution of (4.6) which can be written dµt × dt for alocally 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 ∈ C∞0 (]S, T [×U), all multiindex α and all multiindex η of length |η| = 0 or 1,|〈∂ηt ∂αx h , u〉| ≤ C ‖h‖∞, where C only depends on the data and the support of h.

But if u solves (4.6) in D′, then

〈∂t ∂αx h , u〉 = 〈L?t ∂αx h+ c ∂αx h , u〉 − 〈∂αx h , f〉

hence it is enough to show (4.9) with η = 0, i.e. integrate by parts in the space direction. So we canstate

Theorem 4.10 Assume that 4.3 i) is fulfilled, 4.3 ii) holds on any compact subset of U on the full

time interval [S, T ], and 4.3 iii) holds at each x ∈ U . Then∂

∂t+ Lt + c is parabolic-hypoelliptic in

]S, T [×U .

The proof can be done exactly as in [11] p.433-446, using smoothness of the densities pcst(x, •) ofthe law of the process exponentially killed at rate c, which follows from 4.1, 4.2 and usual tools ofMalliavin calculus. We do not know how to adapt Kusuoka-Stroock method which is based on smalltime precise estimates.In regard with Chaleyat-Maurel and Michel result, Theorem 4.10 holds for any distribution solutioninstead of measure solution, but we have to assume β-Holder continuity in time, while these authorsonly need L∞. However it is quite possible that their proof can be improved, while, as we said, ourapproach really needs Holder continuity.In view of the intricacies of sections 2 and 3, we hope it is possible to get a simpler proof of existenceand regularity for pst. But Malliavin calculus has shown to be a very efficient tool in studying variousproblems, so that Theorem 2.6 and Theorem 3.25 have their own interest. In the next section weshall 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 orderto get smoothness of conditional laws as the ones which appear in filtering theory. Our results givethe correct hypothesis for section 2 of [17]. However the best way to study conditional laws is theuse of the partial Malliavin calculus introduced by Bismut and Michel [8]. The same can be done fortime dependent coefficients, the only difference being that the drift X0 is forbidden in the statementof partial Hormander’s hypotheses.

5.2. Infinitely degenerate diffusions

The terminology is due to Bell and Mohammed [6] (also see [30]). The non homogeneous case isstudied in [33], but with wrong hypotheses and proofs. With the material of the present paper, [33]can be corrected.

5.3. Killed and reflected processes

The study of reflected diffusions in a domain M with non characteristic boundary ∂M or of thekilled 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 boundarypoints 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 changeof coordinates. Indeed it leads to use Ito formula with a time dependent function. If the coefficientsare C1 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. asin [28] (this would correct [34]), manifold values s.d.e and so on.In the numerous applications of Malliavin calculus, small time behaviour of heat kernels takes animportant place (see [22], [2] [3] [23] [24]). We do not know what can be done for non homogeneousheat kernels in the same spirit.Another open question is the study of the set of strict positivity of the density (see [1] or [29] forthe homogeneous case).

References

1. N. Aida, S. Kusuoka, and D. Stroock. On the support of Wiener functionals. In K.D. Elworthy andN.Ikeda, editors, Asymptotic problems in Probability theory, number 284 in Pitman research notes inMathematics, Sanda & Kyoto, 1990. Taniguchi international Symposium.

2. G. Ben Arous. Developpement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus.

Ann. Sci. Ecole Norm. Sup., 21:307–331, 1988.3. G. Ben Arous. Developpement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale.

Ann. Institut Fourier, 39:73–99, 1989.4. V. Bally. On the connection between the Malliavin covariance matrix and Hormander’s condition. J.

Funct. Anal., 96:219–255, 1991.5. D. Bell. The Malliavin Calculus, volume 34 of Pitman Monographs and Surveys in Pure and Applied

Math. Longman and Wiley, 1987.6. D. Bell and S. Mohammed. An extension of Hormander’s theorem for infinitely degenerate second order

operators. Duke Math. J., 78:453–476, 1995.7. J.M. Bismut. Last exit decompositions and regularity at the boundary of transition probabilities. Z.

Wahrsch. verw. Gebiete, 69:65–98, 1985.8. J.M. Bismut and D. Michel. Diffusions conditionnelles I. J. Funct. Anal., 44:174–211, 1981.9. P. Cattiaux. Hypoellipticite et hypoellipticite partielle pour les diffusions avec une condition frontiere.

Ann. Institut Henri Poincare, 22:67–112, 1986.10. P. Cattiaux. Regularite au bord pour les densites et les densites conditionnelles d’une diffusion reflechie

hypoelliptique. Stochastics, 20:309–340, 1987.11. P. Cattiaux. Calcul stochastique et operateurs degeneres du second ordre I. Resolvantes et theoreme de

Hormander. Bull. Sciences. Math., 114:421–462, 1990.12. P. Cattiaux. Calcul stochastique et operateurs degeneres du second ordre II. Probleme de Dirichlet.

Bull. Sciences. Math., 115:81–122, 1991.13. P. Cattiaux. Stochastic calculus and degenerate boundary value problems. Ann. Institut Fourier,

42:541–624, 1992.14. M. Chaleyat-Maurel and D. Michel. Hypoellipticy theorems and conditional laws. Z. Wahrsch. verw.

Gebiete, 65:573–597, 1984.15. M. Chen and X. Zhou. Applications of Malliavin calculus to stochastic differential equations with

time-dependent coefficients. Acta Appl. Math. Sinica, 7(3):193–216, 1991.16. M. Derridj. Un probleme aux limites pour une classe d’operateurs du second ordre hypoelliptiques.

Ann. Institut Fourier, 21:99–148, 1971.17. P. Florchinger. Malliavin calculus with time-dependent coefficients and application to non linear filtering.

Prob. Theor. and Rel. Fields, 86:203–223, 1990.18. L. Hormander. Hypoelliptic second order differential equations. Acta Math., 117:147–171, 1967.19. N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland,

second edition, 1989.20. I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus. Springer-Verlag, second edition,

1997.21. S. Kusuoka and D.W Stroock. Applications of Malliavin calculus, part II. J. Fac. Sc. Univ. Tokyo,

32:1–76, 1985.22. S. Kusuoka and D.W Stroock. Applications of Malliavin calculus, part III. J. Fac. Sc. Univ. Tokyo,

34:391–442, 1987.23. R. Leandre. Integration dans la fibre associee a une diffusion degeneree. Prob. Theor. and Rel. Fields,

76:341–35, 1987.

28 Patrick Cattiaux, Laurent Mesnager: Hypoelliptic non homogeneous diffusions

24. R. Leandre. Developpement asymptotique de la densite d’une diffusion degeneree. Forum Mathe-maticum, 4:45–75, 1992.

25. P. Malliavin. Ck-hypoellipticity with degeneracy. In A. Friedmann and M. Pinsky, editors, StochasticAnalysis, pages 199–214. Acad. Press, New York, 1978.

26. P. Malliavin. Stochastic calculus of variations and hypoelliptic operators. In Conference on StochasticDifferential Equations, Kyoto 1976, pages 195–263. Kinokuniya, Tokyo and Wiley, New York, 1978.

27. P. Malliavin. Stochastic Analysis, volume 313 of Series of comprehensive studies in Mathematics.Springer, 1997.

28. M.D. Nguyen, D. Nualart and M. Sanz. Application of the Malliavin calculus to a class of stochasticdifferential equations. Prob. Theor. and Rel. Fields, 84:549–571, 1990.

29. A. Millet and M. Sanz. A simple proof of the support theorem for diffusion processes. Seminaire deProbabilites XXVIII, Lecture Notes in Math. 1583:36–48, 1994.

30. Y. Morimoto. Hypoellipticity for infinitely degenerate elliptic operators. Osaka J. math, 24:13–35, 1987.31. J.R. Norris. Simplified Malliavin calculus. Seminaire de Probabilites XX, Lecture Notes in Math.

1204:101–130, 1986.32. D. Nualart. The Malliavin Calculus and Related Topics. Probability and its Application. Springer-

Verlag, 1995.33. J. Schiltz. Le theoreme de Hormander pour des operateurs du second ordre infiniment degeneres avec

des coefficients dependant du temps. Stochastics, 59:259–281, 1996.34. J. Schiltz. Malliavin’s calculus with time depending coefficients applied to a class of stochastic differential

equations. Stochastic Analysis and Appl., 17, 1999.35. D.W. Stroock. Some applications of stochastic calculus and partial differential equations. In Ecole d’ete

de Probabilites de Saint-Flour, volume 976, pages 267–280. Lecture Notes in Math., 1983.36. S. Taniguchi. Applications of Malliavin calculus to time-dependent system of heat equations. Osaka J.

Math, 89:457–485, 1991.

Documents

Hypoelliptic non homogeneous di usions · The main tool is Malliavin calculus. Our results extend and correct previous ones ([17] and related works, [15]). Related topics like ltering