The Malliavin Calculus and Hypoelliptic Differential Operators

The Malliavin Calculus and Hypoelliptic Diﬀerential Operators

Denis Bell Department of Mathematics, University of North Florida 4567 St. Johns Bluﬀ Road South,Jacksonville, FL 32224, U. S. A. email: [email protected]

Dedicated to the memory of Paul Malliavin

1 Introduction

The first half of the twentieth century saw the rise of stochastic analysis as an independent branch of probability theory. Remarkable results were obtained. Wiener constructed a rigorous mathematical model of Brownian motion. Kolmogorov discov- ered that the transition probabilities of a diffusion process define the fundamental solution to the associated heat equation. Itôdeveloped a theory of stochastic integration that made it possible to represent a diffusion process with a given infinitesimal generator as the solution of a stochastic differential equation (sde). These developments established a link betwen the classical area of partial differential equations and the nascent area of stochastic calculus, whereby results in the former area could be used to prove results in the latter.

More specifically, let X0,X1,...,Xn denote a collection of smooth vector fields on Rd and define the second-order differential operator1 n 1 X L ≡ X2 + X (1.1) 2 i 0 i=1 Consider the (Stratonovich) sde n X dξt = Xi(ξt) ◦ dwi + Xo(ξt)dt (1.2) i=1 where w = (w1, ..., wn) is an n-dimensional standard Wiener process. Then the solution ξ to (1.2) is a time-homogeneous Markov process with infinitesimal generator L, whose transition probabilities p(t, x, dy) satisfy the following PDE (known as Kol- mogorov’s forward equation) in the weak sense ∂p = L∗p. ∂t y 1The presence of the factor 1/2 in (1.1) is a matter of convenience and is otherwise unimportant.

1 The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 2

A differential operator G is said to be hypoelliptic if, whenever Gu is smooth for some distribution u defined on some open subset of the domain of G, then u is smooth. In 1967, Hörmanderproved that the operator L in (1.1) is hypoelliptic if the Lie algebra generated by X0,...,Xn has dimension d throughout the domain of L. (This hypothesis is known as Hörmander’scondition and will be referred to in the sequel as HC.) It follows immediately that the transition probabilities p of ξ in (1.2) have ∂p ∗ smooth densities at all positive times if the parabolic operator ∂t − L satisfies HC at ξ0. In the reverse direction, if it is possible to establish by a direct probabilistic argument the existence of smooth transition probabilities for ξ under HC, then one can deduce hypoellipticity of L and thereby obtain a probabilistic proof of Hörmander’s theorem. This exciting line of research was initiated in the mid-seventies by Malliavin in his seminal paper [Ma]. The idea underlying Malliavin’s probabilistic approach to hypoellipticity is the following. The measure p(t, x, dy) is the image of Wiener measure under the map gt : w 7→ ξt. Now the Wiener measure has a well-understood analytic structure as an infinite-dimension analogue of the Gaussian distribution in Euclidean space. If the map g were smooth, then regularity properties of p could be obtained by a process of integration by parts on the Wiener space. In fact, this is not the case, it turns out that g is most pathological from the standpoint of standard calculus. Malliavin solved this problem by constructing a calculus applicable to the class of Wiener functionals defined by sde’s with smooth coefficients. Malliavin’s original work was a technical tour de force. It has since been clarified and extended, and has become a powerful tool in stochastic analysis. Different ap- proaches to the subject have been introduced, in particular the functional analytic approach of Kusuoka-Stroock [KS], the variational approach of Bismut [Bi.1], the elementary approach of the author [Be]. Following further pioneering work by Kusuoka and Stroock, the Malliavin calculus has yielded the desired probabilistic proof of Hörmander’stheorem and much more. Achievements of the field to date include new contributions to filtering theory by Michel [Mi] and others, a deeper understanding of the Skorohod integral and the development of an anticipating stochastic calculus by Nualart and Pardoux [NP], an extension of Clark’s formula by Ocone [Oc], Bismut’s probabilistic analysis of the small-time asymptotics of the heat kernel of the Dirac operator on a Riemannian manifold [Bi.2] (which itself has led to striking developments in the area of index theory) and, more recently, numerous applications in the area of mathematical finance related to the computation of Greeks (cf. e.g. Malliavin & Thalmaier [MT]). While these developments fall outside the scope of the present article, the interested reader is strongly encouraged to study them. The article is organized as follows. In Section 2 we derive Malliavin’s integration by parts formula, by which one obtains smoothness of the transition probabilities discussed above. In particular the Malliavin covariance matrix, which plays a central role throughout, is introduced in this section of the article. We also state and prove a result of Bismut that makes explicit the connection between Malliavin’s covariance matrix and Hörmander’scondition. In Section 3, we give the proof of the probabilistic version of Hörmander’stheorem. This result establishes the existence of smooth densities for the diffusion process ξt under the assumption that the vector fields X0,...,Xn satisfy a parabolic version of HC at the point ξ0. The Malliavin Calculus and Hypoelliptic Differential Operators 3

In Section 4 we prove a generalized version of Hörmander’stheorem. This result establishes hypoellipticity of Hörmandertype operators under conditions that allow HC to fail on hypersurfaces of codimension 1. The theorem is sharp within the category of operators with smooth coefficients.

2 Integration by parts and the regularity of measures

As indicated above, the Malliavin calculus has reached a state of maturity, to the point where it is now used routinely by a wide variety of researchers in stochastic analysis and other areas of mathematics. Following the inﬂuential book by Nualart [Nu], the subject is generally formulated in terms of Sobolev spaces. In this section we present a more elementary approach to the theory. Following Malliavin, we will make use of the following result from harmonic analysis.

Lemma 2.1. Let ν denote a ﬁnite Borel measure on Rd. Suppose that for all non- ∞ negative integers k and d1, . . . , dk, there exists a constant C such that for all test (C compact support) functions φ on Rd

Z ∂d1+···+dk

d d φ dν ≤ C||φ∞||. (2.1) d 1 k R ∂x1 . . . ∂xk If this condition holds for k = 1, then ν is absolutely continuous with respect to the Lebesgue measure on Rd. If the condition holds for all k ≥ 1, then ν has a C∞ density.

The next result is included in order to motivate the approach that follows. It is a ﬁnite-dimensional analogue of the inﬁnite-dimensional problem that will be addressed shortly.

Theorem 2.2. Suppose T : Rm 7→ Rd is a C2 map. Let γ denote the standard Gaussian measure on Rm and let ν ≡ γ ◦ T −1 denote the induced measure on Rd by T (also denoted by T (γ)). Deﬁne N to be the set of points in Rm where T has a non-surjective derivative. Then the condition

γ(N) = 0 (2.2) is necessary and suﬃcient for absolute continuity of ν.

Proof. The necessity of condition (2.2) follows immediately from Sard’s theorem, which asserts that the set T (N) has zero Lebesgue measure. To prove suﬃciency, we argue as follows. Deﬁne a d × d matrix

σ(x) = DT (x)DT (x)∗. (2.3)

Let ψk : [0, ∞] 7→ [0, 1] denote a sequence of smooth bump functions such that (i) ( 1, 0 ≤ t ≤ k ψk(t) = 0, t ≥ k + 1 The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 4

p (ii) For all p ≥ 1, supk|D k(t)| < ∞. d d Deﬁne Rk : R ⊗ R 7→ [0, 1] by

( −1 ψk(||α|| , α ∈ GL(d) Rk(α) = 0, α∈ / GL(d)

∞ d d Note that Rk is C since GL(d) is an open subset of R ⊗ R . d We deﬁne sequences of measures γk on C0 and νk on R by dγ k = R (σ(x)), dγ k

−1 νk = γk ◦ T .

Assume (2.2) holds, then σ(x) ∈ GL(d) a.s. This implies νk → ν in variation. Hence it suﬃces to prove absolute continuity of νk for every k. To this end, let e1, . . . ed denote d m m the standard basis of R and deﬁne, for 1 ≤ i ≤ d, hi : R 7→ R by

( ∗ −1 DT (x) σ (x)ei, σ ∈ GL(d) hi(x) = 0, σ∈ / GL(d)

Let φ be a test function on Rd. Then Z Z ∂φ −1 dνk = Dφ(T (x))eiψk(||σ (x)||) dγ(x) Rd ∂xi Rm Z −1 = D(φ ◦ T )(x)hi(x)ψk(||σ (x)||) dγ(x). (2.4) Rm Denote by Div the (Gaussian) divergence operator,

Div(G)(x) ≡< G(x), x > −traceDG(x).

Integrating by parts in (2.4) yields

Z ∂φ Z dνk = φ ◦ T (x)Xi(x) dγ(x) (2.5) Rd ∂xi Rd

−1 where Xi = Div[ψk(||σ ||)hi]. Since Xi is continuous with compact support, it 1 follows that Xi ∈ L (γ). From (2.5), we obtain Z ∂φ

dνk ≤ ||φ||∞||Xi||L1(γ) Rd ∂xi and the absolute continuity of νk follows from Lemma 2.1.

With the above ﬁnite-dimensional case as motivation, we now address the problem at hand. Let A and B denote bounded smooth maps from Rd, into Rd ⊗ Rn and Rd respectively, with bounded derivatives of all orders. Let x ∈ Rd and consider the sde Z s Z s ξt = x + A(ξu)dw + B(ξu)ds, s ≤ t (2.6) 0 0 The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 5

n where (w1, . . . , wn) is a standard Wiener process in R .

We wish to establish regularity of the law ν of ξt. The problem looks similar to the setting of Theorem 2.2. The measure ν is the image of the Wiener measure γ under the map gt : w 7→ ξt. (We denote by g the Itômap, w 7→ ξ.) There are, however, two immediate obstacles in implementing an integration by parts procedure of the type used in the proof of Theorem 2.2. Firstly, we are now working on an infinite-dimensional vector space (the space of continuous paths). Thus there is no Lebesgue measure with which to integrate. It turns out that this is not a major problem since there exists a divergence theorem for the Wiener measure (cf. Gross [G]) which could effect the integration by parts. More problematic is the fact that the map g (and by association gt) is non-differentiable in the classical sense. Prior to Malliavin’s work, this had proved a major obstacle to establishing the distributional regularity of solutions to sde’s. We describe here our own approach to the problem (cf. [Be], Chapter 4). The idea is as follows. Let C0 denote the space of continuous paths on the time interval [0, t] n C0 ≡ {σ : [0, t] 7→ R /σ0 = 0}

2 and H the Hilbert subspace of C0 consisting of the set of absolutely continuous paths h such that Z t 0 2 ||hu|| ds < ∞, 0 with inner product defined by Z t 0 0 < h, k >H ≡ < hu, ku > du, h, k ∈ H. (2.7) 0 The point is that g is differentiable as a map on H. In fact, it is clear that the restriction3 of g to H, which we denote byg ˜, is the map h 7→ k defined by the (ordinary) integral equation Z s 0 ks = x + A(ku)hu + B(ku) du, s ≤ t 0 and this map is C∞ since the coefficients A and B are C∞. This observation allows for an elementary approach to the required integration by parts formula. d For each standard basis vector ei ∈ R , we construct a lift ρi of ei to the Wiener space via the mapg ˜t, i.e. a set of paths ρi(h) satisfying the condition

Dg˜t(h)ρi(h) = ei. (2.8)

Let Pm : C0 7→ H denote a sequence of projections converging strongly to the identity map on C0. Then the aforementioned divergence theorem for Wiener measure, together with (2.8) yields Z ∂φ Z dν = Dφ(gt(w))ei dγ d R ∂xi C0 2The space H is known as the Cameron-Martin space. Its geometry will play a key role in the computations that follow. 3In fact, the term “restriction” requires clarification because g is only defined up to a set of full γ-measure and γ(H) = 0. It is more correct to say that g stochastically extends the smooth mapg ˜ in a sense that will shortly be made clear. The Malliavin Calculus and Hypoelliptic Differential Operators 6

Z = lim Dφ(˜gt(Pmw))ei dγ m C0 Z = lim D(φ ◦ g˜t)(Pmw))hi(Pmw) dγ m C0 Z = lim φ(˜gt(Pmw)Div[hi(Pmw)] dγ m C0 The main task is to show that

C ≡ sup ||Div[hi(Pmw)]|| < ∞. m We will then have Z ∂φ

dν ≤ C||φ||∞ Rd ∂xi and the absolute continuity of ν will follow from Lemma 2.1. The higher order estimates required to establish smoothness of the density of ν are obtained by repeated applications of the divergence theorem. The argument that follows is actually a slightly modified version of this scheme that uses a sequence of piecewise-linear approximations to ξ, rather than the restriction mapg ˜ per se. Being finite-rank operators, these approximations finite-dimensionalise the differential analysis in the argument. Thus at each level of approximation, we need only perform integration by parts in Euclidean space. m We now describe our approximation scheme. Let ∆jw(= ∆j w) denote the Brow- nian increment w((j +1)t/m)−w(jt/m). Define v0, vt/m, . . . , vt inductively by v0 = x and k−1 k−1 X t X v = x + A(v )∆ w + B(v ), k = 1, . . . , m. (2.9) kt/m jt/m j m jt/m j=0 j=0 m Let g (w) denote the piecewise linear path between the points (kt/m, vkt/m), k = m ∞ 0, . . . , m. It is clear that, for all m, the map g : C0 7→ H is C . The following two results are proved in [Be], Chapter 4. Theorem 2.3. For all p ∈ N,

m p lim sup E[|ξs − vs | ] = 0. m→∞ s∈[0,t] Theorem 2.4. Deﬁne

m m m ∗ d d σ (w) ≡ Dgt (w)Dgt (w) ∈ R ⊗ R . As m → ∞, the matrix sequence σm converges a.s. to a limit σ. The matrix σ admits the following representation. Let I denote the d × d identity matrix and consider the sde’s Z s Z s Ys = I + DA(ξu)(Yu, dwu) + DB(ξu)Yu du, 0 0 Z s Z s Zs = I − ZuDA(ξu)(., dwu) − ZuDB(ξu) du. 0 0 Then Z t h ∗ ∗ i ∗ σ = Yt ZsA(ξs)A(ξs) Zs ds Yt (2.10) 0 where ∗ denotes matrix transpose. The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 7

d We note that the matrix Yt is the derivative of the stochastic ﬂow on R (i.e. the −1 random map x 7→ ξt) and Zt = Yt . The matrix σ deﬁned in (2.10) is known as the Malliavin covariance matrix. The next result establishes the link between non-degeneracy of σ and regularity of the law of ξt. It is the main result of this section.

Theorem 2.5. Suppose σ ∈ GL(d) a.s. Then the random variable ξt is absolutely continuous wrt Lebesgue measure on Rd. If, furthermore \ det σ−1 ∈ Lp(γ) p≥1

∞ then the density of ξt is C . We prove only the absolute continuity statement here. (The higher order estimates required to establish the C∞ property of the density follow from iterating the integration by parts step in the argument, then performing estimations in the same spirit as those below. Full details can be found in [Be], Chapter 4). The proof will make use of the following elementary result, which is obtained from a discrete version of Gronwall’s lemma. d d Lemma 2.6. Let X ∈ R ,U0,...,Um−1 denote R -valued random variables, V0,...,Vm−1 d d and Y0,...,Ym−1 random linear maps from R to R , and Z0,...,Zm−1 random bi- linear maps from Rd × Rd to Rd, satisfying the following conditions

(i) for all 0 ≤ i ≤ m − 1,Ui,Vi,Yi,Zi are Fit/m-measurable where {Fs} is the ﬁltration generated by {ws}. p (ii) max{||X||p||, ||Ui||p, ||Vi||p, ||Yi||p, ||Zi||p} ≤ M, where || · ||p denotes the L - norms of the various quantities in their respective spaces.

Deﬁne ηk, 0 ≤ k ≤ m by the relation

k−1 k−1 k−1 k−1 1 X X 1 X X η = X + U + V (∆ w) + Y (η ) + Z (η , ∆ w). k m j j j m j jt/m j jt/m j j=0 j=0 j=0 j=0 Then there exists a constant N, depending only on M and p such that

||ηjt/m||p ≤ N, ∀ 0 ≤ k ≤ m.

Proof of Theorem 2.5 (absolute continuity part).

Let Vm denote the subspace of C0 consisting of paths that are piecewise linear between the times 0, t/m, . . . , t. Following the proof of Theorem 2.2 (and adopting the same notation in places), we deﬁne dγ k ≡ R (σ(w)), dγ k

νk ≡ gt(γk).

As before, the assumption σ ∈ GL(d) a.s. implies νk → ν in variation, so it suﬃces to prove that every νk is absolutely continuous. d m Let e denote an arbitrary unit vector in R and deﬁne h : C0 7→ H by

( −1 Dgm(w)∗σm e, σm ∈ GL(d) hm(x) = t i 0, σm ∈/ GL(d). The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 8

Following the proof of Theorem 2.2 and the outline presented earlier, and integrating by parts on the (ﬁnite-dimensional) Gaussian spaces (Vm,Pm(γ)) yields, for a test function φ Z Z ∂φ m m m dν = lim φ ◦ gt Div[h Rk ◦ σ ] dγ (2.11) d m→∞ R ∂e C0 where Div denotes the divergence operator

(DivG)(w) =< G, w > − traceDG(w), where the inner product is as deﬁned in 2.74. To complete the proof, it is necessary to prove that

m m sup E|Div[h Rk ◦ σ ]| < ∞. (2.12). m First consider the inner product term in the divergence. This is non-zero only if σm ∈ GL(d) and ||σm||−1 ≤ k + 1. In this case

m m m−1 m E| < h Rk ◦ σ , w > | = E| < σ e, Dgt (w)w|

m ≤ (k + 1)E|ηt |. (2.13) where η = Dgm(w)w satisﬁes the equation

k−1 X t ηm = {A(v )∆ w+DA(v )(ηm , ∆ w)+ DB(v )ηm }, k = 1, . . . , m. kt/m it/m i it/m it/m i m it/m it/m i=0

m Since A, DA, and DB are bounded, Lemma 2.6 implies that E|ηt | is bounded in m. m m By (2.13), the same holds for E| < h Rk ◦ σ , w > |. It remains to show that

m m sup E|traceD[h Rk ◦ σ ]| < ∞. (2.14) m

n Let f1, . . . , fn be an orthonormal basis of R . For 1 ≤ r ≤ n and 0 ≤ l ≤ m − 1, deﬁne f rl ∈ H by ( 0, 0 ≤ s < lt/m ψk(t) = p m/t(s − lt/m)fr, lt/m ≤ s < (l + 1)t/m.

This set {f rl} is orthonormal in H and can thus be extended to an orthonormal basis c m Bm of H. Note that for any f ∈ Bm ∩ Sm, we have Dg (w)f = 0. Evaluating the trace in (2.14) on Bm gives

n,m−1 m m X m m rl rl traceD[h Rk ◦ σ ] = < D[h Rk ◦ σ ](w)f , f > r=1,l=0

4The reader may wonder about the appearance of the Wiener process w in this expression, since m m w does not lie in H. Since the path h Rk ◦ σ in (2.11) to which Div is being applied is piecewise C2, the inner product term Z t m m m m 0 0 < h Rk ◦ σ , w >= < [h Rk ◦ σ ]s, ws > ds 0 is well-deﬁned by integration by parts. The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 9

n,m−1 X n m m−1 2 m rl rl m−1 m rl m−1 m rl = Rk◦σ (w)[σ e, D gtσ (w)(f , f ) > − < σ Dσ (w)f σ e, Dgt (w)f ] r=1,l=0

m m rl m−1 m rl o +DRk(σ )Dσ (w)f < σ e, Dgt (w)f > .

In view of the deﬁnition of Rk it suﬃces to show that for all 1 ≤ r ≤ d,

m−1 X 2 m rl rl sup E D gtσ (w)(f , f ) < ∞ (2.15) m l=0 and m−1 h X m rl m rl i sup E Dσ (w)f | × |Dgt (w)f | < ∞. (2.16) m l=0 √ Let ηrl denote the path mDgm(w)f rl. Diﬀerentiation wrt w in (2.9) yields

rl ηjt/m = 0, j ≤ l

j−1 j−1 √ X t X ηrl = tA(v )f + DA(v )(ηrl , ∆ w) + DB(v )ηrl , jt/m it/m r pt/m pt/m p m it/m pt/m p=l p=l j ≥ l + 1. It follows from Lemma 2.6 that

rk sup ||ηjt/m||4 < ∞. (2.17) j0k0m

Let ρrl denote the path mD2gm(w)(f rl, f rl). It satisﬁes

rl ρjt/m = 0, j ≤ l,

j−1 rl rl X 2 rl rl rl ρjt/m = DA(vit/m)(ηlt/m, fr)+ {D A(vpt/m)(ηpt/m, ηpt/m, ∆pw)+DAvpt/m)(ρpt/m, ∆pw)} p=l

j−1 t X + {D2B(v )(ηrl , ηrl )+DB(v )ρrl }, j ≥ l+1. m pt/m pt/m pt/m pt/m pt/m p=l Lemma 2.6 together with (2.15) give

rk sup ||ηjt/m)||4 < ∞ r,j,k,m and this implies (2.15). The ﬁnal condition (2.16) is established by a similar argument. (We omit the details.)

We now focus our attention on the sde (1.2). According to (2.10) (and adopting the same notation as before) the Malliavin covariance matrix for the solution ξt is given by n Z t X ∗ ∗ σ = Yt [ZsXi(ξs)] ⊗ [ZsXi(ξs)] dsYy (2.20) i=1 0 The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 10

where Zs satisﬁes

n X Z s Z s Zs = I − ZuDXi(ξu) ◦ dwi − ZuDX0(ξu) du. (2.21) i=1 0 0

−1 and Yt = Zt . The following result, due to Bismut ([Bi.1]), makes explicit the relationship between the H¨ormandercondition and the non-degeneracy of σ.

Theorem 2.7. Suppose the vector ﬁelds X1,...,Xn together with their iterated Lie d brackets of all orders (denoted Lie{X1,...,Xn}) span R at ξ0. Then σ ∈ GL(d), a.s.

The proof will require the following. Lemma 2.8. Let y ∈ Rd. Suppose B is a smooth vector ﬁeld on Rd and τ is a stopping time such that

< ZsB(ξs), y >= 0, ∀s ∈ [0, τ].

Then for i = 1, . . . , n, we have

< Zs[Xi,B](ξs), y >= 0, ∀s ∈ [0, τ].

Proof. Applying Itˆo’sformula to (2.20) and making use of (2.19), we obtain

0 = ds < ZsB(ξs), y >

n X =< Zs[B,X0](ξs), y > ds + < Zs[B,X0](ξs) ◦ dwi, y >, s ∈ [0, τ]. (2.22) i=1 Writing the Stratonovich integrals as Itˆointegrals plus drift, we have a statement of the form n X G(s)ds + < Zs[B,X0](ξs) ◦ dwi, y >= 0, s ∈ [0, τ]. (2.23) i=1 where G(s) is a continuous adapted process. The conclusion now follows from the Itˆo rules: dwidt = 0, dwidwj = ∆ijdt.

Proof of Theorem 2.7. Denote Θ ≡ Lie {X1,...,Xn}(ξ0). We show that Θ ⊆ Range σ, which clearly implies the result. For 0 < s < t, deﬁne Rs = span{ZuXi(ξu) 0 ≤ u ≤ s, 0 ≤ i ≤ n} and \ R = Rs. 0

Then Rt = Range σ. By the Blumenthal zero-one law, there exists a deterministic set R˜ such that R = R,˜ a.s. ⊥ Suppose y ∈ R˜ . Then with probability 1, there exists τ > 0 such that Rs = R˜ for all s ∈ [0, τ]. Hence The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 11

< ZsXi(ξs), y >= 0, s ∈ [0, τ], i = 1, . . . , n. (2.22) Applying Lemma 2.8 repeatedly to (2.22) gives

< ZsV (ξs), y >= 0, s ∈ [0, τ] for every vector ﬁeld V in Lie {X1,...,Xn}. ⊥ ˜ Setting s = 0, we see that y ∈ Θ . Thus Θ ⊆ R ⊆ Rt as required.

3 The probabilistic analogue of H¨ormander’stheorem

The purpose of this section is to prove the following result.

Theorem 3.1. Consider the sde

n X dξt = Xi(ξt) ◦ dwi + Xo(ξt)dt, i=1

ξ0 = x. Suppose the set of vector ﬁelds

[Xi], [Xj,Xk], [Xj,Xk],Xl],..., 1 ≤ i ≤ n, 0 ≤ j, k, l, . . . (3.1)

d span R at the point x. Then ξt has a smooth density for all t > 0.

The heart of the proof of Theorem 3.1 is the following martingale inequality which, in its original form, is due to Kusuoka & Stroock [KS]. The lemma was given a simpliﬁed proof by Norris [No]. (Norris’ proof can be found in [Be], Lemma 6.5.)

Lemma 3.2. Let x0, y0 ∈ R and suppose as, bs, and vs are continuous adapted processes in R, Rn, and Rnrespectively. Deﬁne

n Z t X Z t xt ≡ x0 + as ds + bi(s) dwi 0 i=1 0 and Z t Z t yt ≡ y0 + xs ds + bi(s) dwi 0 0

Let τ be a bounded stopping tim and C a deterministic constant such that sup{|at|, |bt|, |vt|, |xt|/ t ≤ τ} ≤ C. Then for every q > 17, there exist positive constants α and β (depending only on C) such that Z τ Z τ 2 q 2 2 −α/ P yt dt < , (xt + |vt| ) ≥ ≤ e , ∀ ∈ (0, β). 0 0

The next result is straightforward. The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 12

Lemma 3.3. Suppose X is a non-negative random variable such that

P (X < ) = o(k), → 0, ∀k ∈ N.

Then ∞ \ P (X < ) ∈ Lp. p=1

We introduce the following notation. Denote by Kl the set of vector fields in (3.1) containing at most l − 1 iterated Lie brackets. Choose and fix an integer l such that d Kl spans R and set δ = inf { sup < K(x), v >2}. |v|=1 k∈Kl Note that since the infimum is over a compact set, δ is strictly positive. Let Sd denote the unit sphere in Rd. Finally, for B ∈ R, define the stopping time T by

T = inf{s ≥ 0/ |ξs − x| ≥ 1/B or ||Zs − I|| ≥ 1/B} ∧ t.

d We choose such B large enough to insure that for all v ∈ S , there exist K ∈ Kl and a neighborhood N of v in Sd such that

inf < ZsK(ξs), u > ≥ δ/2. (3.2) s≤T,u∈N

Then {T < } = {sup |ξs − x| ∨ sup ||Zs − I|| ≥ 1/B}. (3.3) s≤ s≤ Jensen’s inequality implies

p p p/2 E sup |ξs − x| ∨ sup ||Zs − I|| = o( ), ∀p ≥ 1. (3.4) s≤ s≤

It follows from (3.3) and (3.4) that

P (T < ) = o(P ), ∀p ≥ 1, so by Lemma 3.3, we have T −1 ∈ Lp, ∀p ≥ 1. (3.5) d Then (3.2) and (3.5) imply: for all v ∈ S , there exist K ∈ Kl and a neighborhood N of v in Sd such that for all p ≥ 1

Z T 2 p sup P < ZsK(ξs), u > ds ≤ P (δT/2 < ) = o( ). (3.6) u∈N 0

The remainder of the proof is structured as follows. The argument is a qualitative version of the proof Theorem 2.7.We need to obtain probabilistic lower bounds on the inverse of the matrix σ in (2.20). Standard moment estimates (cf., e.g. [Be], Theorem −1 p 1.9) show that Yt = Zt ∈ L for all p. Hence it suﬃces to work with the reduced covariance matrix n Z t X ∗ σ = [ZsXi(ξs)] ⊗ [ZsXi(ξs)] ds i=1 0 The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 13 which we will now denote by σ. We shall prove

∞ \ det σ−1 ∈ Lp. (3.7) p=1 from which the the result will follow by Theorem 2.5. We establish (3.7) by proving

∞ \ λ−1 ∈ Lp. p=1 where λ denotes the smallest eigenvalue of σ. In view of Lemma 3.2, it suﬃces to prove P (λ < ) = o(p), ∀p ≥ 1. (3.8) In Lemma 3.5 below we will prove the following result, from which this follows as an easy consequence

n Z T X 2 p P inf < ZsXi(ξs), v > ds < = o( ), ∀p ≥ 1. (3.8) v∈Sd i=1 0

Note that in (3.6) we have a statement similar to this. The diﬀerences between (3.6) and (3.8) are twofold: (i) the presence of the inﬁmum inside the probability in (3.8).

(ii) The fact that in (3.6) we have an assertion about an arbitrary element K ∈ Kl, whereas in (3.8) we require the analogous statement for at least one of the vector fields X1,...,Xn. In the sequel (i) will be addressed by a compactness argument, and (ii) by an inductive argument that turns on Lemma 3.1. We now give the details ((ii) first, then (i)). Lemma 3.4. For all v ∈ Sd there exist i ∈ {1, . . . , n} and a neighborhood N of v in S such that Z T 2 p sup P < ZsXi(ξs), u > ds < = o( ), ∀p ≥ 1. u∈N 0 Proof. Without loss of generality, we may assume that the vector field K in (3.6) has the form

K = [[Xi1 ,Xi2 ],Xi3 ,...,Xir ] where r ≤ l, i1, . . . , ir ∈ {0, 1, . . . , n} and i1 6= 0.

Deﬁne K1 = Xi1 and Kj = [Kj−1,Xij 0], j = 2, . . . , r. We will show by induction on r (decreasing) that for j = 1, . . . , r

Z T 2 p sup P < ZsKj(ξs), u > ds < = o( ), ∀p ≥ 1, (3.9) u∈N 0 which will suﬃce to prove the result. The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 14

Thus, assume (3.9). By Itˆo’sformula

n X d(ZsKj−1(ξs)) = Zs[Xi,Kj−1](ξs)dwi(s). i=1

n n 1 X o +Z [X ,K ](ξ )+ [[X ,K ],X ](ξ ) ds. s 0 j−1 s 2 i j−1 i s i=1 Deﬁne ys ≡< ZsKj−1(ξs), u >, n n 1 X x ≡< Z [X ,K ](ξ ) + [[X ,K ],X ](ξ ), u >, s s 0 j−1 s 2 i j−1 i s i=1

vi(s) =< Zs[Xi,Kj−1](ξs), u >, i = 0, . . . , n.

The deﬁnition of T insures that the processes y, x, and v = (v1, . . . , vn) satisfy the hypotheses of Lemma 3.2 on [0,T ] with the constant C independent of u ∈ N. Then Lemma 3.2 gives

Z T Z T 2 18 2 2 p P ys ds ≤ , xs + |vs| ) ds ≥ = o( ) , ∀p ≥ 1 (3.10) 0 0 uniformly in u ∈ N.

If ij 6= 0, then Kj is one of the vector ﬁelds [Xi,Kj−1], i = 1, . . . , n and it follows from the elementary inequality

c P (Ω1) ≤ P (Ω1 ∩ Ω2) + P (ω2) that (3.9) holds for j − 1.

Suppose, on the other hand, ij = 0. Write

Z T n 2 18o Ω = ys ds < 0 and Yi(s) =< Zs[[Xi,Kj−1],Xi](ξs), u >, i = 1, . . . , n.

Repeating the above argument with [Kj−1,Ai] in place of Kj−1 (for every i = 1, . . . , n) gives Z T n 2 o p P Ω ∩ Yi(s) ds ≥ = o( ), ∀p ≥ 1. (3.11) 0 The desired result will now follow from (3.10) if we show

n n Z T 1 X 2 o P Ω ∩ v (s) + Y (s) ds < = o(p), ∀p ≥ 1. (3.12) 0 2 i 0 i=1 We have n n Z T 1 X 2 o P Ω ∩ v (s) + Y (s) ds < 0 2 i 0 i=1 The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 15

n n Z T 1 X 2 o ≤ P Ω ∩ v (s) + Y (s) ds < 0 2 i 0 i=1 n Z T n Z T \ n 2 o X 2 ∩ Yi (s) ds < + P Yi (s) ds > . (3.13) i=1 0 i=1 0 By (3.11) the ﬁnal term in (3.13) is o(p), while

n n Z T 1 X 2 o P Ω ∩ v (s) + Y (s) ds < 0 2 i 0 i=1 n Z T \ n 2 o ∩ Yi (s) ds < i=1 0 Z T 2 n−1 ≤ P v0(s) ds ≤ (2 + n2 ) 0 = o(p) by (3.9). This establishes (3.12) and the inductive step is complete. Lemma 3.5. n Z T X 2 p P inf < ZsXi(ξs), v > ds < = o( ), ∀p ≥ 1. v∈Sd i=1 0 Proof. By choice of T the quadratic forms Z T 2 v 7→ < ZsXi(ξs), v > ds 0 are uniformly Lipschitz on Sd. Let θ denote the common Lipschitz constant and cover d S with balls of radius /θ and center vj. The number of such balls may be chosen to be less than D(θ/)d for some ﬁnite D. Then Z T 2 < ZsXi(ξs), v > ds < 0 for some v ∈ Sd implies that for some j Z T 2 < ZsXi(ξs), v > ds < 2. 0 Hence n Z T X 2 P inf < ZsXi(ξs), v > ds < v∈Sd i=1 0 Z T d 2 ≤ D(θ/) max P < ZsXi(ξs), vj > ds < 2 j 0 Z T d 2 ≤ D(θ/) sup P < ZsXi(ξs), v > ds < 2 v∈Sd 0 Since Sd is compact, Lemma 3.4 implies this last term is o(p) and this completes the proof of the lemma. The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 16

4 H¨ormander’stheorem for superdegenerate operators

As before, let X0,...,Xn denote smooth vector fields, now defined on an open set D ∈ Rd and consider the differential operator

n 1 X L ≡ X2 + X . (4.1) 2 i 0 i=1 According to Hörmander’stheorem, L is hypoelliptic on D if the vector space Lie(X0,...,Xn)(x) has dimension d for all x ∈ D. It can be shown that Hörmander’s condition is necessary for hypoellipticity for operators of the form (3.1) with analytic coefficients. Such is not the case if the vector fields X0,...,Xn are allowed to be (smooth) non-analytic. This fact is illustrated by a striking result of Kusuoka & Stroock, who studied operators on R3 of the form

2 2 2 ∂ p ∂ ∂ −|x1| Lp ≡ 2 + e 2 + 2 , p < 0. ∂x1 ∂x2 ∂x3 The following theorem is proved in [KS].

Theorem 4.1. The operator Lp is hypoelliptic if and only if p > −1.

Note that in the case p ∈ (−1, 0), Lp fails to satisfy Hörmander’scondition on the hyperplane {x1 = 0}. In this section, we present a sharp form of Hörmander’stheorem due to the author 5 and S. Mohammed [BM] that encompasses the Kusuoka-Stroock operators Lp, p ∈ (−1, 0). The statement of the theorem will require some additional notation. m Let E , m ≥ 0, denote a matrix with columns X0,...,Xn together with all Lie brackets constructed from X0,...,Xn, of length not exceeding m. Define µm to be the smallest eigenvalue of the matrix EmEm∗. Observe that Hörmander’s(general) condition holds for L at the point x ∈ D if and only if µm(x) > 0 for some m ≥ 0. We denote the set of all such x by H. Note that it is an open subset of D. The complement of this set in D (the non-Hörmander set of L) set will be denoted Hc. The main result is the following.

Theorem 4.2. Suppose the set Hc is contained in a C2-hypersurface S ⊂ Rd. Assume that at every point x ∈ Hc

(i) at least one of X1,...,Xn is non-tangential to S. (ii) There exists m, a neighborhood U of x, and p ∈ (−1, 0) such that

p µm(y) ≥ e−|ρ(y,S)| , ∀y ∈ U (4.2) where ρ(y, U) denotes the Euclidean distance between y and S. Then L is hypoelliptic.

5We term diﬀerential operators with this (exponential) level of degeneracy superdegenerate. The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 17

Before discussing the proof of Theorem 4.2, we make a couple of remarks concern- ing the hypotheses of the theorem. By looking at the probabilistic picture, one can see that an assumption such as Theorem 4.2(i) is necessary for the hypoellipticity of L (at least in the case when X0 = 0). Indeed, if this condition fails then the diﬀusion process ξ started from point x will stay in the hypersurface S for small time. Hence ξt will have a singular distribution wrt Lebesgue measure on Rd. This implies L is not hypoelliptic since, as we have observed, hypoellipticity of L implies absolute continuity of ξt. Condition Theorem 4.2(ii) controls the rate at which HC fails fails as one ap- proaches S, in a neighborhood of non-H¨ormander points of L. The non-hypoelliptic operators Lp, p < −1 in Theorem 4.1 show that some such hypothesis is also necessary for hypoellipticity of L. Furthermore, the case L−1 shows that the lower bound −1 on p in (ii) is optimal. The proof of Theorem 4.2 is based on the following result, proved in [KS].

Lemma 4.3. Suppose for all q ≥ 1 and x ∈ D, there exists a neighborhood V of x such that n −1 o lim t log sup ||∆(t, y) ||q = 0, (4.3) t→0+ y∈V where ∆(t, ξ0) denotes the Malliavin covariance matrix corresponding to the process ξt (cf. (2.20)). ∂ Then the parabolic operator ∂t + L is hypoelliptic on R × D.

We prove first that (4.3) holds under the parabolic version of HC (where X0 does not appear explicitly in the spanning set). The proof proceeds via the following steps. (i) A local parameterization φ of the hypersurface S is introduced, and the hypotheses of the theorem are reframed in terms of conditions on φ. q (ii) We derive probabilistic lower bounds on the L norms of the process φ(ξs). These lower bounds become sharp as q → ∞. (iii) We study the rate at which the lower bounds in (ii) are degraded under the exponential type degeneracy allowed in the theorem. This leads to lower bounds on ∆(t, y) from which we are able to deduce (4.3). The statement of the parabolic version of the theorem alluded to above requires a modified form of the notation introduced earlier. m (m) Let F denote the matrix obtained by deleting X0 from the columns in E and define λm to be the smallest eigenvector of the matrix F mF m∗. Define

K ≡ {x ∈ D/λm(x) > 0, for some m ≥ 0}.

The primary objective is to prove the following result. Theorem 4.4. Suppose the hypotheses of Theorem 4.2 hold with the set H replaced by K qnd µm replaced by λm. Then the operator ∂ + L ∂t is hypoelliptic on R × D. The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 18

At this point it is convenient to introduce the following terminology.

Deﬁnition. A non-negative random variable T is exponentially positive (EP) if there exist constants c1, c2 > 0 (which we refer to as the characteristics of T ) such that

−c1/ P (T < ) < e , ∀ ∈ (0, c2).

This definition is motivated by the following well-known result (cf., e.g. [IW], Lemma 10.5). Lemma 4.5. Let y : [0,T ] × Ω 7→ Rd be an Itôprocess n X dy = ai(t)dwi + b(t)dt, (4.4) i=1 where a1, . . . , an, b are measurable adapted processes, bounded by a deterministic constant c1. Let r > 0 and define

τ ≡ {s > 0/|ys − y0 = r} ∧ T. (4.5)

Then τ is an EP stopping time whose characteristics depend only on c1 and r The next two lemmas are key to the proof of Theorem 4.4. )We refer the reader to [BM] for the proofs.) Lemma 4.6. Let y be the Itôprocess in (4.4) and τ an EP stopping time. Suppose that ai(0) 6= 0 for at least one 1 ≤ i ≤ n. Then for all m ≥ 2, there exists T0 > 0 such that Z t∧τ m − 1 P |yu| du < < exp(−t m+1 ), (4.6) 0 m+1 ∀t ∈ (0,T0), ∈ (0, t ), where T0 depends only on m, the constant c1 in Lemma 4.5, and the characteristics of τ. Lemma 4.7. Let τ be an EP stopping time and y the Itôprocess in (4.4). Suppose −p τ and y satisfy (4.6) for some m > p+1 , where p ∈ (−1, 0). Then there exist T1 > 0 and q > 1 such that Z t∧τ p q P exp(|yu| ) du < < exp(−| log | ), (4.7) 0 −1/q ∀t ∈ (0,T1), 0 < < exp(−t ), where T1 and q depend only on p, the constant c1 in Lemma 4.5, and the characteristics of τ. We will need two final Lemmas, whose proofs are straightforward. d Lemma 4.8. For every q ≥ 1 and bounded set V ⊂ R , there exists a constant c2 such that ∞ n 1 o −1 q X − dq ||∆(t, x) ||q ≤ c2 1 + P Q(t, x) < j , ∀t ∈ (0,T ), x ∈ V (4.8) j=1 where n Z t n X x x 2 o Q(t, x) ≡ inf < Zu Xi(ξu), h > du , (4.9) h∈Sd i=1 0 where x = ξ0. The Malliavin Calculus and Hypoelliptic Differential Operators 19

Lemma 4.9. Suppose the hypothesis of Theorem 4.4 holds. Then for all x ∈ D, there exists an integer m ≥ 1 such that one of the following two conditions holds: (a) λm(x) > 0. (b) There exists a neighborhood U ⊂ D of x, a C2 function φ : U 7→ R, and p ∈ (−1, 0) such that

(i) φ(x) = 0 and ∇φ(x) · Xi(x) 6= 0, for at least one i = 1, . . . , n. (ii) λm(y) ≥ exp(−|φ(y)|p), ∀y ∈ U.

With these Lemmas in place, we are ready to prove Theorem 4.4.

d Proof. First we extend the vector ﬁelds X0,...,Xn from D to the whole of R with compact support.6 We use the same notation for the extended vector ﬁelds.

Assume the hypothesis of Theorem 4.4 holds. Let x0 ∈ D and choose m so that the conclusion of Lemma 4.9 holds. Suppose x lies in a ﬁxed bounded neighborhood W of x0. Deﬁne

x x τ1 ≡ inf{s > 0/ |ξs − x| ∨ ||Zs − I|| = 1/2} ∧ t. (4.10)

By Lemma 4.5, τ1 is an EP stopping time with characteristics independent of x ∈ W . Let h ∈ Sd, the unit sphere in Rd. By the same argument as that used to prove Theorem 3.1 we can show the following.

There exist constants c3 and 0 < r1, r2 < 1 (depending only on the characteristics of τ1) such that for all t ∈ (0,T ), x ∈ W , and ∈ (0, c3),

P (Q(t, x) < ) ≤ exp(−r1 )+

N n Z t∧τ1 o −d X x x 2 r2 sup P < Zu Vj(ξu), h > du < (4.11) d h∈S j=1 0 where Q(t, x) is as deﬁned in (4.9) and V1,...,VN denote the columns of the matrix F m. x Since (4.10) implies ||Zu − I|| ≤ 1/2 for all u ∈ (0, τ1), it is easy to deduce from (4.11) that

Z t∧τ1 −r1 −d m x r2 P (Q(t, x) < ) ≤ exp( ) + P λ (ξu) du < . (4.12) 0

We now consider the two cases (a) and (b) delineated in Lemma 4.9. m Suppose first that (a) holds at x0. Then by continuity of λ there exist ρ, δ > 0 such that m λ (y) ≥ δ, ∀y ∈ Bρ(x0). (4.13) x Define V ≡ Bρ/2(x0). Let x ∈ V and τ2 denote the first exit time of ξ from V . Then (4.12) and (4.13) imply7

−r1 −d r2 P (Q(t, x) < ) ≤ exp( ) + P (τ1 ∧ τ2 ∧ t < /δ) (4.14)

6This is is a standard construction using a partition of unity. 7In the estimates hat follow, some multiplicative constants appear outside the exponentials. We omit these constants since they can obviously be eliminated by adjusting the exponents r1 and r2 slightly. The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 20

≤ exp(−r3 ), ∀t < r2 /δ, (4.15) where r3 ≡ r1 ∧ r2. Substituting (4.15) into (4.8) yields the following inequality: for all q ≥ 1, −1 q −dq/r2 ||∆(t, x) ||q ≤ c1 (δt) + A where ∞ X A(t) ≡ 1 + exp(−jr4 ) < ∞ j=k where r4 = r3/dq. −1 We conclude that ||∆(t, x) ||q explodes at polynomial rate as t ↓ 0, uniformly wrt x ∈ V . Hence (4.3) holds.

We now turn to the case where (b) in Lemma 4.9 holds at x0. By this condition, we may choose ρ > 0 small enough to insure that Bρ(x0) ⊂ U and 1 |∇φ(x) · X (x)| ≥ |∇φ(x ) · X (x )| > 0 i 2 0 i 0 for some 1 ≤ i ≤ n and all x ∈ Bρ(x0). x Suppose x ∈ V ≡ Bρ/2(x0) and let τ3 denote the ﬁrst exit time of ξ from V . Combining Lemma 4.5, Condition 4.9(b)(ii), and (4.12), we have

Z t∧τ1∧τ3 −r1 −d x p r2 P (Q(t, x) < ) ≤ exp( ) + P exp(−|yu| ) du < (4.16) 0 x x where yt denotes the process φ(ξt ). By Itˆo’sformula n X x x x dyt = ∇φ(ξt ) · Xi(ξt )dwi + Lφ(ξt )dt. (4.17) i=1 Condition 4.9(b)(i) and (4.17) imply that the process yx and the stopping time τ ≡ τ1 ∧ τ2 satisfy the hypotheses of Lemma 4.6, hence (4.6) holds. Thus, by Lemma 3.6 0 −1/q0 there exist T1 > 0 and q > 1 such that for all t ∈ (0,T1) and ∈ (0, exp(−t ),

Z t∧τ1∧τ3 p q0 P exp(−|yu| ) du < < exp(| log | ). (4.18) 0 Substituting (4.18) into (4.12) gives

0 P (Q(t, x) < ) ≤ exp(−r1 ) + −d exp(| log r2 |q ) (4.19)

−1/q0 for t ∈ (0,T1) and ∈ (0, exp(−t ). Combining (4.19) with (4.8), we arrive at

−1 q −1/q0 ||∆(t, x) ||q ≤ c exp(dqt ) + A), 0 < t < T1, (4.19) where ∞ X n 0 o A = 1 + exp(−jr1/dq) + j1/q exp(−| log jr2/dq|q ) < ∞. j=1 (Note that the constants in (4.19) can all be chosen to be independent of x ∈ V .) Thus the rhs of (4.19) can explode exponentially fast as t ↓ 0, in this case. However, since q0 > 1, we conclude that (4.3) again holds and the proof of Theorem 4.4 is complete. The Malliavin Calculus and Hypoelliptic Diﬀerential Operators 21

Finally, we prove Theorem 4.2.

Proof. Assume L satisfies the hypotheses of Theorem 4.2. We borrow yet another technique from [KS]. The idea is to imbed L in an operator L˜ defined on a (d + 1)- dimensional domain and satisfying the hypotheses of Theorem 4.4. Choose a smooth non-negative real-valued function ρ on (0, 1) such that both ρ and ρ0 are bounded away from zero. Define L˜ on D × (0, 1) by

1 ∂2 L˜ ≡ ρ(s)L + . 2 ∂s2

Then L˜ has the form n+1 1 X L˜ = X˜ 2 + X˜ 2 i 0 i=1 where X˜0(x, s) = ρ(s)X0(x), 1/2 X˜i(x, s) = ρ(s) Xi(x), 1 ≤ i ≤ n, ∂ X˜ (x, s) = . n+1 ∂s ˜m Deﬁne λ , H˜ and K˜ similarly to before (cf. Theorem 4.4) using the vector ﬁelds X˜i in place of the Xi. Then it is easy to check that there exist δm such that

m m λ (x, s) ≥ δmµ (x), ∀(x, s) ∈ D × (0, 1). (4.20)

This implies K˜ c ⊆ Hc × (0, 1). Since Hc is contained in a C2-hypersurface S ⊂ D, it follows that H˜ c is contained in the C2-hypersurface S × (0, 1) ⊂ D × (0, 1). By assumption, at least one of X1,...,Xn is non-tangential to S at every point in c c H . Hence one of X˜1,..., X˜n+1 is non-tangential to S × (0, 1) at every point in H˜ . Hypothesis (ii) in Theorem 4.2 together with (4.20) implies that λ˜m satisﬁes the hypothesis in Theorem 4.4 wrt the set K˜ c. We conclude from Theorem 4.4 that the ˜ ∂ ˜ operator L + ∂t is hypoelliptic on R × D × (0, 1).Consequently L is hypoelliptic on D × (0, 1) which, in turn, implies L is hypoelliptic on D. Stick a fork in us, for we are done!

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