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STOCHASTIC PROCESSES ON SURFACES IN THREE-DIMENSIONAL

CONTACT SUB-RIEMANNIAN MANIFOLDS

DAVIDE BARILARI, UGO BOSCAIN, DANIELE CANNARSA, KAREN HABERMANN

Abstract. We are concerned with stochastic processes on surfaces in three-dimensional contact

sub-Riemannian manifolds. Employing the Riemannian approximations to the sub-Riemannian

manifold which make use of the Reeb vector ﬁeld, we obtain a second order partial diﬀerential

operator on the surface arising as the limit of Laplace–Beltrami operators. The stochastic process

associated with the limiting operator moves along the characteristic foliation induced on the

surface by the contact distribution. We show that for this stochastic process elliptic characteristic

points are inaccessible, while hyperbolic characteristic points are accessible from the separatrices.

We illustrate the results with examples and we identify canonical surfaces in the Heisenberg

group, and in SU(2) and SL(2,R)equipped with the standard sub-Riemannian contact structures

as model cases for this setting. Our techniques further allow us to derive an expression for an

intrinsic Gaussian curvature of a surface in a general three-dimensional contact sub-Riemannian

manifold.

1. Introduction

The study of surfaces in three-dimensional contact manifolds has found a lot of interest, amongst

others, since the so-called oriented singular foliation on the surface provides an important invariant

used to classify contact structures, see Abbas and Hofer [1, Chapter 3], Geiges [17, Chapter 4],

and Giroux [18, 19]. In recent years, there has been an increased activity in studying surfaces

in three-dimensional contact manifolds whose contact distributions additionally carry a metric.

Balogh [4] analyses the Hausdorﬀ dimension of the so-called characteristic set of a hypersurface

in the Heisenberg group. Balogh, Tyson and Vecchi [5] deﬁne an intrinsic Gaussian curvature for

surfaces in the Heisenberg group and an intrinsic signed geodesic curvature for curves on surfaces

to obtain a Gauss–Bonnet theorem in the Heisenberg group. Veloso [27] extends the results in [5]

to general three-dimensional contact manifolds for non-characteristic surfaces. Danielli, Garofalo

and Nhieu [16] discuss the local summability of the sub-Riemannian mean curvature of surfaces in

the Heisenberg group. The contribution of this paper is to introduce a canonical stochastic process

on a given surface in a three-dimensional contact manifold whose contact distribution is equipped

with a metric, to analyse properties of the induced stochastic process and to identify model cases

for this setting.

Let (M, D, g)be a three-dimensional contact sub-Riemannian manifold, that is, we consider a

three-dimensional manifold Mwhich is equipped with a sub-Riemannian structure (D, g)that is

contact. A sub-Riemannian structure on a manifold Mconsists of a bracket generating distribution

D⊂T M and a smooth ﬁbre inner product gdeﬁned on D. Such a sub-Riemannian structure is

said to be contact if the distribution Dis a contact structure on M. Under the assumption that Dis

coorientable, the latter means that there exists a global one-form ωon Msatisfying ω∧dω6= 0 and

such that D= ker ω. The one-form ωis called a contact form and the pair (M, D)is called a contact

manifold. Throughout, we choose the contact form ωto be normalised such that dω|D=−volgfor

volgdenoting the Euclidean volume form on Dinduced by the ﬁbre inner product g. Associated

with the contact form ω, we have the Reeb vector ﬁeld X0which is the unique vector ﬁeld on M

satisfying dω(X0,·)≡0and ω(X0)≡1.

arXiv:2004.13700v1 [math.PR] 28 Apr 2020

2 D. BARILARI, U. BOSCAIN, D. CANNARSA, K. HABERMANN

Let Sbe an orientable surface embedded in the contact manifold (M, D). We call a point x∈S

a characteristic point of Sif the contact plane Dxcoincides with the tangent space TxS. Note

that characteristic points are also called singular points, cf. [1] and [17]. We denote the set of all

characteristic points of Sby Γ(S). If x∈Sis not a characteristic point then Dxand TxSintersect

in a one-dimensional subspace. These subspaces induce a singular one-dimensional foliation on S,

that is, an equivalence class of vector ﬁelds which diﬀer by a strictly positive or strictly negative

function. This foliation is called the characteristic foliation of Sinduced by the contact structure

D. We see that the canonical stochastic process we deﬁne on the surface Smoves along the

characteristic foliation. This process does not hit elliptic characteristic points, whereas a hyperbolic

characteristic point is hit subject to an appropriate choice of the starting point. In the dynamical

systems terminology, an elliptic point corresponds to a node or a focus, and a hyperbolic point is

called a saddle, see Robinson [25].

To construct the canonical stochastic process on S, we consider the Riemannian approximations to

the sub-Riemannian manifold (M, D, g)which make use of the Reeb vector ﬁeld X0. For ε > 0, the

Riemannian approximation to (M, D, g)deﬁned uniquely by requiring √εX0to be unit-length and

to be orthogonal to the distribution Deverywhere induces a Riemannian metric gεon S. This gives

rise to the two-dimensional Riemannian manifold (S, gε)and its Laplace–Beltrami operator ∆ε.

We show that the operators ∆εconverge uniformly on compacts in S\Γ(S)to an operator ∆0on

S\Γ(S), and we study the stochastic process on S\Γ(S)whose generator is 1

2∆0.

To simplify the presentation of the paper, we shall assume that the distribution Dis trivialisable,

that is, globally generated by a pair of vector ﬁelds, and we choose vector ﬁelds X1and X2such

that (X1, X2)is an oriented orthonormal frame for Dwith respect to the ﬁbre inner product g.

By the Cartan formula and due to dω|D=−volg, we have

ω([X1, X2]) = −dω(X1, X2) = 1 .

Since X0is the Reeb vector ﬁeld, we obtain

ω([X0, Xi]) = −dω(X0, Xi)=0 for i∈ {1,2}.

It follows that there exist functions c1

ij , c2

ij :M→R, for i, j ∈ {0,1,2}, such that

[X1, X2] = c1

12X1+c2

12X2+X0,(1.1)

[X0, X1] = c1

01X1+c2

01X2,(1.2)

[X0, X2] = c1

02X1+c2

02X2.(1.3)

In particular, the vector ﬁelds X1,X2and [X1, X2]on Mare linearly independent everywhere. The

Riemannian approximation to (M, D, g)for ε > 0is then obtained by requiring (X1, X2,√εX0)to

be a global orthonormal frame. We further suppose that the surface Sembedded in Mis given by

(1.4) S={x∈M:u(x) = 0}for u∈C2(M)with du6= 0 on S .

While this might deﬁne a surface consisting of multiple connected components, we could always

restrict our attention to a single connected component. A point x∈Sis a characteristic point if

and only if (X1u)(x)=(X2u)(x)=0, that is,

(1.5) x∈Γ(S)if and only if ((X1u)(x))2+ ((X2u)(x))2= 0 .

Consequently, the characteristic set Γ(S)is a closed subset of S. With Hess udenoting the horizontal

Hessian of udeﬁned by

(1.6) Hess u=X1X1u X1X2u

X2X1u X2X2u,

we can classify the characteristic points of Sas follows.

STOCHASTIC PROCESSES ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 3

Deﬁnition. A characteristic point x∈Γ(S)is called non-degenerate if det((Hess u)(x)) 6= 0, it is

called elliptic if det((Hess u)(x)) >0, and it is called hyperbolic if det((Hess u)(x)) <0.

With the notations introduced above, we can explicitly write down the expression of a unit-length

representative of the characteristic foliation of Sinduced by the contact structure D. Let b

XSbe

the vector ﬁeld on S\Γ(S)deﬁned by

(1.7) b

XS=(X2u)X1−(X1u)X2

p(X1u)2+ (X2u)2.

Note that while b

XSis expressed in terms of X1, X2and u, it only depends on the sub-Riemannian

manifold (M, D, g), the embedded surface Sand a choice of sign. It is a vector ﬁeld on S\Γ(S)

whose vectors have unit length and lie in D|S∩T S with a continuous choice of sign. In particular,

the vector ﬁeld b

XSremains unchanged if uis multiplied by a positive function. Let b:S\Γ(S)→R

be the function given by

(1.8) b=X0u

p(X1u)2+ (X2u)2.

Similarly to the vector ﬁeld b

XS, the function bcan be understood intrinsically. Let b

X⊥

Sbe such

that (b

XS,b

X⊥

S)is an oriented orthonormal frame for D|S\Γ(S). The function bis then uniquely given

by requiring bb

X⊥

S−X0to be a vector ﬁeld on S\Γ(S). Set

(1.9) ∆0=b

X2

S+bb

XS,

which is a second order partial diﬀerential operator on S\Γ(S). The operator ∆0is invariant under

multiplications of uby functions which do not change its zero set. As stated in the theorem below,

it arises as the limiting operator of the Laplace–Beltrami operators ∆εin the limit ε→0.

Theorem 1.1. For any twice diﬀerentiable function f∈C2

c(S\Γ(S)) compactly supported in

S\Γ(S), the functions ∆εfconverge uniformly on S\Γ(S)to ∆0fas ε→0.

Since the theorem above only concerns twice diﬀerentiable functions of compact support in S\Γ(S),

we do not have to put any additional assumptions on the set of characteristic points of S.

Following the deﬁnition in Balogh, Tyson and Vecchi [5] for surfaces in the Heisenberg group, we

introduce an intrinsic Gaussian curvature K0of a surface in a general three-dimensional contact

sub-Riemannian manifold as the limit as ε→0of the Gaussian curvatures Kεof the Riemannian

manifolds (S, gε). To derive the expression given in the following proposition, we employ the same

orthogonal frame exhibited to prove Theorem 1.1.

Proposition 1.2. Uniformly on compact subsets of S\Γ(S), we have

K0:= lim

ε→0Kε=−b

XS(b)−b2.

We now consider the canonical stochastic process on S\Γ(S)whose generator is 1

2∆0. Assuming

that it starts at a ﬁxed point then, up to explosion, the process moves along the unique leaf of

the characteristic foliation picked out by the starting point. As shown by the next theorem and

the following proposition, for this stochastic process, elliptic characteristic points are inaccessible,

while hyperbolic characteristic points are accessible from the separatrices. Recall that what we call

hyperbolic characteristic points are known as saddles in the dynamical systems literature, whereas

what is referred to as hyperbolic points in their language are non-degenerate characteristic points

in our terminology.

Theorem 1.3. The set of elliptic characteristic points in a surface Sembedded in Mis inaccessible

for the stochastic process with generator 1

2∆0on S\Γ(S).

4 D. BARILARI, U. BOSCAIN, D. CANNARSA, K. HABERMANN

In Section 4.3, we discuss an example of a surface in the Heisenberg group whose induced stochastic

process is killed in ﬁnite time if started along the separatrices of the characteristic point. Indeed,

this phenomena always occurs in the presence of a hyperbolic characteristic point.

Proposition 1.4. Suppose that the surface Sembedded in Mhas a hyperbolic characteristic point.

Then the stochastic process having generator 1

2∆0and started on the separatrices of the hyperbolic

characteristic point reaches that characteristic point with positive probability.

The Sections 4 and 5 are devoted to illustrating the various behaviours the canonical stochastic

process induced on the surface Scan show. Besides illustrating Proposition 1.4, we show that three

classes of familiar stochastic processes arise when considering a natural choice for the surface Sin

the three classes of model spaces for three-dimensional sub-Riemannian structures, which are the

Heisenberg group H, and the special unitary group SU(2) and the special linear group SL(2,R)

equipped with sub-Riemannian contact structures for ﬁbre inner products diﬀering by a constant

multiple. In all these cases, the orthonormal frame (X1, X2)for the distribution Dis formed by

two left-invariant vector ﬁelds which together with the Reeb vector ﬁeld X0satisfy, for some κ∈R,

the commutation relations

[X1, X2] = X0,[X0, X1] = κX2,[X0, X2] = −κX1,

with κ= 0 in the Heisenberg group, κ > 0in SU(2) and κ < 0in SL(2,R). Associated with each

of these Lie groups and their Lie algebras, we have the group exponential map exp for which we

identify a left-invariant vector ﬁeld with its value at the origin.

Theorem 1.5. Fix κ∈R. For κ6= 0, let k∈Rwith k > 0be such that |κ|= 4k2. Set I= (0,π

k)

if κ > 0and I= (0,∞)otherwise. In the model space for three-dimensional sub-Riemannian

structures corresponding to κ, we consider the embedded surface Sparameterised as

S={exp(rcos θX1+rsin θX2) : r∈Iand θ∈[0,2π)}.

Then the limiting operator ∆0on Sis given by

∆0=∂2

∂r2+b(r)∂

∂r ,

where

b(r) =

2kcot(kr)if κ= 4k2

2

rif κ= 0

2kcoth(kr)if κ=−4k2

.

The stochastic process induced by the operator 1

2∆0moving along the leaves of the characteristic

foliation of Sis a Bessel process of order 3if κ= 0, a Legendre process of order 3if κ > 0and a

hyperbolic Bessel process of order 3if κ < 0.

Notably, the stochastic processes recovered in the above theorem are all related to one-dimensional

Brownian motion by the same type of Girsanov transformation, with only the sign of a parameter

distinguishing between them. For the details, see Revuz and Yor [24, p. 357]. A Bessel process

of order 3arises by conditioning a one-dimensional Brownian motion started on the positive real

line to never hit the origin, whereas a Legendre process of order 3is obtained by conditioning

a Brownian motion started inside an interval to never hit either endpoint of the interval. The

examples making up Theorem 1.5 can be considered as model cases for our setting, and all of them

illustrate Theorem 1.3.

Notice that the limiting operator we obtain on the leaves is not the Laplacian associated with the

metric structure restricted to the leaves as the latter has no drift term. However, the operator ∆0

STOCHASTIC PROCESSES ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 5

restricted to a leaf can be considered as a weighted Laplacian. For a smooth measure µ=h2dxon

an interval Iof the Euclidean line R, the weighted Laplacian applied to a scalar function fyields

divµ∂f

∂x =∂2f

∂x2+2h0(x)

h(x)

∂f

∂x .

In the model cases above, we have

h(r) =

sin (kr)if κ= 4k2

rif κ= 0

sinh (kr)if κ=−4k2

.

We prove Theorem 1.1 and Proposition 1.2 in Section 2, where the proof of the theorem relies on

the expression of ∆ε|S\Γ(S)given in Lemma 2.2 in terms of an orthogonal frame for T(S\Γ(S)).

In Section 3, we prove Theorem 1.3 and Proposition 1.4 using Lemma 3.1 and Lemma 3.2, which

expand the function b:S\Γ(S)→Rfrom (1.8) in terms of the arc length along the integral

curves of b

XS. The results are illustrated in the last two sections. In Section 4, we study quadric

surfaces in the Heisenberg group, whereas in Section 5, we consider canonical surfaces in SU(2) and

SL(2,R)equipped with the standard sub-Riemannian contact structures. The examples establishing

Theorem 1.5 are discussed in Section 4.1, Section 5.1 and Section 5.2, with a uniﬁed viewpoint

presented in Section 5.3.

Acknowledgement. This work was supported by the Grant ANR-15-CE40-0018 “Sub-Riemannian

Geometry and Interactions” of the French ANR. The third author is supported by grants from

Région Ile-de-France. The fourth author is supported by the Fondation Sciences Mathématiques

de Paris. All four authors would like to thank Robert Neel for illuminating discussions.

2. Family of Laplace–Beltrami operators on the embedded surface

We express the Laplace–Beltrami operators ∆εof the Riemannian manifolds (S, gε)in terms of

two vector ﬁelds on the surface Swhich are orthogonal for each of the Riemannian approximations

employing the Reeb vector ﬁeld. Using these expressions of the Laplace–Beltrami operators ∆ε

where only the coeﬃcients and not the vector ﬁelds depend on ε > 0, we prove Theorem 1.1. The

orthogonal frame exhibited further allows us to establish Proposition 1.2.

For a vector ﬁeld Xon the manifold M, the property Xu|S≡0ensures that X(x)∈TxSfor all

x∈S. Therefore, we see that F1and F2given by

F1=(X2u)X1−(X1u)X2

p(X1u)2+ (X2u)2and(2.1)

F2=(X0u)(X1u)X1+ (X0u)(X2u)X2

(X1u)2+ (X2u)2−X0,(2.2)

are indeed well-deﬁned vector ﬁelds on S\Γ(S)due to (1.5) and because we have F1u|S\Γ(S)≡0

as well as F2u|S\Γ(S)≡0. Here, S\Γ(S)is a manifold itself because the characteristic set Γ(S)is a

closed subset of S. We observe that both F1and F2remain unchanged if the function udeﬁning the

surface Sis multiplied by a positive function, whereas F1changes sign and F2remains unchanged

if uis multiplied by a negative function. Since the zero set of the twice diﬀerentiable submersion

deﬁning Sneeds to remain unchanged, these are the only two options which can occur. Observe

that the vector ﬁeld F1on S\Γ(S)is nothing but the vector ﬁeld b

XSdeﬁned in (1.7).

Recalling that gεis the restriction to the surface Sof the Riemannian metric on Mobtained by

requiring (X1, X2,√εX0)to be a global orthonormal frame, we further obtain

gε(F1, F2)=0

6 D. BARILARI, U. BOSCAIN, D. CANNARSA, K. HABERMANN

as well as

(2.3) gε(F1, F1)=1 and gε(F2, F2) = (X0u)2

(X1u)2+ (X2u)2+1

ε.

Thus, (F1, F2)is an orthogonal frame for T(S\Γ(S)) for each Riemannian manifold (S, gε). While

in general, the frame (F1, F2)is not orthonormal it has the nice property that it does not depend

on ε > 0, which aids the analysis of the convergence of the operators ∆εin the limit ε→0. Since

F1and F2are vector ﬁelds on S\Γ(S), there exist functions b1, b2:S\Γ(S)→R, not depending

on ε > 0, such that

(2.4) [F1, F2] = b1F1+b2F2.

Whereas determining the functions b1and b2explicitly from (2.1) and (2.2) is a painful task, we

can express them nicely in terms of, following the notations in [7], the characteristic deviation h

and a tensor ηrelated to the torsion. Let J:D→Dbe the linear transformation induced by the

contact form ωby requiring that, for vector ﬁelds Xand Yin the distribution D,

(2.5) g(X, J (Y)) = dω(X, Y ).

Under the assumption of the existence of the global orthonormal frame (X1, X2)this amounts to

saying that

(2.6) J(X1) = X2and J(X2) = −X1.

For a unit-length vector ﬁeld Xin the distribution D, we use [X, J(X)]|Dto denote the restriction

of the vector ﬁeld [X, J (X)] on Mto the distribution Dand we set

h(X) = −g([X, J (X)]|D, X),

η(X) = −g([X0, X], X),

where the expression for ηis indeed well-deﬁned because according to (1.2) and (1.3), the vector

ﬁeld [X0, X]lies in the distribution D.

Lemma 2.1. For b:S\Γ(S)→Rdeﬁned by (1.8), we have

[F1, F2] = −(bh(F1) + η(F1)) F1−bF2,

that is, b1=−bh(F1)−η(F1)and b2=−b.

Proof. We ﬁrst observe that due to (2.6), we can write

F2=bJ(F1)−X0.

Using (1.2) and (1.3) as well as (2.5), it follows that

ω([F1, F2]) = ω([F1, bJ (F1)−X0]) = −dω(F1, bJ(F1)) = −g(F1, bJ 2(F1)) = b .

On the other hand, from (2.1), (2.2) and (2.4), we deduce

ω([F1, F2]) = ω(b2F2) = −b2,

which implies that b2=−b, as claimed. It remains to determine b1. From (2.5), we see that

g(F1, J (F1)) = −ω([F1, F1]) = 0 .

Together with (2.4) this yields

b1=g([F1, F2], F1) = g([F1, bJ (F1)−X0], F1) = bg ([F1, J(F1)]|D, F1) + g([X0, F1], F1),

and therefore, we have b1=−bh(F1)−η(F1), as required.

STOCHASTIC PROCESSES ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 7

To derive an expression for the Laplace–Beltrami operators ∆εof (S, gε)restricted to S\Γ(S)in

terms of the vector ﬁelds F1and F2, it is helpful to consider the normalised frame associated with

the orthogonal frame (F1, F2). For ε > 0ﬁxed, we deﬁne aε:S\Γ(S)→Rby

(2.7) aε=(X0u)2

(X1u)2+ (X2u)2+1

ε−1

2

and we introduce the vector ﬁelds E1and E2,ε on S\Γ(S)given by

(2.8) E1=F1and E2,ε =aεF2.

In the Riemannian manifold (S, gε), this yields the orthonormal frame (E1, E2,ε)for T(S\Γ(S)).

Lemma 2.2. For ε > 0, the operator ∆εrestricted to S\Γ(S)can be expressed as

∆ε|S\Γ(S)=F2

1+a2

εF2

2+b−F1(aε)

aεF1−a2

ε(bh(F1) + η(F1)) F2.

Proof. Fix ε > 0and let divεdenote the divergence operator on the Riemannian manifold (S, gε)

with respect to the corresponding Riemannian volume form. Since (E1, E2,ε)is an orthonormal

frame for T(S\Γ(S)), we have

(2.9) ∆ε|S\Γ(S)=E2

1+E2

2,ε + (divεE1)E1+ (divεE2,ε)E2,ε .

Let (ν1, ν2,ε)denote the dual to the orthonormal frame (E1, E2,ε ). Proceeding, for instance, in the

same way as in [6, Proof of Proposition 11], we show that, for any vector ﬁeld Xon S\Γ(S),

divεX=ν1([E1, X ]) + ν2,ε ([E2,ε, X]) .

This together with (2.8) and Lemma 2.1 implies that

divεE1=ν2,ε ([aεF2, F1]) = −ν2,ε (aε[F1, F2] + F1(aε)F2) = b−F1(aε)

aε

as well as

divεE2,ε =ν1([F1, aεF2]) = ν1(aε[F1, F2] + F1(aε)F2) = −aε(bh(F1) + η(F1)) .

The desired result follows from (2.8) and (2.9).

Note that ∆ε|S\Γ(S)in Lemma 2.2 can equivalently be written as

∆ε|S\Γ(S)=F2

1+a2

εF2

2+ b−F1a2

ε

2a2

ε!F1−a2

ε(bh(F1) + η(F1)) F2.

Using Lemma 2.2 we can prove Theorem 1.1.

Proof of Theorem 1.1. From (1.8) and (2.7), we obtain that

(2.10) a2

ε=b2+1

ε−1

=ε

εb2+ 1 ,

which we use to compute

F1(aε)

aε

=F1(a2

ε)

2a2

ε

=−εbF1(b)

εb2+ 1 .

It follows that

(2.11) a2

ε≤εas well as

F1(aε)

aε≤ε|bF1(b)|.

Since u∈C2(M)by assumption, both b:S\Γ(S)→Rand F1(b): S\Γ(S)→Rare continuous

and therefore bounded on compact subsets of S\Γ(S). In a similar way, we argue that the function

8 D. BARILARI, U. BOSCAIN, D. CANNARSA, K. HABERMANN

b1=−bh(F1)−η(F1)is bounded on compact subsets of S\Γ(S). Due to (2.11), this implies that,

uniformly on compact subsets of S\Γ(S),

(2.12) lim

ε→0a2

ε= 0 ,lim

ε→0

F1(aε)

aε

= 0 and lim

ε→0a2

ε(bh(F1) + η(F1)) = 0 .

Let f∈C2

c(S\Γ(S)). We then have F1f, F2f∈C1

c(S\Γ(S)) and F2

1f, F 2

2f∈C0

c(S\Γ(S)). Since

the expression (1.9) for ∆0can be rewritten as

∆0=F2

1+bF1

and since the convergence in (2.12) is uniformly on compact subsets of S\Γ(S), we deduce from

Lemma 2.2 that

lim

ε→0k∆εf−∆0fk∞,S\Γ(S)= lim

ε→0

a2

εF2

2f−F1(aε)

aε

F1f−a2

ε(bh(F1) + η(F1)) F2f

∞,S\Γ(S)

= 0 ,

that is, the functions ∆εfindeed converge uniformly on S\Γ(S)to ∆0f.

Using the orthonormal frames (E1, E2,ε), we easily derive the expression given in Proposition 1.2

for the intrinsic Gaussian curvature K0of the surface Sin terms of the vector ﬁeld b

XSand the

function b. Unlike the reasoning presented in [5], which further exploits intrinsic symmetries of the

Heisenberg group H, our derivation does not rely on the cancellation of divergent quantities and

holds for surfaces in any three-dimensional contact sub-Riemannian manifold, cf. [5, Remark 5.3].

Proof of Proposition 1.2. From Lemma 2.1 and due to (2.4) as well as (2.8), we have

[E1, E2,ε]=[F1, aεF2] = aε[F1, F2] + F1(aε)F2=aεb1E1+−b+F1(aε)

aεE2,ε .

According to the classical formula for the Gaussian curvature of a surface in terms of an orthonormal

frame, see e.g. [3, Proposition 4.40], the Gaussian curvature Kεof the Riemannian manifold (S, gε)

is given by

(2.13) Kε=F1−b+F1(aε)

aε−aεF2(aεb1)−(aεb1)2−−b+F1(aε)

aε2

.

We deduce from (2.10) that

aεF2(aε) = 1

2F2a2

ε=−ε2bF2(b)

(εb2+ 1)2

as well as

F1F1(aε)

aε=−F1εbF1(b)

εb2+ 1 =−εF1(bF1(b))

εb2+ 1 +2ε2b2(F1(b))2

(εb2+ 1)2,

which, in addition to (2.11), implies

|aεF2(aε)| ≤ ε2|bF2(b)|and F1F1(aε)

aε≤ε|F1(bF1(b))|+ 2ε2b2(F1(b))2.

By passing to the limit ε→0in (2.13), the desired expression follows.

Notice that, by construction, the function band the intrinsic Gaussian curvature K0are related

by the Riccati-like equation

˙

b+b2+K0= 0 ,

with the notation ˙

b=b

XS(b).

STOCHASTIC PROCESSES ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 9

3. Canonical stochastic process on the embedded surface

We study the stochastic process with generator 1

2∆0on S\Γ(S). After analysing the behaviour

of the drift of the process around non-degenerate characteristic points, we prove Theorem 1.3 and

Proposition 1.4.

By construction, the process with generator 1

2∆0moves along the characteristic foliation of S, that

is, along the integral curves of the vector ﬁeld b

XSon S\Γ(S)deﬁned in (1.7). Around a ﬁxed

non-degenerate characteristic point x∈Γ(S), the behaviour of the canonical stochastic process

is determined by how b:S\Γ(S)→Rgiven in (1.8) depends on the arc length along integral

curves emanating from x. Since the vector ﬁelds X1, X2and the Reeb vector ﬁeld X0are linearly

independent everywhere, the function X0u:S→Rdoes not vanish near characteristic points. In

particular, we may and do choose the function u∈C2(M)deﬁning the surface Ssuch that X0u≡1

in a neighbourhood of x.

Understanding the expression for the horizontal Hessian Hess uin (1.6) as a matrix representation

in the dual frame of (X1, X2), and noting that the linear transformation J:D→Ddeﬁned in (2.5)

has the matrix representation

J=0−1

1 0 ,

we see that

(Hess u)J=X1X2u−X1X1u

X2X2u−X2X1u.

The dynamics around the characteristic point x∈Γ(S)is uniquely determined by the eigenvalues

λ1and λ2of ((Hess u)(x))J. Since x∈Γ(S)is non-degenerate by assumption both eigenvalues are

non-zero, and due to X0u≡1in a neighbourhood of x, we further have

(3.1) λ1+λ2= Tr (((Hess u)(x))J)=(X1X2u) (x)−(X2X1u) (x) = (X0u) (x)=1.

Thus, one of the following three cases occurs, where we use the terminology from [25, Section 4.4] to

distinguish between them. In the ﬁrst case, where the eigenvalues λ1and λ2are complex conjugate,

the characteristic point xis of focus type and the integral curves of b

XSspiral towards the point x. In

the second case, where both eigenvalues are real and of positive sign, we call x∈Γ(S)of node type,

and all integral curves of b

XSapproaching xdo so tangentially to the eigendirection corresponding

to the smaller eigenvalue, with the exception of the separatrices of the larger eigenvalue. In the

third case with the characteristic point xbeing of saddle type, the two eigenvalues are real but of

opposite sign, and the only integral curves of b

XSapproaching xare the separatrices.

Note that an elliptic characteristic point is of focus type or of node type, whereas a hyperbolic

characteristic point is of saddle type. Depending on which of theses cases arises, we can determine

how the function bdepends on the arc length along integral curves of b

XSemanating from x. The

choice of the function u∈C2(M)such that X0u≡1in a neighbourhood of xﬁxes the sign of the

vector ﬁeld b

XS. In particular, an integral curve γof b

XSwhich extends continuously to γ(0) = x

might be deﬁned either on the interval [0, δ)or on (−δ, 0] for some δ > 0. As the derivation

presented below works irrespective of the sign of the parameter of γ, we combine the two cases by

writing γ:Iδ→Sfor integral curves of b

XSextended continuously to γ(0) = x.

The expansion around a characteristic point of focus type is a result of the fact that the real parts

of complex conjugate eigenvalues satisfying (3.1) equal 1

2.

Lemma 3.1. Let x∈Γ(S)be a non-degenerate characteristic point and suppose that u∈C2(M)

is chosen such that X0u≡1in a neighbourhood of x. For δ > 0, let γ:Iδ→Sbe an integral curve

of the vector ﬁeld b

XSextended continuously to γ(0) = x. If the eigenvalues of ((Hess u)(x))Jare

10 D. BARILARI, U. BOSCAIN, D. CANNARSA, K. HABERMANN

complex conjugate then, as s→0,

b(γ(s)) = 2

s+O(1) .

Proof. Since X0u≡1in a neighbourhood of x, we may suppose that δ > 0is chosen small enough

such that, for s∈Iδ\ {0},

b(γ(s)) = 1

q((X1u) (γ(s)))2+ ((X2u) (γ(s)))2.

A direct computation shows

∂

∂s b(γ(s))−1=b

XSb(γ(s))−1= ((Hess u) (γ(s))) Jb

XS(γ(s)),b

XS(γ(s)).

By the Hartman–Grobman theorem, it follows that, for s→0,

∂

∂s b(γ(s))−1= ((Hess u)(x)) Jb

XS(γ(s)),b

XS(γ(s))+O(s).

As complex conjugate eigenvalues of ((Hess u)(x))Jhave real part equal to 1

2and due to b

XSbeing

a unit-length vector ﬁeld, the previous expression simpliﬁes to

(3.2) ∂

∂s b(γ(s))−1=1

2+O(s).

Since (X1u)(x) = (X2u)(x)=0at the characteristic point x, we further have

(3.3) lim

s→0

1

b(γ(s)) = 0 .

A Taylor expansion together with (3.2) and (3.3) then implies that, as s→0,

1

b(γ(s)) =s

2+Os2,

which yields, for s→0,

b(γ(s)) = 2

s(1 + O(s))−1=2

s+O(1) ,

as claimed.

The expansion of the function baround characteristic points of node type or of saddle type depends

on along which integral curve of b

XSwe are expanding. By the discussions preceding Lemma 3.1,

all possible behaviours are covered by the next result.

Lemma 3.2. Fix a non-degenerate characteristic point x∈Γ(S). For δ > 0, let γ:Iδ→Sbe an

integral curve of the vector ﬁeld b

XSwhich extends continuously to γ(0) = x. Assume u∈C2(M)is

chosen such that X0u≡1in a neighbourhood of xand suppose ((Hess u)(x))Jhas real eigenvalues.

If the curve γapproaches xtangentially to the eigendirection corresponding to the eigenvalue λi,

for i∈ {1,2}, then, as s→0,

b(γ(s)) = 1

λis+O(1) .

Proof. As in the proof of Lemma 3.1, we obtain, for δ > 0small enough and s∈Iδ\ {0},

b

XSb(γ(s))−1= ((Hess u) (γ(s))) Jb

XS(γ(s)),b

XS(γ(s)).

Since γis an integral curve of the vector ﬁeld b

XS, we deduce that

∂

∂s 1

b(γ(s))= ((Hess u) (γ(s))) (J(γ0(s)) , γ 0(s)) .

STOCHASTIC PROCESSES ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 11

By Taylor expansion, this together with (3.3) yields, for s→0,

1

b(γ(s)) = ((Hess u) (x)) (J(γ0(0)) , γ0(0)) s+Os2.

By assumption, the vector γ0(0) ∈TxSis a unit-length eigenvector of ((Hess u)(x))Jcorresponding

to the eigenvalue λi, which has to be non-zero because xis a non-degenerate characteristic point.

It follows that

((Hess u) (x)) (J(γ0(0)) , γ0(0)) = λi6= 0 ,

which implies, for s→0,

b(γ(s)) = 1

λis(1 + O(s))−1=1

λis+O(1) ,

as required.

Remark 3.3. We stress Lemma 3.2 does not contradict the positivity of the function bnear the

point xensured by the choice of u∈C2(M)such that X0u≡1in neighbourhood of x. The derived

expansion for bsimply implies that on the separatrices corresponding to the negative eigenvalue

of a hyperbolic characteristic point, the vector ﬁeld b

XSpoints towards the characteristic point for

that choice of u, that is, we have s∈(−δ, 0). At the same time, we notice that

∂2

∂s2+b(γ(s)) ∂

∂s

remains invariant under a change from sto −s. Therefore, in our analysis of the one-dimensional

diﬀusion processes induced on integral curves of b

XS, we may again assume that the integral curves

are parameterised by a positive parameter.

With the classiﬁcation of singular points for stochastic diﬀerential equations given by Cherny and

Engelbert in [15, Section 2.3], the previous two lemmas provide what is needed to prove Theorem 1.3

and Proposition 1.4. One additional crucial observation is that for a characteristic point of node

type both eigenvalues of ((Hess u)(x)) Jare positive and less than one, whereas for a characteristic

point of saddle type, the positive eigenvalue is greater than one.

Proof of Theorem 1.3. Fix an elliptic characteristic point x∈Γ(S). For δ > 0, let γ: [0, δ ]→S

be an integral curve of the vector ﬁeld b

XSextended continuously to x= lims↓0γ(s). Following

Cherny and Engelbert [15, Section 2.3], since the one-dimensional diﬀusion process on γinduced

by 1

2∆0has unit diﬀusivity and drift equal to 1

2b, we set

(3.4) ρ(t) = exp Zδ

t

b(γ(s)) ds!for t∈(0, δ].

If the characteristic point xis of node type the real positive eigenvalues λ1and λ2of ((Hess u)(x))J

satisfy 0< λ1, λ2<1by (3.1). As xis of focus type or of node type by assumption, Lemma 3.1

and Lemma 3.2 establish the existence of some λ∈Rwith 0<λ<1such that, as s↓0,

b(γ(s)) = 1

λs +O(1) .

We deduce, for δ > 0suﬃciently small,

ρ(t) = exp Zδ

t1

λs +O(1)ds!= exp 1

λln δ

t+O(δ−t)=δ

t1

λ

(1 + O(δ−t)) .

12 D. BARILARI, U. BOSCAIN, D. CANNARSA, K. HABERMANN

Due to 1

λ>1, this implies that Zδ

0

ρ(t) dt=∞.

According to [15, Theorem 2.16 and Theorem 2.17], it follows that the elliptic characteristic point

xis an inaccessible boundary point for the one-dimensional diﬀusion processes induced on the

integral curves of b

XSemanating from x. Since x∈Γ(S)was an arbitrary elliptic characteristic

point, the claimed result follows.

Proof of Proposition 1.4. We consider the stochastic process with generator 1

2∆0on S\Γ(S)near

a hyperbolic point x∈Γ(S). Let γbe one of the four separatrices of xparameterised by arc length

s≥0and such that γ(0) = x. Let λ1be the positive eigenvalue and λ2be the negative eigenvalue of

((Hess u)(x))J. From the trace property (3.1), we see that λ1>1. By Lemma 3.2 and Remark 3.3,

we have, for i∈ {1,2}and as s↓0,

b(γ(s)) = 1

λis+O(1) .

As in the previous proof, for δ > 0suﬃciently small and ρ: (0, δ]→Rdeﬁned by (3.4), we have

ρ(t) = δ

t1

λi(1 + O(δ−t)) .

However, this time, due to 1

λi<1for i∈ {1,2}, we obtain

Zδ

0

ρ(t) dt < ∞.

Using 1

λ1>0, we further compute that, on the separatrices corresponding to the positive eigenvalue,

Zδ

0

1 + 1

2|b(γ(t))|

ρ(t)dt=Zδ

0

t1

λ1−1

2λ1δ1

λ1

(1 + O(t)) dt < ∞

and Zδ

0

|b(γ(t))|

2dt=∞.

On the separatrices corresponding to the negative eigenvalue, we have, due to 1

λ2<0,

Zδ

0

1 + 1

2|b(γ(t))|

ρ(t)dt=Zδ

0

t1

λ2−1

2λ2δ1

λ2

(1 + O(t)) dt=∞

as well as

s(t) = Zt

0

ρ(s) ds=λ2δ1

λ2

λ2−1t1−1

λ2(1 + O(t))

and Zδ

0

1 + 1

2|b(γ(t))|

ρ(t)s(t) dt=Zδ

0

1

2 (λ2−1) (1 + O(t)) dt < ∞.

Hence, as a consequence of the criterions [15, Theorem 2.12 and Theorem 2.13], the hyperbolic

characteristic point xis reached with positive probability by the one-dimensional diﬀusion processes

induced on the separatrices. Thus, the canonical stochastic process started on the separatrices is

killed in ﬁnite time with positive probability.

STOCHASTIC PROCESSES ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 13

4. Stochastic processes on quadric surfaces in the Heisenberg group

Let Hbe the ﬁrst Heisenberg group, that is, the Lie group obtained by endowing R3with the group

law, expressed in Cartesian coordinates,

(x1, y1, z1)∗(x2, y2, z2) = x1+x2, y1+y2, z1+z2+1

2(x1y2−x2y1).

On H, we consider the two left-invariant vector ﬁelds

X=∂

∂x −y

2

∂

∂z and Y=∂

∂y +x

2

∂

∂z ,

and the contact form

ω= dz−1

2(xdy−ydx).

We note that the vector ﬁelds Xand Yspan the contact distribution Dcorresponding to ω, that

they are orthonormal with respect to the smooth ﬁbre inner product gon Dgiven by

g(x,y,z)= dx⊗dx+ dy⊗dy ,

and that

dω|D=−dx∧dy=−volg.

Therefore, the Heisenberg group Hunderstood as the three-dimensional contact sub-Riemannian

manifold (R3, D, g)falls into our setting, with X1=X,X2=Yand the Reeb vector ﬁeld

X0=∂

∂z = [X1, X2].

In Section 4.1 and in Section 4.2, we discuss paraboloids and ellipsoids of revolution admitting one

or two characteristic points, respectively, which are elliptic and of focus type. For these examples,

the characteristic foliations can be described by logarithmic spirals in R2lifted to the paraboloids

and spirals between the poles on the ellipsoids, which are loxodromes, also called rhumb lines, on

spheres. The induced stochastic processes are the Bessel process of order 3for the paraboloids and

Legendre-like processes for the ellipsoids moving along the leaves of the characteristic foliation.

In Section 4.3, we consider hyperbolic paraboloids where, depending on a parameter, the unique

characteristic point is either of saddle type or of node type, and we analyse the induced stochastic

processes on the separatrices.

4.1. Paraboloid of revolution. For a∈R, let Sbe the Euclidean paraboloid of revolution given

by the equation z=a(x2+y2)for Cartesian coordinates (x, y, z)in the Heisenberg group H. This

corresponds to the surface given by (1.4) with u:R3→Rdeﬁned as

u(x, y, z) = z−ax2+y2.

We compute

X0u≡1,(X1u) (x, y, z) = −2ax −y

2and (X2u) (x, y, z) = −2ay +x

2,

which yields

(4.1) ((X1u)(x, y, z))2+ ((X2u)(x, y, z))2=1

41 + 16a2x2+y2.

Thus, the origin of R3is the only characteristic point on the paraboloid S. It is elliptic and of focus

type because X0u≡1and

(Hess u)J≡ 1

22a

−2a1

2!

14 D. BARILARI, U. BOSCAIN, D. CANNARSA, K. HABERMANN

has eigenvalues 1

2±2ai. On S\Γ(S), the vector ﬁeld b

XSdeﬁned by (1.7) can be expressed as

(4.2) b

XS=1

p(1 + 16a2) (x2+y2)(x−4ay)∂

∂x + (y+ 4ax)∂

∂y + 2ax2+y2∂

∂z .

Changing to cylindrical coordinates (r, θ, z)for R3\ {0}with r > 0,θ∈[0,2π),z∈Rand using

r∂

∂r =x∂

∂x +y∂

∂y as well as ∂

∂θ =−y∂

∂x +x∂

∂y ,

the expression (4.2) for the vector ﬁeld b

XSsimpliﬁes to

b

XS=1

√1 + 16a2∂

∂r +4a

r

∂

∂θ + 2ar ∂

∂z .

From (4.1), we further obtain that the function b:S\Γ(S)→Rdeﬁned by (1.8) can be written as

b(r, θ, z) = 1

√1 + 16a2

2

r.

Characteristic foliation. The characteristic foliation induced on the paraboloid Sof revolution by

the contact structure Dof the Heisenberg group His described through the integral curves of the

vector ﬁeld b

XS, cf. Figure 4.1. Its integral curves are spirals emanating from the origin which can

be indexed by ψ∈[0,2π)and parameterised by s∈(0,∞)as follows

(4.3) s7→ s

√1 + 16a2,4aln s

√1 + 16a2+ψ, as2

1 + 16a2.

By construction, the vector ﬁeld b

XSis a unit vector ﬁeld with respect to each metric induced on

the surface Sfrom Riemannian approximations of the Heisenberg group. In particular, it follows

that the parameter s∈(0,∞)describes the arc length along the spirals (4.3).

Figure 4.1. Characteristic foliation described by logarithmic spirals

STOCHASTIC PROCESSES ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 15

Remark 4.1. The spirals on Sdeﬁned by (4.3) are logarithmic spirals in R2lifted to the paraboloid

of revolution. In polar coordinates (r, θ)for R2, a logarithmic spiral can be written as

(4.4) r= ek(θ+θ0)for k∈R\ {0}and θ0∈[0,2π).

Therefore, the spirals in (4.3) correspond to lifts of logarithmic spirals (4.4) with k=1

4a. The arc

length s∈(0,∞)of a logarithmic spiral (4.4) measured from the origin satisﬁes

s=r1 + 1

k2r ,

which for k=1

4ayields s=√1 + 16a2r. Note that this is the same relation between arc length and

radial distance as obtained for integral curves (4.3) of the vector ﬁeld b

XS. For further information

on logarithmic spirals, see e.g. Zwikker [29, Chapter 16].

Using the spirals (4.3) which describe the characteristic foliation on the paraboloid of revolution,

we introduce coordinates (s, ψ)with s > 0and ψ∈[0,2π)on the surface S\Γ(S). The vector ﬁeld

b

XSon S\Γ(S)and the function b:S\Γ(S)→Rare then given by

b

XS=∂

∂s and b(s, ψ) = 2

s.

Thus, the canonical stochastic process induced on S\Γ(S)has generator

1

2∆0=1

2b

X2

S+bb

XS=1

2

∂2

∂s2+1

s

∂

∂s .

This gives rise to a Bessel process of order 3which out of all the spirals (4.3) describing the

characteristic foliation on Sstays on the unique spiral passing through the chosen starting point

of the induced stochastic process. In agreement with Theorem 1.3, the origin is indeed inaccessible

for this stochastic process because a Bessel process of order 3with positive starting point remains

positive almost surely. It arises as the radial component of a three-dimensional Brownian motion,

and it is equal in law to a one-dimensional Brownian motion started on the positive real line and

conditioned to never hit the origin. We further observe that the operator ∆0coincides with the

radial part of the Laplace–Beltrami operator for a quadratic cone, cf. [9, 10] for α=−2, where

the self-adjointness of ∆0is also studied.

As the limiting operator ∆0does not depend on the parameter a∈R, the behaviour described

above is also what we encounter on the plane {z= 0}in the Heisenberg group H, where the

spirals (4.3) degenerate into rays emanating from the origin. We note that the stochastic process

induced by 1

2∆0on the rays diﬀers from the singular diﬀusion introduced by Walsh [28] on the

same type of structure, but that it falls into the setting of Chen and Fukushima [14].

4.2. Ellipsoid of revolution. For a, c ∈Rpositive, we study the Euclidean spheroid, also called

ellipsoid of revolution, in the Heisenberg group Hgiven by the equation

x2

a2+y2

a2+z2

a2c2= 1

in Cartesian coordinates (x, y, z ). To shorten the subsequent expressions, we choose u:R3→R

deﬁning the Euclidean spheroid Sthrough (1.4) to be given by

u(x, y, z) = x2+y2+z2

c2−a2.

Proceeding as in the previous example, we ﬁrst obtain

(X0u) (x, y, z) = 2z

c2

16 D. BARILARI, U. BOSCAIN, D. CANNARSA, K. HABERMANN

as well as

(X1u) (x, y, z)=2x−yz

c2and (X2u) (x, y, z) = 2y+xz

c2,

which yields

(4.5) ((X1u)(x, y, z))2+ ((X2u)(x, y, z))2=x2+y24 + z2

c4.

This implies the north pole (0,0, ac)and the south pole (0,0,−ac)are the only two characteristic

points on the spheroid S. We further compute that

(4.6) (X2u)X1−(X1u)X2=2y+xz

c2∂

∂x −2x−yz

c2∂

∂y −x2+y2∂

∂z .

Using adapted spheroidal coordinates (θ, ϕ)for S\Γ(S)with θ∈(0, π )and ϕ∈[0,2π), which are

related to the coordinates (x, y, z )by

x=asin(θ) cos(ϕ), y =asin(θ) sin(ϕ), z =ac cos(θ),

we have

asin(θ)

c

∂

∂θ =xz

c2

∂

∂x +yz

c2

∂

∂y −x2+y2∂

∂z and

∂

∂ϕ =−y∂

∂x +x∂

∂y .

It follows that (4.6) on the surface S\Γ(S)simpliﬁes to

(X2u)X1−(X1u)X2=asin(θ)

c

∂

∂θ −2∂

∂ϕ ,

whereas (4.5) on S\Γ(S)rewrites as

((X1u)(θ, ϕ))2+ ((X2u)(θ, ϕ))2=a2(sin(θ))2 4 + a2(cos(θ))2

c2!.

This shows that the vector ﬁeld b

XSon S\Γ(S)deﬁned by (1.7) is given as

(4.7) b

XS=1

q4c2+a2(cos(θ))2∂

∂θ −2c

asin(θ)

∂

∂ϕ .

For the function b:S\Γ(S)→Rdeﬁned by (1.8), we further obtain that

(4.8) b(θ, ϕ) = 2 cot(θ)

q4c2+a2(cos(θ))2.

As in the preceding example, in order to understand the canonical stochastic process induced by

the operator 1

2∆0deﬁned through (1.9), we need to express the vector ﬁeld b

XSand the function b

in terms of the arc length along the integral curves of b

XS. Since both b

XSand bare invariant under

rotations along the azimuthal angle ϕ, this amounts to changing coordinates on the spheroid S

from (θ, ϕ)to (s, ϕ)where s=s(θ)is uniquely deﬁned by requiring that

∂

∂s =1

q4c2+a2(cos(θ))2∂

∂θ −2c

asin(θ)

∂

∂ϕ and s(0) = 0 .

This corresponds to

(4.9) dθ

ds=1

q4c2+a2(cos(θ))2,

STOCHASTIC PROCESSES ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 17

which together with s(0) = 0 yields

s(θ) = Zθ

0q4c2+a2(cos(τ))2dτ=Zθ

0q(4c2+a2)−a2(sin(τ))2dτfor θ∈(0, π).

Hence, the arc length salong the integral curves of b

XSis given in terms of the polar angle θas a

multiple of an elliptic integral of the second kind. Consequently, the question if θcan be expressed

explicitly in terms of sis open. However, for our analysis, it is suﬃcient that the map θ7→ s(θ)is

invertible and that (4.8) as well as (4.9) then imply

b(s, ϕ) = 2 cot (θ(s)) dθ

ds.

Therefore, using the coordinates (s, ϕ), the operator 1

2∆0on S\Γ(S)can be expressed as

1

2∆0=1

2

∂2

∂s2+cot (θ(s)) dθ

ds∂

∂s ,

which depends on the constants a, c ∈Rthrough (4.9). Without the Jacobian factor dθ

dsappearing

in the drift term, the canonical stochastic process induced by the operator 1

2∆0and moving along

the leaves of the characteristic foliation would be a Legendre process, that is, a Brownian motion

started inside an interval and conditioned not to hit either endpoint of the interval. The reason

for the appearance of the additional factor dθ

dsis that the integral curves of b

XSconnecting the

two characteristic points are spirals and not just great circles. For some further discussions on the

characteristic foliation of the spheroid, see the subsequent Remark 4.3.

The emergence of an operator which is almost the generator of a Legendre process moving along

the leaves of the characteristic foliation motivates the search for a surface in a three-dimensional

contact sub-Riemannian manifold where we do exhibit a Legendre process moving along the leaves

of the characteristic foliation induced by the contact structure. This is achieved in Section 5.1.

Remark 4.2. The northern hemisphere of the spheroid could equally be deﬁned by the function

u(x, y, z) = z−cpa2−x2−y2.

With this choice we have X0u≡1. We further obtain

((Hess u) (0,0, ac)) J= 1

2−c

a

c

a

1

2!,

whose eigenvalues are 1

2±c

ai. A similar computation on the southern hemisphere implies that both

characteristic points are elliptic and of focus type. Thus, by Theorem 1.3, the stochastic process

with generator 1

2∆0hits neither the north pole nor the south pole, and it induces a one-dimensional

process on the unique leaf of the characteristic foliation picked out by the starting point.

Remark 4.3. With respect to the Euclidean metric h·,·i on R3, we have for the adapted spheroidal

coordinates (θ, ϕ)of S\Γ(S)as above that

∂

∂θ ,∂

∂θ =a2(cos(θ))2+a2c2(sin(θ))2and ∂

∂ϕ ,∂

∂ϕ =a2(sin(θ))2.

It follows that the angle αformed by the vector ﬁeld b

XSgiven in (4.7) and the azimuthal direction

satisﬁes

cos (α(θ, ϕ)) = −2c

qa2(cos(θ))2+a2c2(sin(θ))2+ 4c2

.

Notably, on spheres, that is, if c= 1, the angle αis constant everywhere. Hence, the integral curves

of b

XSconsidered as Euclidean curves on an Euclidean sphere are loxodromes, cf. Figure 4.2, which

18 D. BARILARI, U. BOSCAIN, D. CANNARSA, K. HABERMANN

are also called rhumb lines. They are related to logarithmic spirals through stereographic projection.

Loxodromes arise in navigation by following a path with constant bearing measured with respect

to the north pole or the south pole, see Carlton-Wippern [13].

Figure 4.2. Characteristic foliation on spheres described by loxodromes

4.3. Hyperbolic paraboloid. For a∈Rpositive and such that a6=1

2, we consider the Euclidean

hyperbolic paraboloid Sin the Heisenberg group Hgiven by (1.4) with u:R3→Rdeﬁned as

u(x, y, z) = z−axy ,

for Cartesian coordinates (x, y, z ). We compute

(4.10) X0u≡1,(X1u) (x, y, z) = −ay −y

2as well as (X2u) (x, y, z) = −ax +x

2,

and further that

(4.11) (Hess u)J≡ 1

2−a0

01

2+a!.

Due to

((X1u)(x, y, z))2+ ((X2u)(x, y, z))2=1

2−a2

x2+1

2+a2

y2,

the hyperbolic paraboloid Shas the origin of R3as its unique characteristic point. By (4.11), this

characteristic point is elliptic and of node type if 0<a< 1

2, and hyperbolic and therefore of saddle

type if a > 1

2. The reason for having excluded the case a=1

2right from the beginning is that it

gives rise to a line of degenerate characteristic points.

We note that the x-axis and the y-axis lie in the hyperbolic paraboloid S. From (4.10), we see that

the positive and negative x-axis as well as the positive and negative y-axis are integral curves of the

vector ﬁeld b

XSon S\Γ(S). In the following, we restrict our attention to studying the behaviour

of the canonical stochastic process on these integral curves, which nevertheless nicely illustrates

Theorem 1.3 and Proposition 1.4.

STOCHASTIC PROCESSES ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 19

We start by analysing the one-dimensional diﬀusion process induced on the positive y-axis γ+

y,

which by symmetry is equal in law to the process induced on the negative y-axis. For all positive

a∈Rwith a6=1

2, we have

b

XS|γ+

y=∂

∂y ,

implying that the arc length s > 0along γ+

yis given by s=y. This yields, for all s > 0,

bγ+

y(s)=1

1

2+as.

Thus, the one-dimensional diﬀusion process on γ+

yinduced by 1

2∆0has generator

1

2

∂2

∂s2+1

(1 + 2a)s

∂

∂s ,

which gives rise to a Bessel process of order 1 + 2

1+2a. If started at a point with positive value this

diﬀusion process stays positive for all times almost surely if 1 + 2

1+2a>2whereas it hits the origin

with positive probability if 1 + 2

1+2a<2. This is consistent with Theorem 1.3 and Proposition 1.4

because for a > 1

2the positive y-axis is a separatrix for the hyperbolic characteristic point at the

origin and

2<1 + 2

1+2aif 0< a < 1

2as well as 2>1 + 2

1+2aif a > 1

2.

Some more care is needed when studying the diﬀusion process induced on the positive x-axis γ+

x.

As before, this process is equal in law to the process induced on the negative x-axis. We obtain

b

XS|γ+

x=(∂

∂x if 0< a < 1

2

−∂

∂x if a > 1

2

as well as, for x > 0,

b(x, 0,0) =

1

1

2−axif 0<a< 1

2

−1

1

2−axif a > 1

2

.

It follows that the one-dimensional diﬀusion process on γ+

xinduced by 1

2∆0has generator

1

2

∂2

∂x2+1

(1 −2a)x

∂

∂x .

This yields a Bessel process of order 1 + 2

1−2a. In agreement with Theorem 1.3 and Proposition 1.4,

if started at a point with positive value this process never reaches the origin if 0<a< 1

2which

ensures 1 + 2

1−2a>3, whereas the process reaches the origin with positive probability if a > 1

2as

this corresponds to 1 + 2

1−2a<1.

5. Stochastic processes on canonical surfaces in SU(2) and SL(2,R)

In Section 4.1, we establish that for a paraboloid of revolution embedded in the Heisenberg group H,

the operator 1

2∆0induces a Bessel process of order 3moving along the leaves of the characteristic

foliation, which is described by lifts of logarithmic spirals emanating from the origin. As discussed

in Revuz and Yor [24, Chapter VIII.3], the Legendre processes and the hyperbolic Bessel processes

arise from the same type of Girsanov transformation as the Bessel process, where these three cases

only diﬀer by the sign of a parameter. We further recall that in Section 4.2 we encounter a canonical

20 D. BARILARI, U. BOSCAIN, D. CANNARSA, K. HABERMANN

stochastic process which is almost a Legendre process moving along the leaves of the characteristic

foliation induced on a spheroid in the Heisenberg group H. This motivates the search for surfaces

in three-dimensional contact sub-Riemannian manifolds where the canonical stochastic process is

a Legendre process of order 3or a hyperbolic Bessel process of order 3moving along the leaves of

the characteristic foliation.

We consider surfaces in the Lie groups SU(2) and SL(2,R)endowed with standard sub-Riemannian

structures. Together with the Heisenberg group, these sub-Riemannian geometries play the role of

model spaces for three-dimensional contact sub-Riemannian manifolds. In the ﬁrst two subsections,

we ﬁnd, by explicit computations, the canonical stochastic processes induced on certain surfaces

in these groups, when expressed in convenient coordinates. The last subsection proposes a uniﬁed

geometric description, justifying the choice of our surfaces.

5.1. Special unitary group SU(2).One obstruction to recovering Legendre processes moving

along the characteristic foliation in Section 4.2 is that the characteristic foliation of a spheroid in

the Heisenberg group is described by spirals connecting the north pole and the south pole instead of

great circles. This is the reason for considering S2as a surface embedded in SU(2) 'S3understood

as a contact sub-Riemannian manifold because this gives rise to a characteristic foliation on S2

described by great circles.

The special unitary group SU(2) is the Lie group of 2×2unitary matrices of determinant 1, that

is,

SU(2) = z+wiy+xi

−y+xiz−wi:x, y, z, w ∈Rwith x2+y2+z2+w2= 1,

with the group operation being given by matrix multiplication. Using the Pauli matrices

σ1=0 1

1 0, σ2=0−i

i 0 and σ3=1 0

0−1,

we identify SU(2) with the unit quaternions, and hence also with S3, via the map

z+wiy+xi

−y+xiz−wi7→ zI2+xiσ1+yiσ2+wiσ3.

The Lie algebra su(2) of SU(2) is the algebra formed by the 2×2skew-Hermitian matrices with

trace zero. A basis for su(2) is {iσ1

2,iσ2

2,iσ3

2}and the corresponding left-invariant vector ﬁelds on

the Lie group SU(2) are

U1=1

2−x∂

∂z +z∂

∂x −w∂

∂y +y∂

∂w ,

U2=1

2−y∂

∂z +w∂

∂x +z∂

∂y −x∂

∂w ,

U3=1

2−w∂

∂z −y∂

∂x +x∂

∂y +z∂

∂w ,

which satisfy the commutation relations [U1, U2] = −U3,[U2, U3] = −U1and [U3, U1] = −U2. Thus,

any two of these three left-invariant vector ﬁelds give rise to a sub-Riemannian structure on SU(2).

To streamline the subsequent computations, we choose k∈Rwith k > 0and equip SU(2) with the

sub-Riemannian structure obtained by setting X1= 2kU1,X2= 2kU2and by requiring (X1, X2)

to be an orthonormal frame for the distribution Dspanned by the vector ﬁelds X1and X2. The

appropriately normalised contact form ωfor the contact distribution Dis

ω=1

2k2(wdz+ydx−xdy−zdw)

STOCHASTIC PROCESSES ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 21

and the associated Reeb vector ﬁeld X0satisﬁes

X0= [X1, X2] = −4k2U3= 2k2w∂

∂z +y∂

∂x −x∂

∂y −z∂

∂w .

In SU(2), we consider the surface Sgiven by the function u: SU(2) →Rdeﬁned by

u(x, y, z, w) = w .

The surface Sis isomorphic to S2because

S= z y +xi

−y+xiz:x, y, z ∈Rwith x2+y2+z2= 1.

We compute

(X0u)(x, y, z, w) = −2k2z , (X1u)(x, y, z , w) = ky and (X2u)(x, y, z, w ) = −kx ,

which yields

((X1u)(x, y, z, w))2+ ((X2u)(x, y, z , w))2=k2x2+y2.

Due to x2+y2+z2= 1, it follows that a point on Sis characteristic if and only if z=±1. Thus,

the characteristic points on Sare the north pole (0,0,1) and the south pole (0,0,−1). The vector

ﬁeld b

XSon S\Γ(S)deﬁned by (1.7) is given as

(5.1) b

XS=k

px2+y2x2+y2∂

∂z −xz ∂

∂x −yz ∂

∂y ,

and for the function b:S\Γ(S)→Rdeﬁned by (1.8), we obtain

(5.2) b(x, y, z) = −2kz

px2+y2.

We now change coordinates for S\Γ(S)from (x, y, z)with x2+y2+z2= 1 and z6=±1to (θ, ϕ)

with θ∈(0,π

k)and ϕ∈[0,2π)by

x= sin(kθ) cos(ϕ), y = sin(kθ) sin(ϕ)and z= cos(kθ).

We note that ∂

∂θ =kcos(kθ) cos(ϕ)∂

∂x +kcos(kθ) sin(ϕ)∂

∂y −ksin(kθ)∂

∂z

as well as

xz = sin(kθ) cos(kθ) cos(ϕ), yz = sin(kθ) cos(kθ) sin(ϕ)and px2+y2= sin(kθ).

This together with (5.1) and (5.2) implies that

b

XS=−∂

∂θ and b(θ, ϕ) = −2kcot(kθ).

We deduce that the integral curves of b

XSare great circles on Sand that

1

2∆0=1

2

∂2

∂θ2+kcot(kθ)∂

∂θ ,

which indeed, on each great circle, induces a Legendre process of order 3on the interval (0,π

k). These

processes ﬁrst appeared in Knight [23] as so-called taboo processes and are obtained by conditioning

Brownian motion started inside the interval (0,π

k)to never hit either of the two boundary points,

see Bougerol and Defosseux [12, Section 5.1]. As discussed in Itô and McKean [21, Section 7.15],

they also arise as the latitude of a Brownian motion on the three-dimensional sphere of radius 1

k.

22 D. BARILARI, U. BOSCAIN, D. CANNARSA, K. HABERMANN

5.2. Special linear group SL(2,R).The appearance of the Bessel process on the plane {z= 0}in

the Heisenberg group Hand of the Legendre processes on a compactiﬁed plane in SU(2) understood

as a contact sub-Riemannian manifold suggests that the hyperbolic Bessel processes arise on planes

in the special linear group SL(2,R)equipped with a sub-Riemannian structure. This is indeed the

case if we consider the standard sub-Riemannian structures on SL(2,R)where the ﬂow of the Reeb

vector ﬁeld preserves the distribution and the ﬁbre inner product.

The special linear group SL(2,R)of degree two over the ﬁeld Ris the Lie group of 2×2matrices

with determinant 1, that is,

SL(2,R) = x y

z w:x, y, z , w ∈Rwith xw −yz = 1,

where the group operation is taken to be matrix multiplication. The Lie algebra sl(2,R)of SL(2,R)

is the algebra of traceless 2×2real matrices. A basis of sl(2,R)is formed by the three matrices

p=1

21 0

0−1, q =1

20 1

1 0and j=1

20 1

−1 0,

whose corresponding left-invariant vector ﬁelds on SL(2,R)are

X=1

2x∂

∂x −y∂

∂y +z∂

∂z −w∂

∂w ,

Y=1

2y∂

∂x +x∂

∂y +w∂

∂z +z∂

∂w ,

K=1

2−y∂

∂x +x∂

∂y −w∂

∂z +z∂

∂w .

These vector ﬁelds satisfy the commutation relations [X, Y ] = K,[X, K ] = Yand [Y, K] = −X.

For k∈Rwith k > 0, we equip SL(2,R)with the sub-Riemannian structure obtain by considering

the distribution Dspanned by X1= 2kX and X2= 2kY as well as the ﬁbre inner product uniquely

given by requiring (X1, X2)to be a global orthonormal frame. The appropriately normalised contact

form corresponding to this choice is

ω=1

4k2(zdx+wdy−xdz−ydw),

and the Reeb vector ﬁeld X0associated with the contact form ωsatisﬁes

X0= [X1, X2]=4k2K= 2k2−y∂

∂x +x∂

∂y −w∂

∂z +z∂

∂w .

The plane in SL(2,R)passing tangentially to the contact distribution through the identity element

is the surface Sgiven as (1.4) by the function u: SL(2,R)→Rdeﬁned by

u(x, y, z, w) = y−z .

Observe that, on S, we have the relation xw = 1 + y2≥1. Therefore, if a point (x, y, z , w)lies on

the surface Sthen so does the point (−x, y, z , −w), and neither xnor wcan vanish on S. Thus,

the function u: SL(2,R)→Rinduces a surface consisting of two sheets. By symmetry, we restrict

our attention to the sheet containing the 2×2identity matrix, henceforth referred to as the upper

sheet. We compute

(X1u)(x, y, z, w) = −k(y+z)and (X2u)(x, y, z , w) = k(x−w),

as well as

(X0u)(x, y, z, w)=2k2(x+w).

STOCHASTIC PROCESSES ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 23

We note that

((X1u)(x, y, z, w))2+ ((X2u)(x, y, z , w))2=k2(y+z)2+k2(x−w)2

vanishes on Sif and only if y=z= 0 and x=w. From xw = 1 + y2, it follows that the surface S

admits the two characteristic points (1,0,0,1) and (−1,0,0,−1), that is, one unique characteristic

point on each sheet. Following Rogers and Williams [26, Section V.36], we choose coordinates (r, θ)

with r > 0and θ∈[0,2π)on the upper sheet of S\Γ(S)such that

x= cosh (kr) + sinh (kr) cos(θ),

w= cosh (kr)−sinh (kr) cos(θ),and

y= sinh (kr) sin(θ).

On the upper sheet of S\Γ(S), we obtain

(X1u)(r, θ) = −2ksinh (kr) sin(θ)and (X2u)(r, θ) = 2ksinh (kr) cos(θ),

which yields q((X1u)(r, θ))2+ ((X2u)(r, θ))2= 2ksinh (kr),

as well as

(X0u)(r, θ) = 4k2cosh (kr).

A direct computation shows that on the upper sheet of S\Γ(S), we have

b

XS=∂

∂r and b(r, θ)=2kcoth (kr),

which implies that

1

2∆0=1

2

∂2

∂r2+kcoth (kr)∂

∂r .

Hence, we recover all hyperbolic Bessel processes of order 3as the canonical stochastic processes

moving along the leaves of the characteristic foliation of the upper sheet of S\Γ(S), and similarly on

its lower sheet. For further discussions on hyperbolic Bessel processes, see Borodin [8], Gruet [20],

Jakubowski and Wiśniewolski [22], and Revuz and Yor [24, Exercise 3.19]. As for the Bessel process

of order 3and the Legendre processes of order 3, the hyperbolic Bessel processes of order 3can be

deﬁned as the radial component of Brownian motion on three-dimensional hyperbolic spaces.

5.3. A uniﬁed viewpoint. The surfaces considered in the last two examples together with the

plane {z= 0}in the Heisenberg group are particular cases of the following construction.

Let Gbe a three-dimensional Lie group endowed with a contact sub-Riemannian structure whose

distribution Dis spanned by two left-invariant vector ﬁelds X1and X2which are orthonormal for

the ﬁbre inner product gdeﬁned on D. Assume that the commutation relations between X1, X2

and the Reeb vector ﬁeld X0are given by, for some κ∈R,

[X1, X2] = X0,[X0, X1] = κX2,[X0, X2] = −κX1.

Under these assumptions the ﬂow of the Reeb vector ﬁeld X0preserves not only the distribution,

namely etX0

∗D=D, but also the ﬁbre inner product g. The examples presented in Section 4.1

and in Sections 5.1 and 5.2 satisfy the above commutation relations with κ= 0 in the Heisenberg

group, and for a parameter k > 0, with κ= 4k2in SU(2) and κ=−4k2in SL(2,R). These are

the three classes of model spaces for three-dimensional sub-Riemannian structures on Lie groups

with respect to local sub-Riemannian isometries, see for instance [3, Chapter 17] and [2] for more

details.

24 D. BARILARI, U. BOSCAIN, D. CANNARSA, K. HABERMANN

In each of the examples concerned, the surface Sthat we consider can be parameterised as

S={exp(x1X1+x2X2) : x1, x2∈R}

={exp(rcos θX1+rsin θX2) : r≥0, θ ∈[0,2π)}.

Observe that Sis automatically smooth, connected, and contains the origin of the group. Under

these assumptions, the sub-Riemannian structure is of type d⊕sin the sense of [3, Section 7.7.1],

and for θﬁxed, the curve r7→ exp(rcos θX1+rsin θX2)is a geodesic parameterised by length.

Hence, r≥0is the arc length parameter along the corresponding trajectory. It follows that the

surface Sis ruled by geodesics, each of them having vertical component of the initial covector equal

to zero. We refer to [3, Chapter 7] for more details on explicit expressions for sub-Riemannian

geodesics in these cases, see also [11].

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Davide Barilari, Université de Paris, Sorbonne Université, CNRS, Institut de Mathématiques de

Jussieu-Paris Rive Gauche, F-75013 Paris, France.

E-mail address:davide.barilari@imj-prg.fr

Ugo Boscain, CNRS, Laboratoire Jacques-Louis Lions, Sorbonne Université, Université de Paris,

Inria, F-75005 Paris, France.

E-mail address:ugo.boscain@upmc.fr

Daniele Cannarsa, Université de Paris, Sorbonne Université, CNRS, Inria, Institut de Mathématiques

de Jussieu-Paris Rive Gauche, F-75013 Paris, France.

E-mail address:daniele.cannarsa@imj-prg.fr

Karen Habermann, Laboratoire Jacques-Louis Lions, Sorbonne Université, Université de Paris,

CNRS, Inria, F-75005 Paris, France.

E-mail address:karen.habermann@upmc.fr