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# Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds

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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 ${\rm SU}(2)$ and ${\rm SL}(2,\mathbb{R})$ 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.
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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
DT 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 ωdω6= 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 xS
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 xSis 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
20.
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 :MR, 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={xM:u(x) = 0}for uC2(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 xSis a characteristic point if
and only if (X1u)(x)=(X2u)(x)=0, that is,
(1.5) xΓ(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 xΓ(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|ST 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
SX0to 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 ε0.
Theorem 1.1. For any twice diﬀerentiable function fC2
c(S\Γ(S)) compactly supported in
S\Γ(S), the functions εfconverge uniformly on S\Γ(S)to 0fas ε0.
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 ε0of 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
ε0Kε=b
XS(b)b2.
We now consider the canonical stochastic process on S\Γ(S)whose generator is 1
20. 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
20on 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
20and 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 κR,
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 κR. For κ6= 0, let kRwith 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) : rIand 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
20moving 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|S0ensures that X(x)TxSfor all
xS. 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)2X0,(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 ε0. 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:DDbe 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)) F1bF2,
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
ε1
2
and we introduce the vector ﬁelds E1and E2on S\Γ(S)given by
(2.8) E1=F1and E2=aεF2.
In the Riemannian manifold (S, gε), this yields the orthonormal frame (E1, E2)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+bF1(aε)
aεF1a2
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)is an orthonormal
frame for T(S\Γ(S)), we have
(2.9) ε|S\Γ(S)=E2
1+E2
2+ (divεE1)E1+ (divεE2)E2.
Let (ν1, ν2)denote the dual to the orthonormal frame (E1, E2 ). 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([E2, X]) .
This together with (2.8) and Lemma 2.1 implies that
divεE1=ν2([aεF2, F1]) = ν2(aε[F1, F2] + F1(aε)F2) = bF1(aε)
aε
as well as
divεE2=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+ bF1a2
ε
2a2
ε!F1a2
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
ε1
=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
εεas well as
F1(aε)
aεε|bF1(b)|.
Since uC2(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
ε0a2
ε= 0 ,lim
ε0
F1(aε)
aε
= 0 and lim
ε0a2
ε(bh(F1) + η(F1)) = 0 .
Let fC2
c(S\Γ(S)). We then have F1f, F2fC1
c(S\Γ(S)) and F2
1f, F 2
2fC0
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
ε0kεf0fk,S\Γ(S)= lim
ε0
a2
εF2
2fF1(aε)
aε
F1fa2
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), 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]=[F1, aεF2] = aε[F1, F2] + F1(aε)F2=aεb1E1+b+F1(aε)
aεE2.
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=F1b+F1(aε)
aεaεF2(aεb1)(aεb1)2b+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)
aεε|F1(bF1(b))|+ 2ε2b2(F1(b))2.
By passing to the limit ε0in (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
20on 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
20moves 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 xΓ(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:SRdoes not vanish near characteristic points. In
particular, we may and do choose the function uC2(M)deﬁning the surface Ssuch that X0u1
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:DDdeﬁned in (2.5)
has the matrix representation
J=01
1 0 ,
we see that
(Hess u)J=X1X2uX1X1u
X2X2uX2X1u.
The dynamics around the characteristic point xΓ(S)is uniquely determined by the eigenvalues
λ1and λ2of ((Hess u)(x))J. Since xΓ(S)is non-degenerate by assumption both eigenvalues are
non-zero, and due to X0u1in 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 xΓ(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 uC2(M)such that X0u1in 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:IδSfor 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 xΓ(S)be a non-degenerate characteristic point and suppose that uC2(M)
is chosen such that X0u1in a neighbourhood of x. For δ > 0, let γ:IδSbe 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 s0,
b(γ(s)) = 2
s+O(1) .
Proof. Since X0u1in a neighbourhood of x, we may suppose that δ > 0is chosen small enough
such that, for sIδ\ {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 s0,
∂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
s0
1
b(γ(s)) = 0 .
A Taylor expansion together with (3.2) and (3.3) then implies that, as s0,
1
b(γ(s)) =s
2+Os2,
which yields, for s0,
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 xΓ(S). For δ > 0, let γ:IδSbe an
integral curve of the vector ﬁeld b
XSwhich extends continuously to γ(0) = x. Assume uC2(M)is
chosen such that X0u1in 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 s0,
b(γ(s)) = 1
λis+O(1) .
Proof. As in the proof of Lemma 3.1, we obtain, for δ > 0small enough and sIδ\ {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 s0,
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 s0,
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 uC2(M)such that X0u1in 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 xΓ(S). For δ > 0, let γ: [0, δ ]S
be an integral curve of the vector ﬁeld b
XSextended continuously to x= lims0γ(s). Following
Cherny and Engelbert [15, Section 2.3], since the one-dimensional diﬀusion process on γinduced
by 1
20has 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 λRwith 0<λ<1such that, as s0,
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(δt)=δ
t1
λ
(1 + O(δt)) .
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 xΓ(S)was an arbitrary elliptic characteristic
point, the claimed result follows.
Proof of Proposition 1.4. We consider the stochastic process with generator 1
20on S\Γ(S)near
a hyperbolic point xΓ(S). Let γbe one of the four separatrices of xparameterised by arc length
s0and 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 s0,
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(δt)) .
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
λ11
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
λ21
2λ2δ1
λ2
(1 + O(t)) dt=
as well as
s(t) = Zt
0
ρ(s) ds=λ2δ1
λ2
λ21t11
λ2(1 + O(t))
and Zδ
0
1 + 1
2|b(γ(t))|
ρ(t)s(t) dt=Zδ
0
1
2 (λ21) (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(x1y2x2y1).
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= dz1
2(xdyydx).
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)= dxdx+ dydy ,
and that
dω|D=dxdy=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 aR, 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:R3Rdeﬁned as
u(x, y, z) = zax2+y2.
We compute
X0u1,(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 X0u1and
(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)(x4ay)
∂x + (y+ 4ax)
∂y + 2ax2+y2
∂z .
Changing to cylindrical coordinates (r, θ, z)for R3\ {0}with r > 0,θ[0,2π),zRand 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 kR\ {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
20=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 aR, 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
20on 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:R3R
deﬁning the Euclidean spheroid Sthrough (1.4) to be given by
u(x, y, z) = x2+y2+z2
c2a2.
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)=2xyz
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 2xyz
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
20deﬁned 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
20on S\Γ(S)can be expressed as
1
20=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
20and 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) = zcpa2x2y2.
With this choice we have X0u1. We further obtain
((Hess u) (0,0, ac)) J= 1
2c
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
20hits 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 ,·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 aRpositive and such that a6=1
2, we consider the Euclidean
hyperbolic paraboloid Sin the Heisenberg group Hgiven by (1.4) with u:R3Rdeﬁned as
u(x, y, z) = zaxy ,
for Cartesian coordinates (x, y, z ). We compute
(4.10) X0u1,(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
2a0
01
2+a!.
Due to
((X1u)(x, y, z))2+ ((X2u)(x, y, z))2=1
2a2
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
aRwith 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
20has 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
2axif 0<a< 1
2
1
1
2axif a > 1
2
.
It follows that the one-dimensional diﬀusion process on γ+
xinduced by 1
20has generator
1
2
2
∂x2+1
(1 2a)x
∂x .
This yields a Bessel process of order 1 + 2
12a. 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
12a>3, whereas the process reaches the origin with positive probability if a > 1
2as
this corresponds to 1 + 2
12a<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
20induces 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+xizwi: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=0i
i 0 and σ3=1 0
01,
we identify SU(2) with the unit quaternions, and hence also with S3, via the map
z+wiy+xi
y+xizwi7→ 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
2x
∂z +z
∂x w
∂y +y
∂w ,
U2=1
2y
∂z +w
∂x +z
∂y x
∂w ,
U3=1
2w
∂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 kRwith 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+ydxxdyzdw)
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() cos(ϕ), y = sin(kθ) sin(ϕ)and z= cos().
We note that
∂θ =kcos(kθ) cos(ϕ)
∂x +kcos(kθ) sin(ϕ)
∂y ksin(kθ)
∂z
as well as
xz = sin() cos(kθ) cos(ϕ), yz = sin(kθ) cos() sin(ϕ)and px2+y2= sin().
This together with (5.1) and (5.2) implies that
b
XS=
∂θ and b(θ, ϕ) = 2kcot().
We deduce that the integral curves of b
XSare great circles on Sand that
1
20=1
2
2
∂θ2+kcot()
∂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
01, 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
2y
∂x +x
∂y w
∂z +z
∂w .
These vector ﬁelds satisfy the commutation relations [X, Y ] = K,[X, K ] = Yand [Y, K] = X.
For kRwith 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+wdyxdzydw),
and the Reeb vector ﬁeld X0associated with the contact form ωsatisﬁes
X0= [X1, X2]=4k2K= 2k2y
∂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) = yz .
Observe that, on S, we have the relation xw = 1 + y21. 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(xw),
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(xw)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
20=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 κR,
[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, x2R}
={exp(rcos θX1+rsin θX2) : r0, 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 dsin 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, r0is 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
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