The intrinsic hypoelliptic Laplacian on sub-Riemannian manifolds Ugo Boscain (CNRS, CMAP, Ecole Polytechnique, Paris) Robert Neel , (Lehigh University) Luca Rizzi , (CMAP, Ecole Polytechnique, Paris) March 11, 2015
Plan of the Talk definition of the Laplacian in a Riemannian manifold macroscopic Laplacian ∆ (as divergence of the gradient) microscopic Laplacian L (as limit of geodes. rand. walks) we have that ∆ = L up to constants. sub-Riemannian manifolds ( M, � , g ) macroscopic Laplacian ∆ ω (as divergence of the horizontal gradient) → we need a volume ω microscopic Laplacian L ▽ (as limit of geodesic random walks) → we need a measure on the space of geodesics. We will focus on random walks induced by a complement ▽ to � . ▽ ( q ) � ( q ) Q1 Under which conditions on ( ω, ▽ ) we have ∆ ω = L ▽ ? Q2 Since there exists a canonical choice of ω (the Popp volume), does it exist ▽ s.t. ∆ ω = L ▽ ? Is it unique?
The Riemannian macroscopic Laplacian M : smooth, connected, orientable, geodesically-complete n -dim Riemannian manifold. The infinitesimal conservation condition for a smooth scalar quantity φ (the temperature, a concentration, the probability density of a randomly-moving particle, etc.) that flows via a flux F (which says how much of φ is crossing a unit of surface and in a unit of time) is expressed via the equation : ∂ t φ + div( F ) = 0 here div( · ) is computed w.r.t. the Riemannian volume R . If one postulates F = − grad( φ ), one obtains the Riemannian heat equation ∂ t φ = ∆ φ, (1) where ∆( · ) = div(grad( · )) , is the Laplace-Beltrami operator. Since equation (1) has been obtained thinking to φ as to a fluid without a microscopic structure in the following we refer to it as to the macroscopic heat equation and to the corresponding operator ∆ as to the macroscopic Laplacian .
Remarks The divergence form of ∆ with the geodesic completeness of the manifold implies that [Strichartz]: ∆ is self adjoint (evolution well defined in L 2 ( M, R )) it admits a smooth positive symmetric smooth heat kernel � φ ( q, t ) = P t ( q, ¯ q ) φ 0 (¯ q ) d R M If { X 1 , . . . X n } is a local orthonormal frame of the Riemannian manifold we have the nice formulas n n � � ( X 2 grad( φ ) = X i ( φ ) X i , ∆( φ ) = i + div( X i ) X i )( φ ) . i =1 i =1
The Riemannian microscopic view point We consider a particle that at time zero is in q 0 ; at time δt jumps on a point q 1 of the sphere of radius ε centered in q 0 , uniformly on the directions, by following a geodesic; at time 2 δt jumps on a point q 2 of the sphere of radius ε centered in q 1 uniformly on the directions, by following a geodesic; ....... q 2 M q 1 q 0
If φ is the density of probability of finding the particle in q we have that: how much φ is increasing at a point q in time δt is proportional to the difference between the average of φ in a sphere of radious ε centered in q and the value of φ ( q, t ). � � � � � φ ( q, t + δt ) − φ ( q, t ) = S n − 1 φ exp q ( ε, θ ) , t dθ − φ ( q, t ) . Dividing by δt we obtain φ ( q, t + δt ) − φ ( q, t ) = 1 � � � � � S n − 1 φ exp q ( ε, θ ) , t dθ − φ ( q, t ) . δt δt Taking the parabolic scale ( ↔ infinite velocity of propagation) δt = ε 2 /α , and letting δt → 0, ∂ t φ = Lφ (2) where α �� � � � Lφ ( q, t ) = lim S n − 1 φ exp q ( ε, θ ) , t dθ − φ ( q, t ) . (3) ε 2 ε → 0 In the following we refer to the equation (2) as to the “microscopic heat equation” and to L as to the “microscopic Laplacian”.
in Riemannian ∆ is proportional to L If we take α = 2 √ n , then we have ∆ = L. All of this can be viewed in the following way. On one side, the microscopic perspective is a good interpretation of the macroscopic heat equation. On the other side, the microscopic Laplacian L is a good operator because it is essentially self-adjoint with respect to a volume (the Riemannian one): we can study the evolution equation in L 2 ( M, ω ). Indeed we have convergence of the random process.
A remark on the Riemannian volume Recall that the Riemannian volume can be defined equivalently as: the volume R such that R ( X 1 , . . . , X n ) = 1, for any oriented local orthonormal frame { X 1 , . . . , X n } , as the n -dimensional Hausdorff or spherical-Hausdorff volume (up to some constant). The above construction gives an alternative way of characterizing the Riemannian volume: it is the unique volume (up to constant rescaling) such that the microscopic Laplacian can be written in divergence form. These facts are much less trivial in the sub-Riemannian context. Indeed in sub-Riemannian geometry there are several notions of intrinsic volumes and the definition of the microscopic Laplacian requires additional structure.
The Laplace Beltrami vs the sum of squarres Let X 1 , . . . X n be an orthonormal frame. We have two possibilities: geodesic random walk (Ito-on-manifolds view point): q 2 M q 1 q 0 we get L = ∆ = � n i =1 ( X 2 i + div( X i ) X i ) random walk along the integral curves of X 1 , . . . X n (Stratonovich view point): 1 4 q 2 q 1 M q 0 1 1 4 4 we get L = � n i =1 X 2 i (it is not intrinsic)
Sub-Riemannian manifolds Definition A Sub-Riemannian manifold is a triple ( M, � , g ), where ( i ) M is a smooth, connected, orientable manifold of dimension n ; ( ii ) � is a smooth distribution of constant rank k < n , i.e. a smooth map that associates to q ∈ M a k -dimensional subspace � q of T q M satisfying the H¨ ormander condition ( iii ) g q is a Riemannian metric on � q , that is smooth as function of q . x 3 � � � � x 2 x 1
→ Define � 1 ( q ) := � ( q ), � i +1 := � i + [ � i , � ]. If dim( � i ) , i = 1 , . . . , m do not depend on the point, it is called equiregular otherwise non-equiregular . → Locally, ( � , g ) can be given by assigning a set of k smooth vector fields (called a local orthonormal frame ) s.t. � q = span { X 1 ( q ) , . . . , X k ( q ) } , g q ( X i ( q ) , X j ( q )) = δ ij . (4)
The Carnot Carath´ eodory distance is � � T k k � � � � u 2 d ( q 1 , q 2 ) := inf { i ( t ) dt | γ (0) = q 1 , γ ( T ) = q 2 , ˙ γ ( t ) = u i ( t ) X i ( γ ( t )) } � 0 i =1 i =1 Thanks to the Hormander condition, the control system k � q ( t ) = ˙ u i ( t ) X i ( q ( t )) i =1 is completely controllable and ( M, d ) is a metric space having the same topology as the original topology of M (Chow theorem)
Candidate minimizers are computed via the Pontryagin Maximum Principle: Normal extremals Projections on the manifold of solutions of the Hamiltonian system having as Hamiltonian k H ( q, p ) = 1 � � p, X i ( q ) � 2 2 i Arclength parameterized normal extremals belongs to H = 1 2 Abnormal extremals satisfying � p ( t ) , X i ( q ( t )) � ≡ 0 , i = 1 . . . k. → normal extremals are geodesics → abnormal extremals can be or not geodesics
The Macroscopic Laplacian Fix a volume ω is fixed (the intrinsic definition of ω is a subtle question). The conservation of the heat is written similarly to the Riem. case. ∂ t φ + div ω ( F ) = 0 But now one should postulate that the flux is proportional to the horizontal gradient. The horizontal gradient grad H ( · ) of a C ∞ function φ is defined similarly to the Riemannian gradient but is a vector field belonging to the distribution: g q ( v, grad H ( φ ) q ) = d q φ ( v ) , ∀ v ∈ � q We have then for the macroscopic heat equation in the sub-Riemannian context ∂ t φ = ∆ ω φ where ∆ ω ( · ) = div ω (grad H ( · )) is the horizontal Laplacian . In terms of a local orthonormal frame: k k � � ( X 2 grad H ( φ ) = X i ( φ ) X i , ∆ ω ( φ ) = i + div ω ( X i ) X i )( φ ) i =1 i =1
What is the right volume? In the Riemannian context, ∆ = L is correct only if ∆ = div R (grad( · )) Q What is the right volume in SRG playing the role of the Riemannian one? One needs some algorithm to assign (intrinsically), with any sub-Riemannian structure on M , a volume form ω M . Definition An intrinsic volume is a map that associates, with any (orientable) (sub)-Riemannian structure ( M, � , g ) a volume form ω M on M such that if φ : M → N is a sub-Riemannian isometry between ( M, � M , g M ) and ( N, � N , g N ), then φ ∗ ω N = ω M . Surprisingly, even in the Riemannian case, there are many intrinsic definitions of volume. The classical Riemannian volume is the unique volume form R such that R ( X 1 , . . . , X n ) = 1 for any orthonormal (oriented) frame. But R ′ ( X 1 , . . . , X n ) = 1 + κ 2 , where κ is the scalar curvature is intrinsic too. The first, loosely speaking, is more “intrinsic” than the second. In fact, it is true that both depend only on the metric invariant of the structure, but R ′ involves second order informations about the structure.
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