Hypoelliptic heat kernel on 3-step nilpotent Lie groups
aa r X i v : . [ m a t h . A P ] F e b Hypoelliptic heat kernel over 3-step nilpotentLie groups
Ugo Boscain CMAP, ´Ecole Polytechnique CNRS, Route de Saclay, 91128 Palaiseau Cedex, France - [email protected]
Jean-Paul Gauthier
Laboratoire LSIS, Universit´e de Toulon, France - [email protected]
Francesco Rossi
BCAM - Basque Center for Applied Mathematics,Bizkaia Technology Park, Basque Country, Spain - [email protected]
In this paper we provide explicitly the connection between the hypoelliptic heatkernel for some 3-step sub-Riemannian manifolds and the quartic oscillator. Westudy the left-invariant sub-Riemannian structure on two nilpotent Lie groups,namely the (2,3,4) group (called the Engel group) and the (2,3,5) group (called theCartan group or the generalized Dido problem). Our main technique is noncom-mutative Fourier analysis that permits to transform the hypoelliptic heat equationinto a one dimensional heat equation with a quartic potential.
The study of the properties of the heat kernel in a sub-Riemannian manifold drew an increasingattention since the pioneer work of H¨ormander [22]. Since then, many estimates and propertiesof the kernel in terms of the sub-Riemannian distance have been provided (see [5, 6, 15, 26, 32]and references therein). For some particular structures, it is moreover possible to find explicitexpressions of the hypoelliptic heat kernels. In general, this computation can be performed onlywhen the sub-Riemannian structure and the corresponding hypoelliptic heat operator presentsymmetry properties. For this reason, the most natural choice in this field is to considerinvariant operators defined on Lie groups. Results of this kind have been first provided in[16, 23] in the case of the 3D Heisenberg group. Afterwards, other explicit expressions havebeen found first for 2-step nilpotent free Lie groups (again in [16]) and then for general 2-stepnilpotent Lie groups (see [3, 9]). We provide in [1] the expressions of heat kernels for 2-stepgroups that are not nilpotent, namely SU (2) , SO (3) , SL (2) and the group of rototranslationsof the plane SE (2). For other examples, see e.g. [33, 34].In our paper we present the first results, to our knowledge, about the expression of thehypoelliptic heat kernel on the following 3-step Lie groups. The first one is the Engel group G , The first author has been supported by the European Research Council (ERC StG 2009 GeCoMethods). hat is the nilpotent group with growth vector (2 , , L = span { l , l , l , l } ,the generators of which satisfy[ l , l ] = l , [ l , l ] = l , [ l , l ] = [ l , l ] = [ l , l ] = [ l , l ] = 0 . The second example is the Cartan group G , that is the free nilpotent group with growthvector (2 , , L = span { l , l , l , l , l } and generators satisfy[ l , l ] = l , [ l , l ] = l , [ l , l ] = l , [ l , l ] = [ l , l ] = [ l , l ] = [ l , l ] = [ l , l ] = [ l , l ] = [ l , l ] = 0 . In both cases, we consider the heat equation with the so-called intrinsic hypoelliptic Lapla-cian ∆ H (in the sense of [1], see also Section 2.2.1) of the sub-Riemannian structure for which { g l , g l } ( g element of the group) is an orthonormal frame. As it has been proved in [1], since G and G are unimodular, then the intrinsic hypoelliptic Laplacian is the sum of the squareof the Lie derivative with respect to the vector fields g l , g l .One interesting feature of these two sub-Riemannian problems is that they present abnormalminimizers (see [27, 28]) and it is known that in both cases ∆ H is not analytic hypoelliptic[8]. Hence, for these two examples the Tr`eves conjecture holds. Having information about theexpression of the heat kernel can help for further investigations in this direction.Any other left-invariant sub-Riemannian structure of rank 2 on these groups is indeedisometric to the ones we study in this paper, see [27, 28]. Moreover, notice that the sub-Riemannian structures we study on G and G are local approximations (nilpotentizations,see [18]) of arbitrary sub-Riemannian structures at regular points with growth vector (2 , , , , G and G provide approximations of theheat kernels at these points.The goal of this paper is to transform the hypoelliptic heat equations on these Lie groupsinto a family of elliptic heat equations on R , depending on one parameter. To this purpose,we apply the method developed in [1], based on the Generalized Fourier Transform (GFT forshort), and hence on representation theory of these groups (see [11, p. 333–338]).Applying the GFT to the original equation, we get an evolution equation on the Hilbertspace where representations act. For both examples, this is the heat equation over R withquartic potential, the so-called quartic oscillator (see [10, 29]), for which no general explicitsolution is known. Notice that the connection between the quartic oscillator and degenerateelliptic operators has been already noted by previously (see [17]).It is clearly possible to use numerical approximations of the evolution equation with quarticpotential (for which a huge amount of literature is available) to find numerical approximationsof the hypoelliptic heat kernel. However, this analysis is outside the aims of this paper.The organization of the paper is the following. In Section 2 we recall the main definitionsfrom sub-Riemannian geometry, in particular for invariant structures on Lie groups. We thenrecall the definition of the Generalized Fourier Transform and its main properties. Finally, we We recall that Tr`eves conjectured in [31] that the existence of abnormal minimizers on a sub-Riemannianmanifold is equivalent to the loss of analytic-hypoellipticity of the sub-Laplacian. G and G , theiralgebras and their Euclidian and matrix presentations. We then recall results about theirrepresentations. We finally apply the method of computation of hypoelliptic heat kernels tothe two groups G and G , to find explicitly the connection between the heat kernels on thesegroups and the fundamental solution of the 1D heat equation with quartic potential. In this section we recall basic definitions from sub-Riemannian geometry, including the one ofthe intrinsic hypoelliptic Laplacian. Then we recall our method for computing the hypoellipticheat kernel in the case of unimodular Lie groups, using the GFT.
We start by recalling the definition of sub-Riemannian manifold.
Definition 1. A ( n, m ) -sub-Riemannian manifold is a triple ( M, N , g ) , where • M is a connected smooth manifold of dimension n ; • N is a smooth distribution of constant rank m < n satisfying the H¨ormander condition ,i.e. N is a smooth map that associates to q ∈ M a m -dim subspace N ( q ) of T q M and ∀ q ∈ M we have span { [ X , [ . . . [ X k − , X k ] . . . ]]( q ) | X i ∈ Vec H ( M ) } = T q M (1) where Vec H ( M ) denotes the set of horizontal smooth vector fields on M , i.e. Vec H ( M ) = { X ∈ Vec( M ) | X ( p ) ∈ N ( p ) ∀ p ∈ M } . • g q is a Riemannian metric on N ( q ) , that is smooth as function of q .When M is an orientable manifold, we say that the sub-Riemannian manifold is orientable. A Lipschitz continuous curve γ : [0 , T ] → M is said to be horizontal if ˙ γ ( t ) ∈ N ( γ ( t )) foralmost every t ∈ [0 , T ]. Given an horizontal curve γ : [0 , T ] → M , the length of γ is l ( γ ) = Z T q g γ ( t ) ( ˙ γ ( t ) , ˙ γ ( t )) dt. (2)The distance induced by the sub-Riemannian structure on M is the function d ( q , q ) = inf { l ( γ ) | γ (0) = q , γ ( T ) = q , γ horizontal } . (3)3he hypothesis of connectedness of M and the H¨ormander condition guarantee the finitenessand the continuity of d ( · , · ) with respect to the topology of M (Chow’s Theorem, see forinstance [2]). The function d ( · , · ) is called the Carnot-Charateodory distance and gives to M the structure of metric space (see [4, 18]).Locally, the pair ( N , g ) can be given by assigning a set of m smooth vector fields spanning N and that are orthonormal for g , i.e. N ( q ) = span { X ( q ) , . . . , X m ( q ) } , g q ( X i ( q ) , X j ( q )) = δ ij . (4)In this case, the set { X , . . . , X m } is called a local orthonormal frame for the sub-Riemannianstructure. When ( N , g ) can be defined as in (4) by m vector fields defined globally, we say thatthe sub-Riemannian manifold is trivializable .When the manifold is analytic and the orthonormal frame can be assigned through m analytic vector fields, we say that the sub-Riemannian manifold is analytic .We end this section with the definition of regular sub-Riemannian manifold. Definition 2.
Let N be a distribution and define through the recursive formula N := N , N n +1 := N n + [ N n , N ] . The small flag of N is the sequence N ⊂ N ⊂ . . . ⊂ N n ⊂ . . . A sub-Riemannian manifold is said to be regular if for each n = 1 , , . . . the dimension of N n ( q ) does not depend on the point q ∈ M . In this paper we always deal with sub-Riemannian manifolds that are orientable, analytic,trivializable and regular.
In this section we present a natural sub-Riemannian structure that can be defined on Liegroups. All along the paper, we use the notation for Lie groups of matrices. For general Liegroups, by gv with g ∈ G and v ∈ L , we mean ( L g ) ∗ ( v ) where L g is the left-translation of thegroup. Definition 3.
Let G be a Lie group with Lie algebra L and P ⊆ L a subspace of L satisfyingthe Lie bracket generating condition
Lie P := span { [ p , [ p , . . . , [ p n − , p n ]]] | p i ∈ P } = L . Endow P with a positive definite quadratic form h ., . i . Define a sub-Riemannian structure on G as follows: • the distribution is the left-invariant distribution N ( g ) := g P ; • the quadratic form g on N is given by g g ( v , v ) := h g − v , g − v i . n this case we say that ( G, N , g ) is a left-invariant sub-Riemannian manifold. Remark Observe that all left-invariant manifolds ( G, N , g ) are regular.In the following we define a left-invariant sub-Riemannian manifold choosing a set of m vectors { p , . . . , p m } that are an orthonormal basis for the subspace P ⊆ L with respect tothe metric defined in Definition 3, i.e. P = span { p , . . . , p m } and h p i , p j i = δ ij . We thushave N ( g ) = g P = span { g p , . . . , g p m } and g g ( g p i , g p j ) = δ ij . Hence, every left-invariantsub-Riemannian manifold is trivializable. In this section, we recall the definition of intrinsic hypoelliptic Laplacian given in [1] and basedon the Popp volume form in sub-Riemannian geometry presented in [25].Let ( M, N , g ) be a ( n, m )-sub-Riemannian manifold and { X , . . . X m } a local orthonormalframe. The operator obtained by the sum of squares of these vector fields is not a gooddefinition of hypoelliptic Laplacian, since it depends on the choice of the orthonormal frame(see for instance [1]).In sub-Riemannian geometry an invariant definition of hypoelliptic Laplacian is obtainedby computing the divergence of the horizontal gradient, like the Laplace-Beltrami operator inRiemannian geometry. Definition 5.
Let ( M, N , g ) be an orientable regular sub-Riemannian manifold. We define theintrinsic hypoelliptic Laplacian as ∆ H φ := div H grad H φ , where • the horizontal gradient is the unique operator grad H from C ∞ ( M ) to Vec H ( M ) satisfying g q (grad H φ ( q ) , v ) = dφ q ( v ) ∀ q ∈ M, v ∈ N ( q ) . (In coordinates if { X , . . . X m } is alocal orthonormal frame for ( M, N , g ) , then grad H φ = P mi =1 ( L X i φ ) X i .) • the divergence of a vector field X is the unique function satisfying div Xµ H = L X µ H where µ H is the Popp volume form. The construction of the Popp volume form is not totally trivial and we address the readerto [25] or [1] for details. We just recall that the Popp volume form coincide with the Lebesguemeasure in a special system of coordinate related to the nilpotent approximation. In sub-Riemannian geometry one can also define other intrinsic volume forms, like the Hausdorff orthe spherical Hausdorff volume. However, at the moment, the Popp volume form is the onlyone known to be smooth in general. However for left-invariant sub-Riemannian manifolds allthese measures are proportional to the left Haar measure.The hypoellipticity of ∆ H (i.e. given U ⊂ M and φ : U → R such that ∆ H φ ∈ C ∞ , then φ is C ∞ ) follows from the H¨ormander Theorem (see [22]).In this paper we are interested only to nilpotent Lie groups. The next proposition says thatfor all unimodular Lie groups, i.e. for groups such that the left and right Haar measure coincides(and in particular for real connected nilpotent groups) the intrinsic hypoelliptic Laplacian isthe sum of squares. Proposition 6.
Let ( G, N , g ) be a left-invariant sub-Riemannian manifold generated by theorthonormal basis { p , . . . , p m } ⊂ l . If G is unimodular then ∆ H φ = P mi =1 (cid:0) L X i φ (cid:1) where L X i is the Lie derivative w.r.t. the field X i = g p i . .3 Computation of the hypoelliptic heat kernel via the GeneralizedFourier Transform In this section we describe the method, developed in [1], for the computation of the hypoellitpicheat kernel for left-invariant sub-Riemannian structures on unimodular Lie groups.The method is based upon the GFT, that permits to disintegrate a function from a Lie group G to R on its components on (the class of) non-equivalent unitary irreducible representationsof G . For proofs and more details, see [1]. Let f ∈ L ( R , R ): its Fourier transform is defined by the formulaˆ f ( λ ) = Z R f ( x ) e − ixλ dx. If f ∈ L ( R , R ) ∩ L ( R , R ) then ˆ f ∈ L ( R , R ) and one has Z R | f ( x ) | dx = Z R | ˆ f ( λ ) | dλ π , called Parseval or Plancherel equation. By density of L ( R , R ) ∩ L ( R , R ) in L ( R , R ), thisequation expresses the fact that the Fourier transform is an isometry between L ( R , R ) anditself. Moreover, the following inversion formula holds: f ( x ) = Z R ˆ f ( λ ) e ixλ dλ π , where the equality is intended in the L sense. It has been known from more than 50 years thatthe Fourier transform generalizes to a wide class of locally compact groups (see for instance[7, 14, 20, 21, 24, 30]). Next we briefly present this generalization for groups satisfying thefollowing hypothesis: (H ) G is a unimodular Lie group of Type I.For the definition of groups of Type I see [12]. For our purposes it is sufficient to recall thatall groups treated in this paper (i.e. G and G ) are of Type I. Actually, all the real connectednilpotent Lie groups are of Type I [11, 19]. In the following, the L p spaces L p ( G, C ) are intendedwith respect to the Haar measure µ := µ L = µ R .Let G be a Lie group satisfying (H ) and ˆ G be the dual of the group G , i.e. the set ofall equivalence classes of unitary irreducible representations of G . Let λ ∈ ˆ G : in the followingwe indicate by X λ a choice of an irreducible representation in the class λ . By definition, X λ is a map that to an element of G associates a unitary operator acting on a complex separableHilbert space H λ : In this paper, by the dual of the group, we mean the support of the Plancherel measure on the set ofnon-equivalent unitary irreducible representations of G ; we thus ignore the singular representations. λ : G → U ( H λ ) g X λ ( g ) . The index λ for H λ indicates that in general the Hilbert space can vary with λ . Definition 7.
Let G be a Lie group satisfying (H ) , and f ∈ L ( G, C ) . The generalized (ornoncommutative) Fourier transform (GFT) of f is the map (indicated in the following as ˆ f or F ( f ) ) that to each element of ˆ G associates the linear operator on H λ : ˆ f ( λ ) := F ( f ) := Z G f ( g ) X λ ( g − ) dµ. (5)Notice that since f is integrable and X λ unitary, then ˆ f ( λ ) is a bounded operator. Remark ˆ f can be seen as an operator from ⊕ R ˆ G H λ to itself. We also use the notationˆ f = ⊕ R ˆ G ˆ f ( λ )In general ˆ G is not a group and its structure can be quite complicated. In the case in which G is abelian then ˆ G is a group; if G is nilpotent (as in our cases) then ˆ G has the structure of R n for some n .Under the hypothesis (H ) one can define on ˆ G a positive measure dP ( λ ) (called thePlancherel measure) such that for every f ∈ L ( G, C ) ∩ L ( G, C ) one has Z G | f ( g ) | µ ( g ) = Z ˆ G T r ( ˆ f ( λ ) ◦ ˆ f ( λ ) ∗ ) dP ( λ ) . By density of L ( G, C ) ∩ L ( G, C ) in L ( G, C ), this formula expresses the fact that the GFT isan isometry between L ( G, C ) and ⊕ R ˆ G HS λ , the set of Hilbert-Schmidt operators with respectto the Plancherel measure. Moreover, it is obvious that: Proposition 9.
Let G be a Lie group satisfying (H ) and f ∈ L ( G, C ) ∩ L ( G, C ) . We have,for each g ∈ G f ( g ) = Z ˆ G T r ( ˆ f ( λ ) ◦ X λ ( g )) dP ( λ ) . (6) where the equality is intended in the L sense. It is immediate to verify that, given two functions f , f ∈ L ( G, C ) and defining their convo-lution as ( f ∗ f )( g ) = Z G f ( h ) f ( h − g ) dh, (7)then the GFT maps the convolution into non-commutative product: F ( f ∗ f )( λ ) = ˆ f ( λ ) ˆ f ( λ ) . (8)7nother important property is that if δ Id ( g ) is the Dirac function at the identity over G , thenˆ δ Id ( λ ) = Id H λ . (9)In the following, a key role is played by the infinitesimal version of the representation X λ , thatis the map d X λ : X d X λ ( X ) := ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 X λ ( e tp ) , (10)where X = gp , ( p ∈ l , g ∈ G ) is a left-invariant vector field over G . By Stone theorem (see forinstance [30, p. 6]) d X λ ( X ) is a (possibly unbounded) skew-adjoint operator on H λ . We havethe following: Proposition 10.
Let G be a Lie group satisfying (H ) and X be a left-invariant vector fieldover G . The GFT of X , i.e. ˆ X = F L X F − splits into the Hilbert sum of operators ˆ X λ , eachone of them acting on the set HS λ of Hilbert-Schmidt operators over H λ : ˆ X = ⊕ Z ˆ G ˆ X λ . Moreover, ˆ X λ Ξ = d X λ ( X ) ◦ Ξ , for every Ξ ∈ HS λ , (11) i.e. the GFT of a left-invariant vector field acts as a left-translation over HS λ . Remark
From the fact that the GFT of a left-invariant vector field acts as a left-translation,it follows that ˆ X λ can be interpreted as an operator over H λ . In this section we provide a general method to compute the kernel of the hypoelliptic heat equa-tion on a left-invariant sub-Riemannian manifold ( G, N , g ) such that G satisfies the assumption (H ) .We begin by recalling some existence results (for the semigroup of evolution and for thecorresponding kernel) in the case of the sum of squares. We recall that for all the examplestreated in this paper the invariant hypoelliptic Laplacian is the sum of squares.Let G be a unimodular Lie group and ( G, N , g ) a left-invariant sub-Riemannian manifoldgenerated by the orthonormal basis { p , . . . , p m } , and consider the hypoelliptic heat equation ∂ t φ ( t, g ) = ∆ H φ ( t, g ) . (12)Since G is unimodular, then ∆ H = L X + . . . + L X m , where L X i is the Lie derivative w.r.t.the vector field X i := g p i ( i = 1 , . . . , m ). Following Varopoulos [32, pp. 20-21, 106], since ∆ H is a sum of squares, then it is a symmetric operator that we identify with its Friedrichs (self-adjoint) extension, that is the infinitesimal generator of a (Markov) semigroup e t ∆ H . Thanks8o the left-invariance of X i (with i = 1 , . . . , m ), e t ∆ H admits a a right-convolution kernel p t ( . ),i.e. e t ∆ H φ ( g ) = φ ∗ p t ( g ) = Z G φ ( h ) p t ( h − g ) µ ( h ) (13)is the solution for t > φ (0 , g ) = φ ( g ) ∈ L ( G, R ) with respectto the Haar measure.Since the operator ∂ t − ∆ H is hypoelliptic, then the kernel is a C ∞ function of ( t, g ) ∈ R + × G .Notice that p t ( g ) = e t ∆ H δ Id ( g ).The main results of the paper are based on the following key fact. Theorem 12.
Let G be a Lie group satisfying (H ) and ( G, N , g ) a left-invariant sub-Riemannianmanifold generated by the orthonormal basis { p , . . . , p m } . Let ∆ H = L X + . . . + L X m be theintrinsic hypoelliptic Laplacian where L X i is the Lie derivative w.r.t. the vector field X i := g p i .Let (cid:8) X λ (cid:9) λ ∈ ˆ G be the set of all non-equivalent classes of irreducible representations of thegroup G , each acting on an Hilbert space H λ , and dP ( λ ) be the Plancherel measure on the dualspace ˆ G . We have the following:( i) the GFT of ∆ H splits into the Hilbert sum of operators ˆ∆ λH , each one of which leaves H λ invariant: ˆ∆ H = F ∆ H F − = ⊕ Z ˆ G ˆ∆ λH dP ( λ ) , where ˆ∆ λH = m X i =1 (cid:16) ˆ X λi (cid:17) . (14) ( ii) The operator ˆ∆ λH is self-adjoint and it is the infinitesimal generator of a contractionsemi-group e t ˆ∆ λH over HS λ , i.e. e t ˆ∆ λH Ξ λ is the solution for t > to the operator equation ∂ t Ξ λ ( t ) = ˆ∆ λH Ξ λ ( t ) in HS λ , with initial condition Ξ λ (0) = Ξ λ .( iii) The hypoelliptic heat kernel is p t ( g ) = Z ˆ G T r (cid:16) e t ˆ∆ λH X λ ( g ) (cid:17) dP ( λ ) , t > . (15) Remark
As a consequence of Remark 11, it follows that ˆ∆ λH and e t ˆ∆ λH can be consideredas operators on H λ .The following corollary gives a useful formula for the hypoelliptic heat kernel in the casein which for all λ ∈ ˆ G each operator e t ˆ∆ λH admits a convolution kernel Q λt ( ., . ). Below by ψ λ ,we intend an element of H λ . Corollary 14.
Under the hypotheses of Theorem 12, if for all λ ∈ ˆ G we have H λ = L ( X λ , dθ λ ) for some measure space ( X λ , dθ λ ) and h e t ˆ∆ λH ψ λ i ( θ ) = Z X λ ψ λ (¯ θ ) Q λt ( θ, ¯ θ ) d ¯ θ, then p t ( g ) = Z ˆ G Z X λ X λ ( g ) Q λt ( θ, ¯ θ ) (cid:12)(cid:12) θ =¯ θ d ¯ θ dP ( λ ) , where in the last formula X λ ( g ) acts on Q λt ( θ, ¯ θ ) as a function of θ . Hypoelliptic heat kernels on G and G G and G and we provide their matrix and Euclideanpresentations. We define left-invariant sub-Riemannian structures on them and the corre-sponding hypoelliptic Laplacian.We then provide representations of the groups and compute the GFT of the hypoellipticLaplacian. We apply the method presented in Section 2.3.2 to compute the fundamentalsolution of the hypoelliptic heat equation. G and G In our paper we deal with two 3-step Lie groups. The first one is the nilpotent group G withgrowth vector (2 , , L = span { l , l , l , l } , whose generators satisfy[ l , l ] = l , [ l , l ] = l , [ l , l ] = [ l , l ] = [ l , l ] = [ l , l ] = 0 . The second one is the free nilpotent group G with growth vector (2 , , L = span { l , l , l , l , l } , whose generators satisfy[ l , l ] = l , [ l , l ] = l , [ l , l ] = l , [ l , l ] = [ l , l ] = [ l , l ] = [ l , l ] = [ l , l ] = [ l , l ] = [ l , l ] = 0 . Both G and G are 3-step nilpotent, as a direct consequence of the definition. G In this section we first give the matrix and Euclidean presentations of the Lie group G . Wethen define a sub-Riemannian structure on it. We give explicitly the representations of thegroup, that we use at the end to compute the hypoelliptic kernel in terms of the kernel of thequartic oscillator.We start with the Lie algebra L , that can be presented as the follow matrix space L ≃ − a a − a a a | a i ∈ R . We present each l i as the matrix with a j = δ ij . It is straightforward to prove that thesematrices satisfy the commutation rules for L , where the bracket operation is the standard[ A, B ] := BA − AB .A matrix presentation of the group G is thus the matrix exponential of L : G ≃ exp − a a − a a a | a i ∈ R = − x x x − x x x | x i ∈ R , x = a , x = a , x = a − a a , x = a + a a − a a . We now define the isomorphism Π between G and R given byΠ − x x x − x x x = ( x , x , x , x ) . This isomorphism is a group isomorphism when R is endowed with the following product (see[13, p. 330]):( x , x , x , x ) · ( y , y , y , y ) := (cid:18) x + y , x + y , x + y − x y , x + y + 12 x y − x y (cid:19) The isomorphism Π induces an isomorphism of tangent spaces T g g ≃ T Π ( g ) R , that is explic-itly g l i ≃ X i , with X i given by X ( x ) = ∂∂x , X ( x ) = ∂∂x − x ∂∂x + x ∂∂x , (16) X ( x ) = ∂∂x − x ∂∂x , X ( x ) = ∂∂x , where x = ( x , x , x , x ). G We endow G with a left-invariant sub-Riemannian structure as presented in Section 2.2. Wedefine the sub-Riemannian manifold ( G , N , g ) where N ( g ) = g p with p = span { l , l } and g g ( g l i , g l j ) = δ ij with i, j = 1 or 2.Since G is nilpotent, then it is unimodular, thus the intrinsic hypoelliptic Laplacian ∆ H is the sum of squares (see [1, Proposition 17]). In terms of the Euclidean presentation of G ,the hypoelliptic Laplacian is thus ∆ H = X + X , with the X i given by (16).We thus want to find the fundamental solution for the following heat equation: ∂ t φ ( t, x ) = ∆ H φ ( t, x ) . (17) G We now recall the representations of the group G , as computed by Dixmier in [13, p. 333]. Asstated before, we may consider only representations on the support of the Plancherel measure. Proposition 15.
The dual space of G is ˆ G = (cid:8) X λ,µ | λ = 0 , µ ∈ R (cid:9) , where λ,µ ( x , x , x , x ) : H → H ψ ( θ ) exp (cid:18) i (cid:18) − µ λ x + λx − λx θ + λ x θ (cid:19)(cid:19) ψ ( θ + x ) whose domain is H = L ( R , C ) , endowed with the standard product < ψ , ψ > := R R ψ ( θ ) ψ ( θ ) dθ where dθ is the Lebesgue measure.The Plancherel measure on ˆ G is dP ( λ, µ ) = dλdµ , i.e. the Lebesgue measure on R . Remark
Notice that in this case the domain H of the representation X λ,µ does not dependon λ, µ . Consider the representation X λ,µ of G and let d X λ,µi be the corresponding representations ofthe differential operators L X i with i = 1 ,
2. Recall that d X λ,µi are operators on H . Again from[13, p. 333], or by explicit computation, we have h d X λ,µ ψ i ( θ ) = ddθ ψ ( θ ) , [ d X λ,µ ψ ]( θ ) = (cid:18) − i µλ + i λθ (cid:19) ψ ( θ ) , thus h ˆ∆ λ,µH ψ i ( θ ) = (cid:18) d dθ − (cid:16) λθ − µλ (cid:17) (cid:19) ψ ( θ ) . The GFT of the hypoelliptic heat equation is thus ∂ t ψ = (cid:18) d dθ − (cid:16) λθ − µλ (cid:17) (cid:19) ψ ( θ ) . (18)We rewrite it as ∂ t ψ = (cid:18) d dθ − (cid:0) αθ + β (cid:1) (cid:19) ψ ( θ ) , (19)with α = λ , β = − µ λ .The operator d dθ − ( αθ + β ) is the Laplacian with quartic potential, see e.g. [29]. Asalready stated, no general explicit solutions are known for this equation. We callΨ t (cid:0) θ, ¯ θ ; α, β (cid:1) the solution of ( ∂ t ψ ( t, θ ) = (cid:16) d dθ − ( αθ + β ) (cid:17) ψ ( t, θ ) ,ψ (0 , θ ) = δ ¯ θ , i.e. the solution of (19) evaluated in θ at time t , with initial data δ ¯ θ and parameters α, β .Applying Corollary 14 and after straightforward computations, one gets the kernel of thehypoelliptic heat equation on the group G : p t ( x , x , x , x ) = Z R \{ } dλ Z R dµ Z R dθ e i ( − µ λ x + λx − λx θ + λ x θ )Ψ t (cid:18) θ + x , θ ; λ , − µ λ (cid:19) . (20)12 .3 Hypoelliptic heat kernel on G The Lie algebra L of the group G can be presented as the following matrix space L ≃ (cid:26)(cid:18) M ( a , a , a , a ) × × M ( a , a , a , a ) (cid:19) | a i ∈ R (cid:27) , where M ( a , a , a , a ) = − a a − a a a , M ( a , a , a , a ) = a a a − a − a . We present each l i as the matrix with a j = δ ij . It is straightforward to prove that thesematrices satisfy the commutation rules for L , where the bracket operation is the standard[ A, B ] := BA − AB .A matrix presentation of the group G is thus the matrix exponential of L : G ≃ (cid:26) exp (cid:18)(cid:18) M ( a , a , a , a ) × × M ( a , a , a , a ) (cid:19)(cid:19) | a i ∈ R (cid:27) == (cid:26)(cid:18) exp ( M ( a , a , a , a )) × × exp ( M ( a , a , a , a )) (cid:19) | a i ∈ R (cid:27) == (cid:26)(cid:18) N ( x , x , x , x ) × × N ( x , x , x , x ) (cid:19) | x i ∈ R (cid:27) with N ( x , x , x , x ) = − x x x − x x x , N ( x , x , x , x ) = x x x − x x x − x − x x − x ,x = a , x = a , x = a − a a , x = a + a a − a a , x = a + a a − a a . We now define the isomorphism Π between G and R given byΠ (cid:18)(cid:18) N ( x , x , x , x ) × × N ( x , x , x , x ) (cid:19)(cid:19) = ( x , x , x , x , x ) . This isomorphism is a group isomorphism when R is endowed with the following product (see[13, p. 331]): ( x , x , x , x , x ) · ( y , y , y , y , y ) := (cid:18) x + y , x + y , x + y − x y , x + y + 12 x y − x y , x + y + 12 x y − x y + x x y (cid:19) . induces an isomorphism of tangent spaces T g g ≃ T Π ( g ) R , that is explic-itly g l i ≃ X i , with X i given by X ( x ) = ∂∂x , X ( x ) = ∂∂x − x ∂∂x + x ∂∂x + x x ∂∂x , (21) X ( x ) = ∂∂x − x ∂∂x − x ∂∂x , X ( x ) = ∂∂x , X ( x ) = ∂∂x , where x = ( x , x , x , x , x ).We endow G with a left-invariant sub-Riemannian structure as presented in Section 2.2,where p = span { l , l } and g g ( g l i , g l j ) = δ ij with i, j = 1 or 2.We thus want to find the fundamental solution for the following heat equation: ∂ t φ ( t, x ) = ∆ H φ ( t, x ) , (22)with ∆ H = X + X . G We now recall the representations of the group G , as computed by Dixmier in [13, p. 338]. Asstated before, we may consider only representations on the support of the Plancherel measure. Proposition 17.
The dual space of G is ˆ G = (cid:8) X λ,µ,ν | λ + µ = 0 , ν ∈ R (cid:9) , where X λ,µ,ν ( x , x , x , x , x ) : H → H ψ ( θ ) exp (cid:0) iK λ,µ,νx ,x ,x ,x ,x ( θ ) (cid:1) ψ (cid:18) θ + λx + µx λ + µ (cid:19) with K λ,µ,νx ,x ,x ,x ,x ( θ ) = − νλ + µ ( µx − λx ) + λx + µx + − µλ + µ (cid:0) λ x + 3 λµx x + 3 µ x x − λµx (cid:1) + µ x x θ + λµ ( x − x ) θ ++ 12 (cid:0) λ + µ (cid:1) ( µx − λx ) θ . The domain of X λ,µ,ν ( x , x , x , x , x ) is H = L ( R , C ) , endowed with the standard product < ψ , ψ > := R R ψ ( θ ) ψ ( θ ) dθ where dθ is the Lebesgue measure.The Plancherel measure on ˆ G is dP ( λ, µ, ν ) = dλdµdν , i.e. the Lebesgue measure on R . Remark
Notice that in this case the domain H of the representation X λ,µ,ν does not dependon λ, µ, ν . 14 .3.2 The kernel of the hypoelliptic heat equation Consider the representation X λ,µ,ν of G and let d X λ,µ,νi be the corresponding representationsof the differential operators L X i with i = 1 ,
2. Recall that d X λ,µ,νi are operators on H . Againfrom [13, p. 338], or by explicit computation, we have h d X λ,µ,ν ψ i ( θ ) = (cid:18) − i µηλ + µ + λλ + µ ddθ − i µ (cid:0) λ + µ (cid:1) θ (cid:19) ψ ( θ ) h d X λ,µ,ν ψ i ( θ ) = (cid:18) i λµλ + µ + µλ + µ ddθ + i λ (cid:0) λ + µ (cid:1) θ (cid:19) ψ ( θ ) , thus h ˆ∆ λ,µ,νH ψ i ( θ ) = 1 λ + µ d ψ ( θ ) dθ − ( ν + ( λ + µ ) θ ) λ + µ ) . The GFT of the hypoelliptic heat equation is thus ∂ t ψ = 1 λ + µ d ψ ( θ ) dθ − ( ν + ( λ + µ ) θ ) λ + µ ) ψ ( θ ) . (23)We rewrite it as ∂ τ ψ = (cid:18) d dθ − (cid:0) αθ + β (cid:1) (cid:19) ψ ( θ ) , (24)with τ = t ( λ + µ ) , α = λ + µ , β = − ν .The operator d dθ − ( αθ + β ) is the Laplacian with quartic potential, see e.g. [29]. Asalready stated, no general explicit solutions are known for this equation. We callΨ τ (cid:0) θ, ¯ θ ; α, β (cid:1) the solution of ( ∂ τ ψ ( τ, θ ) = (cid:16) d dθ − ( αθ + β ) (cid:17) ψ ( τ, θ ) ,ψ (0 , θ ) = δ ¯ θ , i.e. the solution of (24) evaluated in θ at time τ , with initial data δ ¯ θ and parameters α, β .Applying Corollary 14 and after straightforward computations, one gets the kernel of thehypoelliptic heat equation on the group G : p t ( x , x , x , x , x ) == Z λ + µ =0 dλdµdν Z R dθ exp (cid:0) iK λ,µ,νx ,x ,x ,x ,x ( θ ) (cid:1) Ψ tλ µ (cid:18) θ + λx + µx λ + µ , θ ; λ + µ , − ν (cid:19) . (25)15 eferences [1] A. Agrachev, U. Boscain, J.-P. Gauthier, F. Rossi , The intrinsic hypoellipticLaplacian and its heat kernel on unimodular Lie groups,
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