Probab. Theory Relat. Fields (2008) 142:285–311 DOI 10.1007/s00440-007-0105-y
Semimartingales and geometric inequalities on locally symmetric manifolds H. Le · D. Barden
Received: 14 December 2006 / Revised: 10 September 2007 / Published online: 13 October 2007 © Springer-Verlag 2007
Abstract We generalise, to complete, connected and locally symmetric Riemannian manifolds, the construction of coupled semimartingales X and Y given in Le and Barden (J Lond Math Soc 75:522–544, 2007). When such a manifold has non-negative curvature, this makes it possible for the stochastic anti-development of the correspon (Y ) to be a time-changed Brownian motion with ding semimartingale exp X t α exp−1 t Xt drift when X and Y are. As an application, we use the latter result to strengthen, and extend to locally symmetric spaces, the results of Le and Barden (J Lond Math Soc 75:522–544, 2007) concerning an inequality involving the solutions of the parabolic 1 equation ∂ψ ∂t = 2 ψ − h ψ with Dirichlet boundary condition and an inequality involving the first eigenvalues of the Laplacian, both on three related convex sets. Keywords Brownian motion · First eigenvalue of the Laplacian · Jacobi field · Parabolic equation · Parallel translation Mathematics Subject Classification (2000) 58J65 · 58J32 1 Introduction Pioneered by W. S. Kendall, coupled semimartingales, X and Y , on manifolds M have been used to transfer analytical results for Euclidean space onto Riemannian manifolds, for example for the study of harmonic maps (cf. [3,6–8]). Recently, this construction has been explored further in [10] in studying Z t = exp X t α exp−1 X t (Yt ) H. Le (B) School of Mathematical Science, University of Nottingham, University Park, NG7 2RD Nottingham, UK e-mail:
[email protected] D. Barden University of Cambridge, Cambridge, UK
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for a given constant α ∈ (0, 1). Clearly, Z so constructed is a semimartingale and, for all t such that Z t is defined, it keeps the distances to X t and Yt in a constant ratio α : 1 − α. In [10], we considered the case where M is complete and connected with constant sectional curvature, and used two versions of the semimartingale Y : one involving parallel translation along geodesics and the other involving that parallel translation followed by reflection in the hyperplane orthogonal to the geodesic. The novelty of this approach lies in that the coupling between X and Y is so chosen that the resulting Z will have its stochastic anti-development a time-changed Brownian motion with drift when the stochastic anti-developments of X and Y are. The SDEs for X , Y and Z can then be combined to derive an inequality involving the solutions of 1 the parabolic equation ∂ψ ∂t = 2 ψ − h ψ, with Dirichlet boundary condition, on three interrelated convex sets, and an inequality for the first eigenvalues of the Laplacian on those sets with the Dirichlet boundary condition, analogous to similar results in Euclidean space (cf. [1]). In this paper, we generalise the construction of X and Y in [10] to complete, connected and locally symmetric Riemannian manifolds M: instead of using entirely correlated Brownian motions to drive X and Y , we add to Yt in each of naturally chosen directions orthogonal to the geodesic between X t and Yt an independent Brownian motion component that may vary from direction to direction. When M has nonnegative sectional curvature, this may be done in such a way that the time-changes for X and Y may be chosen to be the same and that the resulting Z has its stochastic anti-development a Brownian motion with drift and with the same time-change. Note that, even for the case of constant curvature, the choice of Y and consequent Z here differ from those in [10]. This enables us to improve the results of [10] mentioned above and to extend them to locally symmetric spaces. This paper is organised as follows. In Sect. 2, we recall relevant definitions for, and basic facts concerning, locally symmetric manifolds and then obtain expressions for certain covariant derivatives that are involved in our analysis. In Sect. 3, we construct our coupled semimartingales X and Y , obtain the Itô differential of ρt , their distance apart at time t, and their Stratonovich differentials. In Sect. 4, we show how to choose X and Y in such a way that the semimartingale Z = exp X (α exp−1 X Y ) is a time-changed Brownian motion with drift. Section 5 applies this to derive an inequality concerning solutions of the parabolic equation on interrelated convex sets in the manifolds and, as a corollary, a similar inequality for the first eigenvalues of the Laplacian.
2 Basic geometry of locally symmetric Riemannian manifolds We first recall the definition and basic properties of locally symmetric spaces, following O’Neill [11] and Jost [5]. On a d-dimensional Riemannian manifold M we take the Levi-Civitá connection, the unique torsion free metric connection, expressed by covariant derivation where, for vector fields ν1 and ν2 , Dν1 ν2 is the vector field that is the covariant derivative of ν2 with respect to ν1 . This has unique extension to all tensor fields as a derivation that commutes with contractions. Then we denote by R the Riemannian curvature (1,3)-tensor given by R(ν1 , ν2 )ν3 = [Dν1 , Dν2 ]ν3 − D[ν1 ,ν2 ] ν3 for vector fields ν1 , ν2 and ν3 , and by D the covariant differential mapping tensor
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fields of a given type to those with one higher covariant degree, by treating the ν1 in Dν1 as the first argument. The most natural definition of M being locally symmetric is that for each x in M there should exist a local isometry fixing x and reversing all geodesics through x. That is equivalent to any one of the following conditions: D R ≡ 0; the sectional curvature R(ν1 , ν2 )ν2 , ν1 /(ν1 2 ν2 2 − ν1 , ν2 2 ) is invariant under parallel translation; R(ν1 , ν2 )ν3 is parallel along any curve for which ν1 , ν2 and ν3 are. From the first definition, joining an arbitrary pair of points by a broken geodesic, it is clear that a connected locally symmetric space is locally homogeneous. From the second equivalent definition it follows that manifolds of constant sectional curvature are locally symmetric, as are the Kaehler manifolds of constant holomorphic curvature described below. For each x ∈ M and ω in the tangent space τx (M) to M at x, we define the map Rω : τx (M) → τx (M); ν → R(ν, ω)ω. For a geodesic γ (s), the symmetries of the Riemannian curvature tensor imply that Rγ (s) is a self-adjoint map of the orthogonal complement of γ (s) in τγ (s) (M) onto itself. Throughout this paper, we shall assume that γ has unit speed and let e1 (s) = γ (s) and then take {ek (0) | 2 k d} to be the completion of an orthonormal frame of eigenvectors of Rγ (0) with corresponding eigenvalues κ 2 κ 3 · · · κ d . Thus, κ j is the sectional curvature of the plane spanned by e1 (0) and e j (0). Then, if {ek (s) | 2 k d} is the parallel translation of {ek (0) | 2 k d} along γ , it follows from the third equivalent definition of local symmetry that, for all j 2 and all s, e j (s) is a unit eigenvector of Rγ (s) acting on γ (s)⊥ with eigenvalue κ j . Note that, although the κ j remain constant along the geodesic, they will vary in some interrelated fashion according to the direction of the geodesic through γ (0). Also, when any of them are repeated, and so the corresponding eigenspaces have dimension greater than one, we should make a smooth choice of eigenvectors as x or y vary. We shall require covariant derivatives that we denote by De j (s) ek (s) which, for k > 1, are not a priori well-defined since the ek are not vector fields on M. However, it will be clear from the context that the derivatives we require are partial ones, calculated holding the points x or y fixed. Then, for example, holding x fixed and letting y vary in a sufficiently small neighbourhood of y the vector field ek will be that obtained, in that neighbourhood, by the above construction along the various geodesics from x to y . Note that, since ek is parallel along γ and e1 is its tangent field, De1 (s) ek (s) = 0. In general, we write jk (s) = De j (s) ek (s), e (s) ,
(1)
noting that, although these are not the classical Christoffel symbols arising from a local chart, since ek and e are orthogonal for k = , we do have jk = − kj and, in particular, kjk = 0. The orthogonality of the vector fields ek also implies that the Koszul formula reduces to give 2 jk (s) = e j (s), ek (s) , e (s) − ek (s), e (s) , e j (s) + e (s), e j (s) , ek (s) . (2)
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Noting that κ j = R(e j , e1 )e1 , e j and recalling that R is parallel, for j > 1 we have d k R(e j , e1 )ek , e j , 1 e κ j ≡ De (κ j ) = R(e j , e1 )De e1 , e j = 2
(3)
k=1
using the symmetry R(ν1 , ν2 )ν3 , ν4 = R(ν4 , ν3 )ν2 , ν1 of the curvature tensor and the fact that R(e j , e1 )e1 , De e j = κ j e j , De e j = 0. An important class of locally symmetric manifolds comprises the Kaehler ones of constant holomorphic sectional curvature. A complex structure on a manifold M determines a tensor field J of endomorphisms Jx of the tangent spaces τx (M) such that Jx2 = −identity. Then a Hermitian metric g on M is Riemannian metric g such that g(J ν1 , J ν2 ) = g(ν1 , ν2 ) for all vector fields ν1 , ν2 on M. It follows that g(ν, J ν) = 0 and the holomorphic sectional curvature is the restriction of the Gaussian sectional curvature to the tangent planes spanned by such orthogonal pairs of vectors {ν, J ν} and, hence, invariant under J . If the fundamental two-form, given by ϕ(ν1 , ν2 ) = g(ν1 , J ν2 ), is closed then M is said to be a Kaehler manifold. That such a manifold is locally symmetric follows since dϕ = 0 implies that J is parallel and then the expression for R, that we quote in the proof of Lemma 3 below, shows that D R ≡ 0. In this special case, many of the jk defined by (1) above vanish. Lemma 1 If M is a Kaehler manifold of constant holomorphic sectional curvature k and 2 vanish for all κ0 , then e2 (0) = J e1 (0) is an eigenvector of Re1 (0) , and 22 kk k > 2. Proof Under the hypotheses on M, the Riemannian curvature tensor may be expressed as R(ν1 , ν2 )ν3 =
κ0 {g(ν2 , ν3 ) ν1 − g(ν1 , ν3 ) ν2 − g(ν2 , J ν3 ) J ν1 4 + g(ν1 , J ν3 ) J ν2 + 2 g(ν1 , J ν2 ) J ν3 }
(cf. [9, Prop. 7.3] where the authors write R(X, Y, Z , W ) for g(R(X, Y )W, Z )). Then, the equality R(J e1 (0), e1 (0))e1 (0) = κ0 J e1 (0) shows that J e1 (0) is an eigenvector of Re1 (0) with eigenvalue κ0 , so that we may indeed take it to be e2 (0). We similarly find that e (0) for > 2 all have eigenvalues κ0 /4. We complete the details of the proof for the case κ0 > 0 to which we shall refer in Sect. 4: for κ0 < 0, since our calculations are local, we may work in the universal cover which is an open ball where the analogous details are straightforward. Recall then, that a complete Kaehler manifold of constant positive holomorphic sectional curvature is necessarily homothetic with complex projective space (cf. [9, p. 171]) and, since the jk are invariant under homothety, we may assume that κ0 = 4. Then, M = CP(d/2) is the Riemannian quotient space of the unit sphere Sd+1 in Cd/2+1 and the Riemannian submersion from Sd+1 to M may be explicitly realised as the projection π of the principal S1 -bundle obtained from the diagonal action of S1 , regarded as the complex numbers of modulus one, on vectors in Cd/2+1 .
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Identifying Cd/2+1 with Rd+2 , the hermitian inner product h on the former is given, in terms of the Riemannian inner product g on the latter, by h(z, w) = g(z, w) + i g(z, iw) (cf. [9, pp. 273–275]). In particular, since g(z, i z) = 0, h(z, z) = g(z, z) so that the two possible definitions of z coincide. Then, the tangent space at a point z in Sd+1 is the, translation of the, g-orthogonal complement of z and the horizontal subspace of this with respect to the submersion is the h-orthogonal complement of z. The unit speed geodesic between orthogonal vectors w and z in Sd+1 is given by γz (s) = cos s w + sin s z and, if z is horizontal at w, then this remains horizontal at all points. We recall (loc.cit.) that the standard almost complex structure J on τ (M) lifts to multiplication by i in the horizontal subspaces. We also recall (cf. [11, p. 212]) that to compute Dν1 ν2 for vector fields ν1 , ν2 on M, we may take the, projection of the, horizontal component of Dν˜1 ν˜ 2 where ν˜ 1 , ν˜ 2 are horizontal lifts of ν1 , ν2 , respectively. Then to compute Dν˜1 ν˜ 2 on Sd+1 we take respective smooth extensions ν¯ 1 , ν¯ 2 of ν˜ 1 , ν˜ 2 to a neighbourhood of Sd+1 in Rd+2 and compute the tangential component of Dν¯1 ν¯ 2 : in our case it will be convenient to take extensions over Rd+2 \{0} that are constant on lines through the origin. For any x, y in M, we choose h-orthogonal w, z in Sd+1 such that x = π(w) and y = π( p) for some point p = γz (s) on the horizontal geodesic through w with initial direction z. Then the horizontal lift e˜1 ( p) of the vector e1 (y) is just γz (s) and e˜2 ( p) is i e˜1 ( p) = i {− sin s w + cos s z}. Writing, temporarily, X for the complex span of a set X and X ⊥ for its h-orthogonal complement, we note that the horizontal subspace H( p) at p contains γz (s) and also H0 ( p) = w, z⊥ , so that H( p) is the h-orthogonal direct sum H0 ( p) ⊕ γz (s). We project the point ps (t) = p + t e¯2 ( p) = p + t e˜2 ( p) = (cos s − i t sin s) w + (sin s + i t cos s) z onto Sd+1 and rotate it so that the coefficient of w is real to obtain a point qs (t). Then, qs (t) is of the form γλz (u) where λ ∈ S1 and so e˜2 (qs (t)) = (u) ∈ w, z. Thus, as q (t) and p both lie in Sd+1 , i γλz s De˜2 e˜2 ( p) = the tangential component of De¯2 e¯2 ( p) 1 2 e˜ (qs (t)) − e˜2 ( p) = the tangential component of lim t→0 t lies in w, z and its horizontal component must lie in γz (s), which is the real span k = 0 for k > 2 as required. of {e1 ( p), e2 ( p)}, so that 22 2 are zero for k > 2, we consider p (t) = p + t e˜ k ( p) Similarly, to show that the kk s d+1 whose projection onto S is qs (t) ≡ γv (λ) = cos λ w + sin λ v where, writing u √ for e˜k ( p) ∈ w, z⊥ = H0 ( p), v = cos µ z + sin µ u so that cos λ = cos s/ 1 + t 2 and sin(µ(t)) = t/ sin2 s + t 2 . Now, e˜k (qs (t)) = ξ(t) ∈ H0 (qs (t)) = w, v⊥ and v(t) ˆ = cos(µ(t)) u − sin(µ(t)) z ∈ H0 (qs (t)) so, as ξ(t) → u and v(t) ˆ → u as ˆ − sin(θ (t)) η(t)} where η(t) is a t → 0, we may write ξ(t) = eiψ(t) {cos(θ (t)) v(t) ˆ ⊥ and ψ(t) → 0 and θ (t) → 0 as t → 0. Thus, as unit vector in H0 (qs (t)) ∩ v u, w and z are mutually h-orthogonal and sin(µ(t))/t → cosec s as t → 0,
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1 g De˜k e˜k , e˜2 ( p) = lim g e˜k (qs (t)), e˜2 ( p) t→0 t
1 = lim h eiψ(t) {cos(θ (t)) cos(µ(t)) u − sin(µ(t)) z}, t→0 t i (− sin s w + cos s z) = − h(z, i cot s z) = 0, 2 = 0 as required. so that kk
For the purposes of this paper, we shall assume that the locally symmetric manifold M has dimension d at least two, and is complete and connected. We write ρ for the distance function on M induced by the Riemannian metric. Our analysis will involve Jacobi fields along geodesics which, for constant curvature, may be expressed in terms of the function Sκ (s) defined, for s 0, by ⎧ √ √ ⎨ sin( κs)/ κ Sκ (s) = s √ √ ⎩ sinh( |κ|s)/ |κ|
if κ > 0 if κ = 0 if κ < 0.
Taking the value of Sκ (s)/s at s = 0 to be its limit as s ↓ 0, namely 1, √we see that Sκ (s)/s is a strictly decreasing continuous function of s for 0 s π/ κ, if κ > 0. This function remains relevant in our case since, for our choice of orthonormal frame, each sectional curvature function κi remains constant along the geodesic. As in [10], the coupling of our semimartingales X and Y will require the isomorphism px,y between the tangent spaces, at distinct points x and y of M that are not on each other’s cut locus, obtained by parallel translation along the shortest geodesic γx,y from x to y, as well as the mirror-map m x,y = r x,y ◦ px,y that is px,y composed . Then, in order to with the reflection of τ y (M) in the hyperplane orthogonal to γx,y express the Stratonovich differentials of X and Y , we require the covariant derivatives of px,y and m x,y or, more precisely, of the cross-sections that they determine of the bundle of isomorphisms over, most of, M × M. We obtain the following analogues of the results in [10]. The apparent difference, apart from the varying curvature, between the following lemma and the corresponding result in [10] is due to our using in the latter the explicit expression for the Riemannian curvature when the sectional curvature is constant. Lemma 2 With the conventions and notation introduced above for a locally symmetric manifold M, De (0) px,y (ek (0)) = De (ρ(x,y)) px,y (ek (0)) 1 − S (ρ(x, y)) R γx,y (ρ(x, y)), e (ρ(x, y)) ek (ρ(x, y)). = κ κ Sκ (ρ(x, y))
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Proof Fixing distinct points x, y in M that are not on each other’s cut locus and working first on De (0) px,y , we define a geodesic variation q(s, t) = exp y ((1 − s/ρ(x(t), y)) exp−1 y (x(t))), where x (0) = e (0) for 2 d, and, for each ek (0), we define a vector field V over the image of q by V (s, t) = px(t),q(s,t) ◦ px,x(t) (ek (0)). Writing ∂t for q∗ ∂t∂ and Dt for the covariant derivative with respect to ∂t , and similarly for ∂s and Ds , then De (0) px,y (ek (0)) is, as explained in [10], given by ∂t V (s, t) s=ρ(x,y) which, since all V (ρ(x, y), t) lie in τ y (M), is identical with t=0 Dt V (s, t) s=ρ(x,y) . The definition of V ensures that Ds V = 0 and, since [∂s , ∂t ] = 0 t=0
implies that R(∂s , ∂t ) = Ds Dt − Dt Ds , we get Ds (Dt V ) = R(∂s , ∂t )V . Since ∂t t=0 is the Jacobi field j (s) along γx,y determined by j (0) = x (0) = e (0) and j (ρ(x, y)) = 0, it is given by Sκ (ρ(x, y) − s)/Sκ (ρ(x, y)) px,γx,y (s) (e (0)). Also, (s) for t = 0 and V (s, 0) = p k ∂s = γx,y x,γx,y (s) (e (0)). This gives S (ρ(x, y) − s) R γx,y (s), px,γx,y (s) (e (0)) px,γx,y (s) (ek (0)), Ds (Dt V )t=0 = κ Sκ (ρ(x, y)) so that De (0) px,y (ek (0)) = Dt V (s, t) s=ρ(x,y) t=0
=
1 − Sκ (ρ(x, y)) κ Sκ (ρ(x, y))
R γx,y (ρ(x, y)), e (ρ(x, y)) ek (ρ(x, y))
as required. Since Dν y px,y (νx ) = − px,y (Dν y p y,x ( px,y (νx ))) for any νx ∈ τx (M) and ν y ∈ τ y (M) (cf. [10]), De (ρ(x,y)) px,y (ek (0))
= − px,y De (ρ(x,y)) p y,x (ek (ρ(x, y))) =−
1 − Sκ (ρ(x, y))
px,y R −γx,y (0), e (0) ek (0)
κ Sκ (ρ(x, y)) 1 − S (ρ(x, y)) R γx,y (ρ(x, y)), e (ρ(x, y)) ek (ρ(x, y)) = κ κ Sκ (ρ(x, y))
(s), e (s) ek (s) is parallel. as R γx,y
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In particular, since R e (ρ(x, y)), e1 (ρ(x, y)) e1 (ρ(x, y)) = κ e (ρ(x, y)) for > 1, it follows from Lemma 2 that
De (0) px,y (ek (0)), e1 (ρ(x, y))
1 − Sκ (ρ(x, y)) 1 k 1 R e (ρ(x, y)), e (ρ(x, y)) e (ρ(x, y)), e (ρ(x, y)) κ Sκ (ρ(x, y)) 1 − S (ρ(x, y)) R e (ρ(x, y)), e1 (ρ(x, y)) e1 (ρ(x, y)), ek (ρ(x, y)) = κ κ Sκ (ρ(x, y)) 1 − Sκ (ρ(x, y)) δ k = Sκ (ρ(x, y))
=
and De (0) px,y (e (0)), e (ρ(x, y)) =
1 − Sκ (ρ(x, y)) 1 R e (ρ(x, y)), e (ρ(x, y)) e (ρ(x, y)), e (ρ(x, y)) = 0. κ Sκ (ρ(x, y))
In the Kaehler case, we have the following result, continuing with the conventions and notation introduced above. Lemma 3 If M is a Kaehler manifold of constant holomorphic sectional curvature κ0 then, for > 1, De (0) px,y (e (0)) = De (ρ(x,y)) px,y (e (0)) =
1 − Sκ (ρ(x, y)) Sκ (ρ(x, y))
γx,y (ρ(x, y)),
where κ 2 = κ0 and κ = κ0 /4 for > 2. Proof As shown in the proof of Lemma 1, e2 (0) = J e1 (0) with eigenvalue κ0 and the eigenvalues of e (0) for > 2 are all κ0 /4. Using the fact that g(ν, J ν) = 0 and , we get recalling our choice of e1 = γx,y R(γx,y (ρ(x, y)), e2 (ρ(x, y)))e2 (ρ(x, y)) = κ0 γx,y (ρ(x, y))
and, for > 2, R(γx,y (ρ(x, y)), e (ρ(x, y)))e (ρ(x, y)) =
which by Lemma 2 give the stated results.
κ0 γ (ρ(x, y)), 4 x,y
Applying the general formula obtained in [10] and noting that, for our choices of e when M is locally symmetric,
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r x,y (R(γx,y (ρ(x, y)), e (ρ(x, y)))e (ρ(x, y))) = R(γx,y (ρ(x, y)), e (ρ(x, y)))e (ρ(x, y)) −2 R(γx,y (ρ(x, y)), e (ρ(x, y)))e (ρ(x, y)), γx,y (ρ(x, y)) γx,y (ρ(x, y)) = R(γx,y (ρ(x, y)), e (ρ(x, y)))e (ρ(x, y)) − 2 κ γx,y (ρ(x, y))
for 2 d, we also have the following result relating the covariant derivatives of the mirror-map m x,y = r x,y ◦ px,y to those of px,y in Lemma 2. Lemma 4 If M is locally symmetric, then De (0) m x,y (e (0)) = De (0) px,y (e (0)) +
2 Sκ (ρ(x, y))
γ (ρ(x, y)) Sκ (ρ(x, y)) x,y 2 De (ρ(x,y)) m x,y (e (0)) = De (ρ(x,y)) px,y (e (0)) − γ (ρ(x, y)) Sκ (ρ(x, y)) x,y for 2 d. 3 Construction of coupled semimartingales on M Let B and B˜ be two independent Brownian motions on Rd , both starting from the origin, and V be the class of bounded progressively measurable processes on Rd . Write θk , k = 2, . . . , d, for d − 1 differentiable functions on R2 with |θk | 1. Then, for any u and v inV and any two differentiable non-zero functions ξ and η on R with ∞ ∞ the property that 0 ξ(t)2 dt = 0 η(t)2 dt = ∞, we define the partially coupled semimartingales on M, X ξ,u starting at x and Y η,v starting at y, where y = x and y is not on the cut locus of x, by ξ,u
∂t
ξ,u
ξ,u
= Hξ,u ∂ X t , t
ξ,u
ξ,u
ξ,u
∂ X t = t ∂ At , X 0 = x, ξ,u d At = ξ(ρt ) d Bt + ξ(ρt )2 u t dt,
(4)
for X ξ,u and, for Y η,v , by η,v
∂t η,v ∂Yt η,v d At y η,v d Bt , {t }−1 et1 (ρt ) y η,v d Bt , {t }−1 et (ρt )
= = =
η,v
Htη,v ∂Yt , η,v η,v η,v t ∂ At , Y0 = y, y 2 η(ρ t ) d Bt + η(ρt ) vt dt ξ,u (5) = TX ξ,u ,Y η,v t d Bt , et1 (ρt ) t t ξ,u = θ κt , ρt TX ξ,u ,Y η,v t d Bt , et (ρt ) t t η,v + 1−θ (κt , ρt )2 d B˜ t , {t }−1 et (ρt ) , 2 d,
where ∂ and d denote the Stratonovich and Itô differentials, respectively, Tx,y is either the parallel transport px,y or the mirror-map m x,y = r x,y ◦ px,y and H denotes the
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horizontal lift of the tangent space of M to the tangent space of the orthonormal frame j η,v ξ,u ξ,u bundle O(M) of M. We write ρt for ρ(X t , Yt ) and the et ’s are defined for X t η,v j and Yt in a similar manner to the e ’s for x and y in the previous section. Note that, if we choose θ j ≡ 1 for 2 j d, then Y η,v defined here will be identical with that defined in [10], so that this construction is a generalisation of that in [10]. In [10], the component of Brownian motion in Y η,v is either the parallel translation, or the mirror-map, of that of X ξ,u , so that it is entirely correlated with the Brownian motion that drives X ξ,u . Here, in each of the chosen directions orthogonal to the geodesic γt , we add to Y an independent Brownian motion component. Since this additional component varies generally from direction to direction according to the value of θ j , it will allow us to take account of the different curvatures of the planes tangent to the geodesic when we make an explicit choice of the θ j as a function of κ j in the next section. Nevertheless, since we require |θ j | 1 and since B and B˜ are independent, B y defined above is still a Brownian motion on Rd , so that the stochastic anti-developments Aξ,u and Aη,v , respectively, of X ξ,u and Y η,v on Rd both remain time-changed Brownian motions with drifts u and v, respectively. We first study the behaviour of ρt before the minimum of the first collision time of the processes X ξ,u and Y η,v and the first time that each is on the cut locus of the other, which time we shall denote by τe . Lemma 5 For any x, y ∈ M such that y = x and y is not on the cut locus of x, let X ξ,u and Y η,v be the semimartingales defined by (4) and (5), respectively, and γt be η,v ξ,u the unit speed (minimal) geodesic from X t to Yt . Then, for t < τe ,
η,v ξ,u dρt = {ξ(ρt ) ∓ η(ρt )} d Bt1 − ξ(ρt )2 t u t , γt (0) −η(ρt )2 t vt , γt (ρt ) dt +
d
1
G κ j (ρt , ρt ; ξ, η θ j ) + G κ j ρt , ρt ; 0, η 1 − θ 2j dt, 2 j=2
where B 1 = −
t 0
ξ,u −1 1 er (0) is a Brownian motion on R, d Br , r
G κ (u, s; ξ, η) = gκ (u, s; ξ, η)
∂gκ (u, s; ξ, η) ∂gκ (u, 0; ξ, η) − gκ (u, 0; ξ, η) ∂s ∂s
with gκ (u, s; ξ, η) =
ξ(u) Sκ (u − s) + η(u) Sκ (s) , Sκ (u)
(6)
and ∓ is − if Tx,y = px,y and is + if Tx,y = m x,y . Note that g0 is a linear function of s so that, in particular, G 0 (u, αs; ξ, η) = α G 0 (u, s; ξ, η) for any constant α. Proof As in [10], the main ingredient of this proof is an adaption of that for Theorem 1 of [3]. We repeat it to check that the above functions G κ j have the correct form for this context.
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Following the argument in [6] and in [10], the Itô formula for ρt has the expression η,v ξ,u dρt = d X t , −et1 (0) + dYt , et1 (ρt ) +
d 2 1 j j ξ,u ξ,u (H et (0))2 ρt d At , {t }−1 et (0) 2 j=2
j j j j η,v η,v ξ,u ξ,u + 2 H et (0) H et (ρt ) ρt d At , {t }−1 et (0) d At , {t }−1 et (ρt ) 2 j η,v η,v −1 j 2 + (H et (ρt )) ρt d At , {t } et (ρt )
η,v ξ,u = {ξ(ρt ) ∓ η(ρt )} d Bt1 − ξ(ρt )2 t u t , γt (0) −η(ρt )2 t vt , γt (ρt ) dt 2 1
j j j ξ(ρt ) H et (0) + η(ρt ) θ j (κt , ρt ) H et (ρt ) ρt dt 2 d
+
j=2
d
1 j j 2 1 − θ j (κt , ρt )2 (H et (ρt ))2 ρt dt, + η(ρt ) 2 j=2
η,v η,v · η,v η,v · ξ,u ξ,u ξ,u ξ,u and dYt = t d 0 {r }−1 ∂Yr t . where d X t = t d 0 {r }−1 ∂ X r t Then, it follows from the use of the second variation of arclength as in [6] and in [3] that
η,v ξ,u dρt = {ξ(ρt ) ∓ η(ρt )} d Bt1 − ξ(ρt )2 t u t , γt (0) − η(ρt )2 t vt , γt (ρt ) dt 1 + 2
ρt d 2 1 1 Det1 (r ) jk (r, t) − R(jk (r, t), et (r ))et (r ), jk (r, t) dr dt
1 + 2
ρt d 2 1 1 Det1 (r ) j˜k (r, t) − R(j˜k (r, t), et (r ))et (r ), j˜k (r, t) dr dt
0 k=2
0 k=2
η,v ξ,u = {ξ(ρt ) ∓ η(ρt )} d Bt1 − ξ(ρt )2 t u t , γ˙t (0) −η(ρt )2 t vt , γ˙t (ρt ) dt 1 jk (ρt , t), jk (ρt , t) − jk (0, t), jk (0, t) dt 2 d
+
k=2
1 j˜k (ρt , t), j˜k (ρt , t) dt, 2 d
+
k=2
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where the j (s, t) = g (s) et (s) and j˜ (s, t) = g˜ (s) et (s) are the Jacobi fields along g (0) = ξ(ρt ), g (ρt ) = η(ρt ) θ (κt , ρt ), g˜ (0) = 0 and g˜ (ρt ) = γt satisfying η(ρt ) 1 − θ (κt , ρt )2 and where the integral is along the geodesic γt . Also j (s, t)
d j (s, t), since s is the denotes the covariant derivative De1 (s) j (s, t) which is given by ds t 1 parameter along the geodesic with tangent field et . Under the given conditions for our manifolds, g (s) = gκ (ρt , s; ξ, η θ ) and g˜ (s) = gκ ρt , s; 0, η 1 − θ (κ , ·)2 , where gκ (u, s; ξ, η) is given by (6). Thus, since Sκ (0) = 0, we have
(j (ρt , t), j (ρt , t) − j (0, t), j (0, t) = G κ (ρt , ρt ; ξ, η θ (κ , ·)) and
j˜ (ρt , t), j˜ (ρt , t) = G κ ρt , ρt ; 0, η 1 − θ (κ , ·)2
so that the required result follows. It is clear from the result, as well as the proof, of Lemma 5 that ξ,u
dρt t d Bt = −{ξ(ρt ) ∓ η(ρt )} γt (0) dt, where ∓ is − if Tx,y = px,y and is + if Tx,y = m x,y . Thus, the conversion from the Itô to the Stratonovich differential for X ξ,u gives that ξ,u
∂ Xt
1 ξ,u ξ,u ξ,u = ξ(ρt ) t ∂ Bt − ξ (ρt ) dρt t d Bt + ξ(ρt )2 t u t dt 2 1 ξ,u ξ,u = ξ(ρt ) t ∂ Bt + ξ (ρt ) {ξ(ρt ) ∓ η(ρt )} γt (0) dt + ξ(ρt )2 t u t dt. 2 (7)
To obtain the Stratonovich differential for Y η,v we define, for each x and distinct y not on its cut locus, E γ1x,y (s)
: τγx,y (s) (M) → τγx,y (s) (M);
v →
d
θ j (κ j , ρ) e j (s), v e j (s);
j=1
E γ2x,y (s) : τγx,y (s) (M) → τγx,y (s) (M);
v →
d
1 − θ j (κ j , ρ)2 e j (s), v e j (s),
j=1
where θ1 ≡ 1, ρ is ρ(x, y) and the e j (s) along γx,y also vary with x and y. Clearly, E γ1x,y (s) ◦ Tx,γx,y (s) = Tx,γx,y (s) ◦ E x1 and the Itô differential for Y η,v can now be expressed as
ξ,u η,v η,v dYt = η(ρt ) E Y1 η,v TX ξ,u ,Y η,v t d Bt + E Y2 η,v t d B˜ t t
+η(ρt )
123
2
η,v t vt
t
dt.
t
t
(8)
Semimartingales and geometric inequalities on locally symmetric manifolds
297
For each x and distinct y not on its cut locus, we also define the following tangent vectors at γx,y (s): ν p (s, ρ; ξ, η, θ2 , . . . , θd ) d
ξ(ρ) E γ1x,y (s) De j (0) px,γx,y (s) (e j (0)) = η(ρ) j=2
+ η(ρ) θ j (κ j , ρ) E γ1x,y (s) De j (s) px,γx,y (s) (e j (0))
d
ξ(ρ) px,γx,y (s) De j (0) E x1 (e j (0)) + η(ρ) j=2
+ η(ρ) θ j (κ j , ρ)De j (s) E γ1x,y (s) (e j (s))
+ 1 − θ j (κ j , ρ)2 De j (s) E γ2x,y (s) (e j (s))
and ν m (s, ρ; ξ, η, θ2 , . . . , θd ) d
ξ(ρ) E γ1x,y (s) De j (0) m x,γx,y (s) (e j (0)) = η(ρ) j=2
+ η(ρ)
+ η(ρ) θ j (κ j , ρ) E γ1x,y (s) De j (s) m x,γx,y (s) (e j (0))
d
ξ(ρ) m x,γx,y (s) De j (0) E x1 (e j (0)) j=2
+ η(ρ) θ j (κ j , ρ)De j (s) E γ1x,y (s) (e j (s))
+ 1 − θj
(κ j , ρ)2 D
2 j e j (s) E γx,y (s) (e (s))
.
j
It follows from Lemmas 2 and 4 and from the definition of E γx,y (s) and Eq. (3) that ν p (s, ρ; ξ, η, θ2 , . . . , θd ) d
1 − Sκ j (s)
j 1 1 j j ξ(ρ) + η(ρ) θ R e (κ , ρ) E (s), e (s) e (s) = η(ρ) j γ (s) x,y κ j Sκ j (s) j=2
+ ξ(ρ) η(ρ)
d d
j
(θk − θ j ) kk e j (s)
j=1 k=2
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298
H. Le, D. Barden
+ η(ρ)2
d d j 1 − θk θ j − 1 − θk2 1 − θ 2j kk e j (s) j=1 k=2
+ 2 ξ(ρ) η(ρ)
d d
j
∂1 θk k1 R(e1 (0), ek (0))ek (0), e j (0) ek (s),
(9)
j=1 k=2
where jk are defined by (1) and ∂1 θk denotes the partial derivative with respect to the first argument of θk , and where the terms involving the derivative of θ j coming from the last line in the definition of ν p cancel. Similarly, ν m (s, ρ; ξ, η, θ2 , . . . , θd ) = ν p (s, ρ; ξ, η, θ2 , . . . , θd ) +2 ξ(ρ) η(ρ)
d 1 − θ j 1j j e1 (s) j=2
+ 2 η(ρ)
d j=2
Sκ j (s) ξ(ρ) − η(ρ) θ j (κ j , ρ) Sκ j (s)
e1 (s). (10)
Note that, for j > 1, ν m , e j ≡ ν p , e j . Lemma 6 The Stratonovich differential of Y η,v is determined by
η,v
∂Yt
ξ,u , et1 (ρt ) = η(ρt ) TX ξ,u ,Y η,v (t ∂ Bt ), et1 (ρt ) + η(ρt )2 η,v vt , et1 (ρt ) dt t
t
1 − η (ρt ) [η(ρt ) ∓ ξ(ρt )] 2 + ν(ρt , ρt ; ξ, η, θ2 , . . . , θd ), et1 (ρt ) dt
and, for 2 j d, j ξ,u j j η,v ∂Yt , et (ρt ) = η(ρt ) θ j κt , ρt TX ξ,u ,Y η,v t ∂ Bt , et (ρt ) t t j 2 η,v j ˜ + 1 − θ j κt , ρt t ∂ Bt , et (ρt ) j 2 η,v + η(ρt ) vt , et (ρt ) 1 j − ν(ρt , ρt ; ξ, η, θ2 , . . . , θd ), et (ρt ) dt 2 where, if Tx,y = px,y , then ν = ν p and ∓ is − and, if Tx,y = m x,y , then ν = ν m and ∓ is +, and where ν p and ν m are defined by (9) and (10), respectively. Proof We prove the case when Tx,y = px,y . The proof for Tx,y = m x,y is similar.
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Using the expression (8) for the Itô differential for Y η,v and noting that B and B˜ are independent, the conversion from the Itô to the Stratonovich differentials for Y η,v gives that η,v ∂Yt
= η(ρt )
E Y1 η,v t
p
ξ,u
ξ,u η,v ˜ 2 + E Y η,v t ∂ Bt η,v t ∂ Bt
X t ,Yt
t
η,v
+ η(ρt )2 t
vt dt
ξ,u 1 η,v ˜ 1 2 − η (ρt ) dρt E Y η,v p ξ,u η,v t d Bt + E Y η,v t d Bt X t ,Yt t t 2
ξ,u 1 − η(ρt ) p ξ,u η,v Dd X ξ,u E Y1 η,v t d Bt X t ,Yt t t 2
ξ,u 1 η,v ˜ 1 2 η,v η,v − η(ρt ) DdYt E Y η,v p ξ,u η,v t d Bt + DdYt E Y η,v t d Bt X t ,Yt t t 2 ξ,u 1 − η(ρt ) E Y1 η,v Dd X ξ,u p ξ,u η,v t d Bt X t ,Yt t t 2 ξ,u 1 . (11) +E Y η,v DdYtη,v p ξ,u η,v t d Bt X t ,Yt
t
η,v
Now, dρt dYt , et (ρt ) = δ1 j {η(ρt ) − ξ(ρt )} γt (ρt ) dt by Lemma 5. Since Dν1 ν2 j is linear in ν1 , we have by the definition of E γx,y (s) and Eq. (3) that j
ξ,u Dd X ξ,u E 1 ξ,u t d Bt Xt
t
=
d
ξ,u ξ,u d X t , etk (0) Dek (0) E 1 ξ,u t d Bt
=
Xt
t
k=2 d
ξ,u
d X t , etk (0)
⎧ d ⎨
k=2 j
⎩
j j ξ,u θ j Dek (0) et (0) t d Bt , et (0) t
j=1
ξ,u
j
+ et (0) t d Bt , Dek (0) et (0) t
= ξ(ρt )
⎧ d ⎨ k=2
⎩
+ 2 θk ∂1 θk
+
d
ξ,u
j
∂1 θ j etk (0)κ j t d Bt , et (0) et (0)
j=1
θk
j
⎫ ⎬ ⎭
d j j (θk − θ j ) kk et (0)
d j=1
j=1 j
j
⎫ ⎬
k1 R(et1 (0), etk (0)) etk (0), et (0) etk (0) dt. ⎭
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H. Le, D. Barden
Similarly, DdYtη,v E Y1 η,v t =
d
p
ξ,u η,v t d Bt
ξ,u
X t ,Yt
η,v dYt , etk (ρt )
k=2
= η(ρt )
⎧ d ⎨ k=2
⎩
θk
Dek (ρt ) E Y1 η,v t t
p
ξ,u
ξ,u η,v t d Bt
X t ,Yt
d j j (θk − θ j ) kk et (ρt ) j=1
+ 2 θk ∂1 θk
d j=1
j
j
⎫ ⎬
k1 R(et1 (0), etk (0)) etk (0), et (0) etk (ρt ) dt ⎭
and η,v DdYtη,v E Y2 η,v t d B˜ t t ⎧ d d ⎨ j j 2 2 2 = η(ρt ) 1 − θk 1 − θk − 1 − θ j kk et (ρt ) ⎩ k=2
j=1
− 2 θk ∂1 θk
d j=1
j
j
⎫ ⎬
k1 R(et1 (ρt ), etk (ρt )) etk (ρt ), et (ρt ) etk (ρt ) dt, ⎭
so that D
η,v dYt
E Y1 η,v t
= η(ρt )
p
ξ,u
ξ,u η,v 2 ˜ + E Y η,v t d Bt η,v t d Bt
X t ,Yt
d d
t
j j 2 2 1 − θk θ j − 1 − θk 1 − θ j kk et (ρt ) dt.
j=1 k=2
On the other hand, noting that Dν px,y (ν) = 0 if ν ∈ τx (M) is tangent to γx,y at x and that Dν1 px,y (ν2 ) is linear in ν1 and ν2 , we also have by Lemma 2 that
Dd X ξ,u p t
D
η,v dYt
d 1 − Sκ j (ρt ) 1 ξ,u j j R e = ξ(ρ d B ) (ρ ), e (ρ ) et (ρt ) dt t t t t ξ,u η,v t t t X t ,Yt κ j Sκ j (ρt ) j=2
d ξ,u j 1 − Sκ j (ρt ) p ξ,u η,v t d Bt = η(ρt ) θ j κt , ρt X t ,Yt κ j Sκ j (ρt ) j=2
j j ×R et1 (ρt ), et (ρt ) et (ρt ) dt.
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Semimartingales and geometric inequalities on locally symmetric manifolds
301
Thus, the required result follows from (11) and from
ξ,u j η,v ˜ 2 p ξ,u η,v t ∂ Bt + E Y η,v t ∂ Bt , et (ρt ) X t ,Yt t ⎧ ξ,u 1 ⎪ ⎨ TX tξ,u ,Ytη,v t ∂ Bt , et (ρt ) = ξ,u j 2 η,v j j ⎪ ˜ ⎩ θ j κt , ρt T ξ,u η,v t ∂ Bt + 1−θ j κt , ρt t ∂ Bt , et (ρt ) X ,Y
E Y1 η,v t
t
t
j =1 j > 1.
4 The SDEs for exp X ξ,u t
η,v α exp−1ξ,u (Yt ) Xt
For the remainder of this paper, we shall further assume that M has non-negative sectional curvature bounded above by κ0 > 0. Thus, in particular, the eigenvalues of Rγ (0) are always bounded above by κ0 and below by zero. It is possible for the smallest eigenvalues κ d , etc., to be zero when, for example, the rank of the universal covering space of M, a simply connected symmetric space, is at least 2. By definition, up to the first time that X ξ,u and Y η,v lie on each other’s conjugate η,v −1 locus, Z t = exp X ξ,u α exp ξ,u (Yt ) is a semimartingale whose path lies at time t on t
Xt
ξ,u
η,v
the geodesic γt between the paths of the semimartingales X t and Yt at respective distances α ρt and (1 − α) ρt . In order to obtain its SDEs recall that, if x(t) and y(t) are two differentiable curves on M such that, for each t, y(t) = x(t) and y(t) is not on the cut locus of x(t) and if cα (t) is the curve such that, for all t, cα (t) lies on the geodesic γt from x(t) to y(t) at respective distances α ρt and (1 − α) ρt , where ρt = ρ(x(t), y(t)), then since, for fixed t and varying α, cα (t) is the Jacobi field along γt determined by this one-parameter family of geodesics, its components jt (αρt ) in the directions of et (α ρt ) are, respectively, given by s jt1 (s) = x (t), et1 (0) + y (t), et1 (ρt ) − x (t), et1 (0) ρt
(12)
and, for 2 d, by jt (s) =
1 x (t), et (0) Sκ (ρt − s) + y (t), et (ρt ) Sκ (s) Sκ (ρt )
(13)
(cf. [5]). Then, by the Stratonovich transfer principle, the Stratonovich SDE for Z is obtained by replacing x (t) and y (t) in the Eqs. (12) and (13) for cα (t) with the η,v ξ,u Stratonovich differentials ∂ X t and ∂Yt , respectively. However, as in [10], for our purposes we shall require the anti-development of Z to be a time-changed Brownian motion with drift. To achieve this, given X ξ,u , we make the following choices of the various parameters occurring in the definition of Y η,v : we take the operator Tx,y that occurs there to be the parallel translation px,y ; η(s) to be the same as ξ(s),
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H. Le, D. Barden
the corresponding function in the definition of X , and the functions θ (κ , s), for 2 d, to be θ (κ , s) where θ (κ, s) =
1 Sκ (s)2 − Sκ ((1 − α)s)2 − Sκ (α s)2 . (14) 2 Sκ (α s) Sκ ((1 − α)s)
It is clear that, if κ = 0, then θ (0, s) ≡ 1 and, if κ = κ j , then θ (κ , s) = θ j (κ j , s). In particular, if M has constant curvature κ0 , then κ = κ0 and so θ (κ , ·) = θ (κ0 , ·) for 2 d. We check that the θ satisfy the bound required for (5). Lemma 7 For√any fixed κ > 0, the functions defined by (14) satisfy |θ (κ, s)| 1 for all s ∈ [0, π/ κ]. Proof |θ (κ, s)| 1 if and only if {Sκ ((1 − α)s) − Sκ (αs)}2 Sκ (s)2 {Sκ ((1 − α)s) + Sκ (αs)}2 . √ For κ > 0, Sκ (s)/s is decreasing for s ∈ [0, π/ κ] so Sκ (α s) √ α Sκ (s) for α ∈ [0, 1]. Hence, Sκ ((1 − α)s)√+ Sκ (αs) Sκ (s) for all s ∈ [0, π/ κ]. Consider, for fixed s ∈ [0, π/ κ], f (α) = Sκ ((1 − α)s) − Sκ (αs). Then, f (α) = for α ∈ [0, 1] since, fixing α, Sκ ((1 − α)s) + −s {Sκ ((1 − α)s) +Sκ (αs)} is negative √ Sκ (αs) is positive for s ∈ [0, π/ κ] as it decreasing with respect to s and Sκ ((1 − α)π ) + Sκ (απ ) = 0. Hence, Sκ (s) = f (0) Sκ ((1 − α)s) − Sκ (αs) f (1) = −Sκ (s) for all α ∈ [0, 1]. The SDEs for Z , analogous to (5), will then involve the functions ζ (s) = ξ(s) and θ˜ j (κ j , s) = θ˜ (κ j , s) where θ˜ (κ, s) =
Sκ (αs) Sκ ((1 − α)s) + θ (κ, s). Sκ (s) Sκ (s)
(15)
Since ˜ s)2 + {1 − θ (κ, s)2 } θ(κ,
Sκ (αs)2 = 1, Sκ (s)2
√ ˜ s)| 1 for s ∈ [0, π/ κ]. Note that θ j and θ˜ j , |θ (κ, s)| 1 ensures that |θ(κ, 2 j d, depend implicitly, via κ j , on x and y. Additional to that determined by u and v, the drift for Z contains an extra term, arising from the covariant derivative of px,y , in terms of the tangent vector χ (x, y; α) at γx,y (αρ(x, y)) defined by 1 χ (x, y; α), e1 (α ρ) = − α ν p (ρ, ρ; ξ, η, θ2 , . . . , θd ), e1 (ρ) 2 1 + ν p (αρ, ρ; ξ, ζ, θ˜2 , . . . , θ˜d ), e1 (αρ) 2
123
(16)
Semimartingales and geometric inequalities on locally symmetric manifolds
303
and, for 2 j d, 1 S j (αρt ) χ (x, y; α), e j (α ρ) = − ν p (ρ, ρ; ξ, η, θ2 , . . . , θd ), e j (ρ) κ 2 Sκ j (ρ) 1 p (17) + ν (αρ, ρ; ξ, ζ, θ˜2 , . . . , θ˜d ), e j (αρ), 2 where ρ = ρ(x, y) and ν p is defined by (9). Theorem 1 For any fixed α ∈ (0, 1) and any x, y ∈ M such that y = x and y is not on the cut locus of x, let X ξ,u and Y η,v be the semimartingales defined by (4) and (5), respectively, with Tx,y = px,y , η(s) = ξ(s) and θ j (κ j , s) = θ (κ j , s), where θ is defined by (14). Then, up to the first time that X ξ,u and Y η,v lie on each other’s conjugate locus, ζ,w˜ Zt
η,v −1 = exp ξ,u α exp ξ,u (Yt ) Xt
Xt
is the solution of the stochastic differential equations ζ,w˜
∂t
ζ,w˜ ∂ Zt ζ,w˜ d At
ζ,w˜
= Hζ,w˜ ∂ Z t
= = z ζ,w˜ −1 1 d B , {t } et (α ρt ) = tz j ζ,w˜ d Bt , {t }−1 et (α ρt ) =
t
,
ζ,w˜ t
ζ,w˜ ζ,w˜ ∂ At , Z 0 = expx (α exp−1 x (y)), z 2 ζ (ρt ) d Bt + ζ (ρt ) wt dt + χt dt ξ,u d Bt , {t }−1 et1 (0) j , ρ ) d B , {ξ,u }−1 e j (0) θ˜ j (κ t t t t j η,v + 1 − θ˜ j (κ j , ρt )2 d B˜ t , {t }−1 et (ρt ) ,
(18)
2 j d, where ζ (s) = ξ(s), θ˜ j (κ j , s) = θ˜ (κ j , s) are defined via (15), χt is the progressively η,v ζ,w˜ ξ,u measurable process in Rd defined by χt = {t }−1 χ (X t , Yt ; α) with χ given by (16) and (17) and w is the progressively measurable process in Rd defined by the equations
ζ,w˜ −1 1 } et (α ρt )
wt , {t
ξ,u η,v = (1 − α) t u t , et1 (0) + α t vt , et1 (ρt )
and, for 2 j d, S j ((1 − α)ρ ) t j j ζ,w˜ ξ,u wt , {t }−1 et (α ρt ) = t u t , et (0) κ Sκ j (ρt ) S j (αρ ) t j η,v , + t vt , et (ρt ) κ Sκ j (ρt ) j
and where {et | 1 j d} is the orthonormal moving frame which is parallel along γt comprising the eigenvectors of Rγt (0) as defined in Sect. 2.
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H. Le, D. Barden
The occurrence of w˜ in the Eq. (18) is merely a notational convention in order to parallel the definitions (4) and (5). Granted the result, however, it is clear that w˜ must be given by w˜ t = wt + χt /ζ (ρt )2 . Proof As in [10], let the stochastic vector field Jt (s) be obtained by replacing x (t) η,v ξ,u ζ,w˜ and y (t) in (12) and (13) with ∂ X t and ∂Yt , respectively. Then, ∂ Z t = Jt (αρt ) ξ,u η,v and Y , given by (7) and the result of and the Stratonovich differentials of X Lemma 6, respectively, together lead to the following expression for the components Jt of Jt with respect to our chosen frame: 1 ξ,u Jt1 (αρt ) = ζ (ρt ) t ∂ Bt , et1 (0) − α ν p (ρt , ρt ; ξ, η, θ2 , . . . , θd ), et1 (ρt ) dt 2
η,v ξ,u 2 + ξ(ρt ) (1 − α) t u t , et1 (0) + α t vt , et1 (ρt ) dt and, for 2 d, Jt (αρt )
ξ,u η,v ˜ 2 ˜ ˜ = ζ (ρt ) θ (κ , ρt ) t ∂ Bt , et (0) + 1 − θ (κ , ρt ) ∂ Bt , et (ρt )) S ((1 − α)ρ ) S (αρ ) t t η,v ξ,u + t vt , et (ρt ) κ +ξ(ρt )2 t u t , et (0) κ dt Sκ (ρt ) Sκ (ρt ) S (αρ ) 1 t dt. − ν p (ρt , ρt ; ξ, η, θ2 , . . . , θd ), et (ρt ) κ 2 Sκ (ρt )
Thus, ζ,w˜
∂ At = ζ (ρt ) ∂ Btz + ζ (ρt )2 wt dt ζ,w˜ 1 − α ν p (ρt , ρt ; ξ, η, θ2 , . . . , θd ), e1 (ρt ) {t }−1 et1 (α ρt ) dt 2 d S j (αρt ) ζ,w˜ −1 j 1 p j {t } et (α ρt ) dt ν (ρt , ρt ; ξ, η, θ2 , . . . , θd ), et (ρt ) κ − 2 Sκ j (ρt ) j=2 z ζ,w˜ ξ,u d Bt , {t }−1 et1 (α ρt ) = d Bt , {t }−1 et1 (0) z ζ,w˜ ξ,u d Bt , {t }−1 et (α ρt ) = θ˜ (κ , ρt ) d Bt , {t }−1 et (0) η,v + 1 − θ˜ (κ , ρt )2 d B˜ t , {t }−1 et (ρt ) , 2 d. ζ,w˜
The required SDEs for d At of Lemma 6.
then follow from a similar argument to that of the proof
Remarks (i) Since B z is a Brownian motion on Rd , the stochastic anti-development ˜ Similarly to Aζ,w˜ of Z ζ,w˜ on Rd is a time-changed Brownian motion with drift w.
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the case for Y η,v , B z is not entirely correlated with the Brownian motion B that drives X ξ,u . The independent Brownian motion added in each direction depends on the curvature of the plane determined by the geodesic and the associated direction. (ii) With our choices of η and θ j , it can be checked that, if M has constant sectional with curvature κ0 , χ (x, y; α) defined by (16) and (17) will be tangent to γx,y χ(x, y; α) = (d − 1)
1 − Sκ 0 (ρ) ξ(ρ)2 −α (1 + θ (κ0 , ρ)) 2 Sκ0 (ρ) 1 − Sκ 0 (αρ) ˜ 1 + θ (κ0 , ρ) γx,y + (αρ) Sκ0 (αρ)
and, if M is a Kaehler manifold of constant holomorphic sectional curvature κ0 , Lemmas 1 and 3 lead to # 1 − Sκ 0 (ρ) ξ(ρ)2 {1 + θ (κ0 , ρ)} χ(x, y; α) = −α 2 Sκ0 (ρ) $ 1 − Sκ 0 /4 (ρ) (d − 2) {1 + θ (κ0 /4, ρ)} + Sκ0 /4 (ρ) 1 − Sκ 0 (αρ)
1 + θ˜ (κ0 , ρ) + Sκ0 (αρ) %
1 − Sκ 0 /4 (αρ) (d − 2) 1 + θ˜ (κ0 /4, ρ) γx,y (αρ), + Sκ0 /4 (αρ) where ρ = ρ(x, y) in both cases. (iii) To make Z ζ,w˜ a time-changed Brownian motion for a given X ξ,u , the choice of parameters for Y η,v is not unique. For example, we may choose η(s) = ξ(s) σα (s), where σα has the expression given in [10] with κ there replaced by κ 2 , and choose θ j (κ j , s) =
1 2 Sκ j (α s) Sκ j ((1 − α)s) σα (s)
2 2 × Sκ j (s) σ˜ α (s) − Sκ j ((1 − α)s)2 − Sκ j (α s) σα (s)
for 2 j d where, for α ∈ (0, 1), σ˜ α (s) = 1 − α − α σα (s), s ∈ R+ . In particular, this choice gives θ2 ≡ 1 and, if κ = κ j , then θ (κ , s) = θ j (κ j , s) and so it generalises the construction of [10]. It can be checked that the θ j so defined have bounds −1 and 1 when s is small enough and that, if Tx,y = m x,y , the bounds √ η,v hold for s ∈ [0, π/ κ ]. Then, the corresponding exp ξ,u α exp−1ξ,u (Yt ) will be Xt
Xt
a time-changed Brownian motion with drift; its SDEs, analogous to (5), will involve
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the functions ζ (s) = ξ(s) σ˜ α (s) and θ˜ j (κ j , s) =
1 σ˜ α (s)
Sκ j ((1 − α)s) S j (αs) + κ σα (s) θ j (κ j , s) , 2 j d; Sκ j (s) Sκ j (s)
and the expression for its drift resembles that given in Theorem 1. However, since κ 2 depends generally on x and y, both σα and σ˜ α so defined usually depend, implicitly via apart, so that the drift in the stochastic antiκ 2 , on x and y and not just on their distance η,v development of exp ξ,u α exp−1ξ,u (Yt ) involves an additional term arising from the Xt
Xt
derivatives of σα and σ˜ α with respect to x and y. 5 Two geometric inequalities on M Let K 0 and K 1 be two non-empty compact convex subsets of M such that, for any x in K 0 and distinct y in K 1 , x and y do not lie on each other’s cut locus so that there is always a unique shortest geodesic between them. Writing d K 0 ,K 1 = max{ρ(x, y) | x ∈ K 0 , y ∈ K 1 }, √ we also assume that d K 0 ,K 1 < π/ κ 0 to ensure that the θ j and θ˜ j defined in the previous section are all bounded between −1 and 1. For a fixed α ∈ (0, 1), define K α to consist of all points z in M such that, for some x ∈ K 0 and y ∈ K 1 , z lies on the shortest geodesic joining x and y at a distance α ρ(x, y) from x and (1 − α) ρ(x, y) from y. Then, K α = {z ∈ M | z = expx (α exp−1 x (y)) for some x ∈ K 0 and y ∈ K 1 }. We now apply the result of Theorem 1 to obtain an inequality, analogous to that in [10], relating the solutions on these three sets of the initial-boundary value problem for the parabolic equation 1 ∂ψ = ψ − h(x) ψ ∂t 2 ψ(t, x) = 0, lim ψ(t, x) = f (x), t↓0
(t, x) ∈ (0, ∞) × int(K ); (t, x) ∈ (0, ∞) × ∂ K ; x ∈ int(K );
(19)
where, on int(K ), f is positive and continuous and h is non-negative and continuous. Theorem 2 Fix α ∈ (0, 1). For j = 0, 1, α, let ψ j (t, x) be the solution of Eq. (19) with K replaced by K j and with f (x) = f j (x) and h(x) = h j (x) for x ∈ K j . Assume that f α and h α satisfy the conditions that, for some constant β ∗ > 0 and for any x ∈ K 0 , y ∈ K 1 and z = expx (α exp−1 x (y)), f α (z) β ∗ f 0 (x)kα,0 f 1 (y)kα,1 h α (z) kα,0 h 0 (x) + kα,1 h 1 (y),
123
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307
where, for j = 0, 1,
kα, j
2 Sκ0 (1 − j − α) d K 0 ,K 1 =2
2 . |1 − j − α| Sκ0 d K 0 ,K 1
Then, ψα (t, z) β ∗ ψ0 (t, x)kα,0 ψ1 (t, y)kα,1 e−Cα t , where
Cα = max |χ (x, y; α ∧ (1 − α))|2 | x ∈ K 0 , y ∈ K 1 with the corresponding ξ and η chosen to be 1 and with θ j and θ˜ j defined via (14) and (15), respectively. −1 Proof Since expx α exp−1 x (y) = exp y (1 − α) exp y (x) , without loss of generality we assume, in the proof, that α ∈ (0, 1/2]. We also assume that x = y since the case for x = y can be deduced by taking appropriate limits. In a similar manner to that developed in [10], the proof makes use of the solution of (19) in terms of Brownian motion together with X t1,u , Yt1,v and Z t1,w˜ as constructed in Theorem 1 with ξ ≡ 1. Hence, we quote results from the proof of Theorem 2 in [10], giving proof only so far as is necessary to make our current account comprehensible. For any x ∈ K 0 , y ∈ K 1 and any given u, v ∈ V, consider the semimartingales X ξ,u and Y η,v constructed by (4) and (5) with Tx,y = px,y , ξ = η ≡ 1 and θ j given by (14), and the semimartingale Z t1,w˜ = exp X 1,u α exp−11,u (Yt1,v ) . By Theorem 1, Xt
t
Z 1,w˜ is well defined a.s. before it exits from K α , under the assumptions we made at the beginning of the section. By choosing appropriate u h , vn ∈ V such that lim u n = u ∗ and lim vn = v ∗ , where n→∞
n→∞
∗ −1 1,u ∗ u ∗s = 1,u t − s, X and ∇ log ψ 0 s s
∗ −1 ∗ vs∗ = s1,v ∇ log ψ1 t − s, Ys1,v , the analysis in [10], using the Girsanov theorem and involving results on stochastic optimal control, shows that ⎡ − log ψ0 (t, x) = E ⎣ ⎡ − log ψ1 (t, y) = E ⎣
t 0
t
1 ∗2 |u | + h 0 X s1,u 2 s
∗
⎤
*
+ ∗ ds ⎦ − E log f 0 X t1,u
⎤
*
+ ∗ 1 ∗2 ∗ |vs | + h 1 Ys1,v ds ⎦ − E log f 1 Yt1,v , 2
0
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and
⎡
− log ψα (t, z) E ⎣
t
⎤
*
+ ∗ 1 ∗ 2 ∗ w˜ s + h α Z s1,w˜ ds ⎦ − E log f α Z t1,w˜ , 2
0
since there is no time-change involved for either Y 1,v or Z 1,w˜ in our current setup. Thus, using the definition of w from Theorem 1 and the expression for w, ˜ we have t
|w˜ s∗ |2 ds
0
t 2
t |χs | ds + 2
0
+
η,v ∗ ∗ 1 2 ∗ ∗ 1 2 (1 − α) 1,u ds s u s , es (0) + α s vs , es (ρs )
0
d t
2
∗ ∗ j 1,u s u s , es (0)
j=2 0
⎧ ⎨
S j (αρs ) Sκ j ((1 − α)ρs ) η,v ∗ ∗ j + s vs , es (ρs ) κ Sκ j (ρs ) Sκ j (ρs )
2 ds
⎫ 2⎬ d S ((1 − α)ρ ) ∗ j 2 2 s j κ ∗ (1 − α) 2 s u ∗s , es1 (0) + 1,u ds s u s , es (0) ⎩ (1 − α) Sκ j (ρs ) ⎭ j=2 0 ⎧ ⎫ t ⎨ d η,v ∗ ∗ j 2 2 Sκ j (α ρs ) 2⎬ η,v ∗ ∗ 1 +α 2 s vs , es (ρs ) + s vs , es (ρs ) ds +2 Cα t. ⎩ α Sκ j (ρs ) ⎭ t
1,u ∗
j=2
0
Since M has non-negative sectional curvature, Sκ (α ρ)/Sκ (ρ), for any√fixed α ∈ [0, 1], is greater than α and increasing with respect to ρ for ρ ∈ [0, π/ κ], and so the above inequality implies that t
|w˜ s∗ |2 ds
t 2 Cα t + kα,0
0
|u ∗s |2 ds
t + kα,1
0
|vs∗ |2 ds.
0
Putting these results, and the assumption (20), together gives − log ψα (t, z) − log β ∗ + Cα t ⎡
1,u ∗
+ kα,0 E ⎣− log f 0 X t ⎡ + kα,1 E ⎣− log f 1 ∗
t +
1 ∗2 |u | ds + h 0 X s1,u 2 s
∗
0
∗ Yt1,v
t +
1 ∗2 ∗ |vs | ds + h 1 Ys1,v 2
⎤ ds ⎦ ⎤ ds ⎦
0
= − log β + Cα t − kα,0 log ψ0 (t, x) − kα,1 log ψ1 (t, y) as required.
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To deduce, from Theorem 2, an inequality involving the first eigenvalues of the Laplacian on the three sets, as in [10], we choose, in Theorem 2, h j ≡ 0 and the constant β ∗ to satisfy
β ∗ max Fκ,α (ρ(x, y))d−1 x ∈ K 0 , y ∈ K 1 , where Fκ,α (s) =
(1 − α)1−α α α Sκ (s) . Sκ ((1 − α) s)1−α Sκ (α s)α
Then, for any fixed t, we apply the Riemannian version of the Prékopa-Leindler inequality of Cordero-Erausquin et al. (cf. [2]), as well as the Hölder inequality, to the ψ j . Since the first eigenvalues of the Laplacian on K j with the Dirichlet boundary condition can be given in terms of the ψ j by 1 log t→∞ t
λ(K j ) = − lim
ψ j (t, x) dvol(x) Kj
(cf. [4, pp. 171–172]), the result of Theorem 2 establishes the following. Corollary 1 Let λ(K j ), j = 0, α, 1, denote the first eigenvalue of the Laplacian on K j with the Dirichlet boundary condition. If K j for j = 0, 1 satisfy the conditions set in the preamble to Theorem 2, then λ(K α ) kα,0 λ(K 0 ) + kα,1 λ(K 1 ) + Cα , where kα, j and Cα are as given in Theorem 2. Remarks (i) The constants kα, j , appearing in Theorem 2 and Corollary 1, are larger than those in [10]. However, it is clear from the proof of Theorem 2 that, if ζ,w˜ χt appearing in the SDEs for Z 1,w˜ always has the same direction as γt (αρt ), for example, when M has constant sectional curvature κ0 or M is a Kaehler manifold of constant holomorphic sectional curvature κ0 as we noted at the end of Sect. 4, the constants kα, j , j = 0, 1, can be further improved to
kα, j
⎧
2 ⎫ ⎪ ⎨ ⎬ Sκ0 (1 − j − α) d K 0 ,K 1 ⎪ = max 2|1 − j − α|, ,
2⎪ ⎪ ⎩ |1 − j − α| Sκ0 d K 0 ,K 1 ⎭
which are identical with those in [10]. (ii) Since there is no time change involved for any of the X 1,u , Y 1,v and Z 1,w˜ used in the proof of Theorem 2, the constant cα , which is less than 1, involved in the resulting inequalities in [10] disappears in our current setting.
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Fig. 1 Graph of the difference between Cα in [10] and Cα in this paper, multiplied by 4/(d − 1)2 , as a function of d K 0 ,K 1 when κ0 = 1. The curves, reading from the bottom up, correspond to α = 0.3, 0.4, and 0.5, respectively
3
2.5
2
1.5
1
0.5
0 0.5
1
1.5
2
2.5
3
d
(iii) In the special case when M has constant curvature κ0 , the magnitude of χ given at the end of Sect. 4 differs from the corresponding |χ | in [10] and so does Cα here from Cα there. However, it can be checked that |χ(x, y; α)| is an increasing function of ρ(x, y) and so it is still the case that the dependence of Cα here on K 0 and K 1 is √ via d K 0 ,K 1 . In particular, the limit of Cα as ρ(x, y) ↑ π/ κ0 is (d − 1)2 (1 − cos(απ )2 )2 (1 − cos((α ∧ (1 − α))π ))2 κ0 < (d − 1)2 κ0 2 4 sin(απ ) sin((α ∧ (1 − α))π )2 for α ∈ (0, 1), where the right hand side of the above inequality is the limit of Cα in √ [10] as ρ(x, y) ↑ π/ κ0 . In fact, as shown in Fig. 1, as a function of d K 0 ,K 1 , Cα here is always less than that in [10]. (iv) When M is a Kaehler manifold of constant holomorphic sectional curvature κ0 , the constant Cα again depends on K 0 and K 1 via d K 0 ,K 1 in an increasing manner √ and its limit as ρ(x, y) ↑ π/ κ0 is 1 κ0 4
#
√ %2 1 − cos(απ )2 d − 2 1 − cos(απ/ 2)2 + √ . √ sin(απ ) 2 sin(απ/ 2)
(v) When d K 0 ,K 1 ↓ 0, |χ| ∼ O(d K 0 ,K 1 ) and kα, j ≈ |1 − j − α|. A modification of the estimation of the bound for w˜ in the proof of Theorem 2 then suggests that the resulting inequality is approximately log ψα (t, z) log β ∗ + (1 − α) log ψ0 (t, x) + α log ψ1 (t, y)
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up to O(d K 0 ,K 1 ), recovering the corresponding result of [1] for Euclidean space. A similar approximation holds for the eigenvalues of Laplacian. (vi) A similar argument to that in (v) shows that when κ0 ↓ 0, our inequalities reduce to the corresponding ones for Euclidean space. Acknowledgments This research was supported by the Leverhulme Trust.
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