A stochastic look at geodesics on the sphere

07/11/2017
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We describe a method allowing to deform stochastically the completely integrable (deterministic) system of geodesics on the sphere S2 in a way preserving all its symmetries.

A stochastic look at geodesics on the sphere

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application/pdf A stochastic look at geodesics on the sphere Marc Arnaudon, Jean-Claude Zambrini
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A stochastic look at geodesics on the sphere

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A stochastic look at geodesics on the sphere Marc Arnaudon, Jean-Claude Zambrini Inst. de Mathématiques de Bordeaux, Univ. de Bordeaux, Grupo de Fı́sica-Matemática Univ. Lisbon marc.arnaudon@math.u-bordeaux.fr , jczambrini@fc.ul.pt Abstract. We describe a method allowing to deform stochastically the completely integrable (deterministic) system of geodesics on the sphere S2 in a way preserving all its symmetries. Keywords: Geodesic flow; stochastic deformation; integrable systems. 1 Introduction Free diffusions on a sphere S2 are important case studies in applications, for instance in Biology, Physics, Chemistry, Image processing etc..., where they are frequently analysed with computer simulations. However, as for most diffusions on curved spaces, no closed form analytical expressions for their probability densities are available for such simulations.Another way to express the kind of difficulties one faces is to observe that one cannot define Gaussian functions on S2 . If, instead of free diffusions on S2 we consider their deterministic counterpart, the classical geodesic flow, a famous integrable system whose complete solution dates back to the 19th century, the situation is much simpler. Indeed, one can use the conservation of angular momentum and energy to foliate the phase space (the cotangent bundle of its configuration space). We describe here a method allowing to construct free diffusions on S2 as stochastic deformations of the classical geodesic flow, including a probabilistic counterpart of its conservation laws. 2 Classical geodesics The problem of geodesic on the sphere S2 is a classical example of completely integrable elementary dynamical system [1]. For a unit radius sphere and using spherical coordinates (qi ) = (θ, φ) ∈ ]0, π[×[0, 2π] where φ is the longitude, the Lagrangian L of the system is the scalar defined on the tangent bundle TS2 of the system by L(θ, φ, θ̇, φ̇) = (θ̇2 + sin2 θ φ̇2 ) (1) (where θ̇ = dθ dt etc ...), since it coincides with ds2 = gij dqi dqj , here g = (gij ) =  1 0 0 sin2 θ  . The Euler-Lagrange equations d dt  ∂L ∂q̇i  − ∂L ∂qi i = 1, 2 (2) in these coordinates are easily solved. They describe the dynamics of the ex- tremals (here minimal) curves of the action functional SL[q(·)] = Z Q2 Q1 L(q, q̇)dt (3) computed, for instance, between two fixed configurations Q1 = (θ1, φ1) and Q2 = (θ2, φ2) in the configuration space. Those equations are θ̈ = φ̇2 sin θ cos θ, φ̈ = −2θ̇φ̇ cotg θ (4) Defining the Hamiltonian H : T∗ S2 → R as the Legendre transform of L, we have H = 1 2 gij pipj , where pi = ∂L ∂q̇i = gij q̇j denote the momenta, here H(θ, φ, pθ, pφ) = 1 2 (p2 θ + 1 sin2 θ p2 φ), (5) with pθ = θ̇, pφ = sin2 θφ̇. It is clear that the energy H is conserved during the evolution. There are three other first integrals for this system, corresponding to the three components of the angular momentum L. They can be expressed as differential operators of the form Xθ j ∂ ∂θ + Xφ j ∂ ∂φ , j = 1, 2, 3, namely L1 = sin φ ∂ ∂θ + cos φ tan θ ∂ ∂φ L2 = − cos φ ∂ ∂θ + sin φ tan θ ∂ ∂φ , L3 = − ∂ ∂φ (6) In geometrical terms, written as Lj = (Xθ j , Xφ j ), j = 1, 2, 3, they are the three Killing vectors for S2 , forming a basis for the Lie algebra of the group of isometries SO(3) of S2 . L3 corresponds to the conservation of the momentum pφ. The integrability of this dynamical system relies on the existence of the two first integrals H and pφ. They allow to foliate the phase space by a two-parameter family of two-dimensional tori. Let us recall that the list of first integrals of the system is the statement of Noether’s Theorem, according to which the invariance of the Lagrangian L under the local flow of vector field v(1) = Xi (q, t) ∂ ∂qi + dXi dt ∂ ∂q̇i (7) associated with the group of transformations (qi , t) → (Qi α = qi + αXi (q, t), τα = t + αT(t)) for α a real parameter, provides a first integral along extremals of SL of the form d dt (Xi pi − TH) = 0. (8) The coefficients Xi , T must, of course, satisfy some relations between them called “determining equations” of the symmetry group of the system [2]. For instance, for our geodesics on S2 , T = 1, X = (Xθ , Xφ ) = (0, 0) corresponds to the conservation of the energy H, and T = 0, X = (0, −1) to the conservation of pφ In fact, the three vectors Xj must satisfy the Killing equations in the (θ, φ) coordinates, ∇θ Xφ j + ∇φ Xθ j = 0 , j = 1, 2, 3 (9) where ∇· denotes the covariant derivatives. 3 Stochastic deformation of the geodesics on the sphere Many ways to construct diffusions on S2 are known. In the spirit of K. Itô [3], we want to deform the above classical dynamical system in a way preserving the essential of its qualitative properties. Let us start from the backward heat equation for the Laplace-Beltrami “Hamil- tonian” operator H (without potentials). in local coordinates (qi ) it can be writ- ten ∂η ∂t = Hη, where g = det(gij ) and H = − 1 2 ∆LB = − 1 2 √ det g ∂ ∂qi p det g gij ∂ ∂qj  . (10) A more revealing form in terms of the Christoffel symbols of the Riemannian connection is − 1 2 ∆LB = − gij 2 ∂2 ∂qi ∂qj + 1 2 Γi jk(q)gjk (q) ∂ ∂qi . (11) Indeed, the extra first order term, of purely geometric origin, will coincide with the drift of the simplest diffusion on our manifold, the Brownian motion; this was observed by K. Itô , as early as 1962 [3]. In our spherical case, one finds Γθ jkgjk = −cotg θ , Γφ jkgjk = 0. (12) Now we shall consider general diffusions zi on S2 solving SDEs of the form dzi (τ) = (Bi − 1 2 Γi jkgjk )dτ + dWi (τ) , τ > t (13) for Bi an unspecified vector field, where dWi (τ) = σi kdβk (τ) with σi k the square root of gij , i. e. gij = σi kσj k, in our case σ =  1 0 0 1 sin θ  and β is a two dimensional Wiener process. Here is the stochastic deformation of the extremality condition for dynamical trajectories in terms of the classical action SL. It will be convenient to consider SL as a function of starting configurations q at a time t. For convenience, we shall add a final boundary condition to Su to SL. Let SL(q, t) be defined now by SL(q, t) = − ln η(q, t), where η(q, t) is a positive solution of the backward heat equation for a (smooth) final boundary condition Su(q) , u > t. Let Bi in (13) be adapted to the increasing filtration Pτ , bounded but otherwise arbitrary. Then SL(q, t) ≤ Eqt{ Z u t 1 2 Bi Bi(z(τ), τ)dτ + Su(z(u))} (14) where Eqt denotes the conditional expectation given z(t) = q. The equality holds on the extremal diffusion on S2 , of drift Bi (q, t) = ∂iη η (q, t) = −∇i SL. (15) This means, on S2 , that SL minimizes the r.h.s. functional of (14) for the La- grangian L = 1 2 [( ∂θη η )2 + sin2 θ( ∂φη η )2 ] (16) where, manifestly, (∂θη η ) and ( ∂φη η ) plays the roles of θ̇ and φ̇ in the deterministic definition (1). Let us observe that after the above logarithmic change of variable, it follows from the backward heat equation that the scalar field SL solves − ∂SL ∂t + 1 2 k∇SLk2 − 1 2 ∇i ∇iSL = 0, (17) with t < u and SL(q, u) = Su(q). This is an Hamilton-Jacobi-Bellman equation, whose relation with heat equa- tions is well known and used in stochastic control [4]. The Laplacian term rep- resents the collective effects of the irregular trajectories τ → zi (τ) solving (13). A second order in time dynamical law like (4) requires the definition of the parallel transport of our velocity vector field Bi . In [3] Itô had already mentioned that there is some freedom of choice in this, involving the Ricci tensor Ri k on the manifold. One definition is known today in Stochastic Analysis as “Damped parallel transport” [5]. Then the generator of the diffusion zi acting on a vector field V on S2 is given by DtV i = ∂V i ∂t + Bk ∇kV i + 1 2 (∆V )i (18) where, instead of the Laplace-Beltrami operator, one has now ∆V i = ∇k ∇kV i + Ri kV k (19) When acting on scalar fields ϕ, Dt reduces to the familiar form Dtϕ = ∂ϕ ∂t + Bk ∇kϕ + 1 2 ∇k ∇kϕ (20) When ϕ = qk , Dtϕk = Bk (z(t), t) = −∇k SL, so the r.h.s. Lagrangian of (14) is really 1 2 kDtzk2 , for k · k the norm induced by the metric, as it should. For the vector field Bi , we use (17) and the integrability condition ∂ ∂t ∇i SL = ∇i ∂SL ∂t , following from the definition of SL, to obtain DtDtzi = 0 (21) i.e., the stochastic deformation of both O.D.E.s (4) when z(t) = (θ(t), φ(t)) solve Eq. (13) namely, in our case, dθ(t) = ∂θη η + cotgθ 2  dt + dWθ (t) , dφ(t) = 1 sin2 θ ∂φη η  dt + dWφ (t) (22) The bonus of our approach lies in the study of the symmetries of our stochas- tic system. The symmetry group of the heat equation, in our simple case with constant positive curvature, is generated by differential operators of the form [6] N̂ = Xi (q)∇i + T ∂ ∂t + α (23) where T and α are constants, and the Xi are three Killing vectors on (S2 , g). Besides a one dimensional Lie algebra generated by the identity, another one corresponds to T = 1 and X = (Xθ , Xφ ) = (0, 0). This provides the conservation of energy defined here, since SL = − ln η, by h(θ(t), φ(t)) = −1 η ∂η ∂t or, more explicitly, h = 1 2 gij BiBj + 1 2 gij ∂ ∂qi Bj − 1 2 Γi jkgjk Bi (24) Using (20), one verifies that Dth(z(t), t) = 0 (25) in other words, h is a martingale of the diffusion z(t) extremal of the Action func- tional in (14). This is the stochastic deformation of the corresponding classical statement (8) when X = (0, 0), T = 1. Analogously, our (deformed) momentum pφ is a martingale. In these conditions, one can define a notion of integrability for stochastic systems (not along Liouville’s way, but inspired instead by Ja- cobi’s classical approach) and show that, in this sense, our stochastic problem of geodesics on the sphere is as integrable as its deterministic counterpart. This will be done in [7]. To appreciate better in what sense our approach is a stochastic deformation of the classical problem of geodesics in S2 , replace our metric (gij ) by ~(σij ) for σij the Riemannian metric, where ~ is a positive constant, and take into account that our underlying backward heat equation now becomes ∂η ∂t = − ~ 2 ∆LBη (26) then, one verifies easily that, when ~ → 0 , Dt → d dt , the Lagrangian of (14) reduces to the classical one (1) and the conditional expectation of the action (1) disappears. The Hamilton-Jacobi-Bellman equation (17) reduces to the one of the classical dynamical system and our martingales to its first integrals. In this respect, observe that general (positive) final conditions for Eq. (26) may depend as well on ~. They provide analogues of Lagrangian submanifolds in the semiclassical limit of Schrödinger equation (Cf Appendix 11 of [13]). We understand better, now, the role of the future boundary condition Su in (1): when Su is constant, the extremal process z(·) coincides with the Brownian motion on S2 but, of course, in general this is not the case anymore. Stochastic deformation on a Riemannian manifold was treated in [8]. For another approach c.f. [9]. In spite of what was shown here, our approach can be made invariant under time reversal, in the same sense as our underlying classical dynamical system. The reason is that the very same stochastic system can be studied as well with respect to a decreasing filtration and an action functional on the time interval [s, t], with an initial boundary condition S∗ s (q). This relates to the fact that to any classical dynamical systems like ours are associated, in fact, two Hamilton-Jacobi equations adjoint with respect to the time parameter. The same is true after stochastic deformation. So, a time-adjoint heat equation, with initial positive boundary condition, is involved as well. The resulting (“Bernstein reciprocal” ) diffusions, built from these past and future boundary conditions, are invariant under time reversal on the time interval [s, u]. C.f. [10], [12]. In particular, Markovian Bernstein processes are uniquely determined from the data of two (stictly positive) probability densities at different times s and u, here on S2 . They solve a “Schrödinger’s optimization problem”, an aspect very reminiscent of foundational questions of Mass Transportation theory [14]. The close relations between this theory and our method of Stochastic Deformation have been carefully analysed in [11], where many additional references can be found as well. References [1] Dubrovin, B.A., Krichever, J.M., Novikov, S.P.: Integrable systems I. Encyclop. of Mathematical Sciences, Vol. 4, Dynamical Systems IV, Ed. V.I. Arnold, S.P. Novikov, Springer (1990). [2] Olver, P.: Applications of Lie groups to differential equations. 2nd ed., Springer (1993). [3] Itô, K.: The Brownian motion and tensor fields on a Riemannian manifold. Proc. Int. Congress Math. (Stockholm), 536–539 (1962). [4] Fleming, W.H., Soner, H.M.: Controlled Markov processes and viscosity solutions. 2nd ed., Springer (2005). [5] Malliavin, P.: Stochastic Analysis. Springer (1997). [6] Kuwabara, R.: On the symmetry algebra of the Schrödinger wave equation. Math. Japonica 22, p. 243 (1977). [7] Léonard, C., Zambrini, J.-C.: Stochastic deformation of Jacobi’s integrability The- orem in Hamiltonian mechanics. In preparation. [8] Zambrini, J.-C.: Probability and Quantum Symmetries in a Riemannian manifold. Progress in Probability, Vol. 45, Birkhäuser (1999). [9] Kolsrud, T.: Quantum and classical conserved quantities: Martingales, conservation law and constants of motion. Stochastic Analysis and Applications: Abel Symposium, Springer (2005). [10] Zambrini, J.-C.: The research program of Stochastic Deformation (with a view to- ward Geometric Mechanics). Stochastic Analysis, a series of Lectures, Ed. R. Dalang, M. Dozzi, F. Flandoli, F. Russo, Birkhäuser (2015). [11] Léonard, C.: A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst. A, 34(4) (2014). [12] Léonard, C. Roelly, S., Zambrini, J.-C.: Reciprocal processes. A measure- theoretical point of view. Probability Surveys, Vol. 11, 237–269 (2014). [13] Arnold, V.: Méthodes Mathématiques de la Mécanique Classique. Ed. Mir, Moscou (1976). [14] Villani, C.: Optimal Transport, old and new. Springer, Grundlehren der math. Wissens. (2009).