Page - 406 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 Euclideanspace. Thedistance is thusweightedby theanisotropyofu, andu canbe interpretedas asquarerootcovariancematrixΣ1/2. However, the above approachdoesnot specifyhowu changeswhenmoving away from the basepoint x0. TheuseofLogx0x effectively linearises thegeometryaround x0, butageometrically naturalwaytorelateuatpointsnearbytox0willbe toparallel transport it, equivalentlyspecifying thatuwhentransporteddoesnotchangeasmeasuredfromthecurvedgeometry. Thisconstraint is nonholonomic, andit implies thatanypathfromx0 tox carrieswith itaparallel transportofu lifting paths fromM topaths in the framebundleFM. It thereforebecomesnatural toequipFMwitha form ofmetric thatencodestheanisotropyrepresentedbyu. Theresult is thesub-RiemannianmetriconFM definedbelowthatweights infinitesimalmovementsonMusingtheparallel transportof the frameu. Optimalpaths for thismetricaresub-Riemanniangeodesicsgiving the familyofmostprobablepaths for thedrivingprocess that thispaperconcerns. Figure1showsonesuchpathforananisotropicnormal distributionwithManellipsoidembeddedinR3. 2. FrameBundles,StochasticDevelopment,andAnisotropicDiffusions LetMbeafinitedimensionalmanifoldofdimensiondwithconnectionC, andletx0 beafixed point inM.WhenaRiemannianmetric ispresent,andC is itsLevi–Civitaconnection,wedenote the metricgR. Foragiven interval [0,T],we letW(M)denote theWienerspaceofcontinuouspaths inM startingatx0. Similarly,W(Rd) is theWienerspaceofpaths inRd.WeletH(Rd)denote thesubspace ofW(Rd)offiniteenergypaths. Let now u = (u1, . . . ,ud) be a frame for TxM, x ∈ M; i.e., u1, . . . ,ud is an ordered set of linearly independent vectors in TxM with span{u1, . . . ,ud} = TxM. We can regard the frame as an isomorphismu :Rd → TxMwithu(ei) = ui,where e1, . . . ,ed denotes the standardbasis inRd. Stochasticdevelopment (e.g., [4])providesan invertiblemapϕu fromW(Rd) toW(M). Throughϕu, Euclidean semi-martingales map to stochastic processes on M. When M is Riemannian and u orthonormal, theresult is theEells–Elworthy–MalliavinconstructionofBrownianmotion[17].Wehere outline thegeometrybehinddevelopment, stochasticdevelopment, theconnection, andcurvature, focusing inparticularonframebundlegeometry. 2.1. TheFrameBundle For eachpoint x ∈ M, let FxMbe the set of frames forTxM (i.e., the set of orderedbases for TxM). Theset{FxM}x∈M canbegivenanaturaldifferential structureasafiberbundleonM called the framebundleFM. It canequivalentlybedefinedas theprincipalbundleGL(Rd,TM).Welet the mapπ :FM→Mdenote thecanonicalprojection. Thekernelofπ∗ :TFM→TM is thesub-bundle ofTFM thatconsistsofvectors tangent to thefibersπ−1(x). It isdenotedthevertical subspaceVFM. Wewilloftenwork ina local trivializationu=(x,u1, . . . ,ud)∈FM,wherex=π(u)∈Mdenotes the basepoint,andforeach i=1,. . . ,d,ui∈TxM is the ith framevector. Forv∈TxMandu∈FMwith π(u)= x, thevectoru−1v∈Rd expressesv incomponents in termsof the frameu.Wewilldenote the vectoru−1v framecoordinatesofv. Foradifferentiable curve xt inMwith x= x0, a frameu forTx0M canbeparallel transported along xt by parallel transporting each vector in the frame, thus giving a path ut ∈ FM. Such paths are calledhorizontal, andhave zero acceleration in the sense C(u˙i,t) = 0. For each x ∈ M, their derivatives form a d-dimensional subspace of the d+d2-dimensional tangent space TuFM. Thishorizontal subspaceHFMandthevertical subspaceVFM togethersplit the tangentbundleof FM (i.e.,TFM=HFM⊕VFM). Thesplit inducesamapπ∗ : HFM→TM, seeFigure3. Forfixed u∈FM, therestrictionπ∗|HuFM :HuFM→TxM isanisomorphism. Its inverseiscalledthehorizontal lift and isdenotedhu in the following. Usinghu, horizontalvectorfieldsHe onFMaredefinedfor vectors e∈RdbyHe(u)= hu(ue). Inparticular, thestandardbasis (e1, . . . ,ed)onRdgivesdglobally definedhorizontal vector fields Hi ∈ HFM, i = 1,. . . ,dby Hi = Hei. Intuitively, thefields Hi(u) model infinitesimal transformations inMofx0 indirectionui=ueiwithcorresponding infinitesimal 406