Page - 412 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 Therefore, the metric on HFM can be extended to a Riemannian metric on FM. Such metrics incorporate theanisotropicallyweightedmetriconHFM,however,allowingverticalvariationsand thus thatcovariancescanchangeunrestricted. WhenM isRiemannian, themetricgFM is inadditionrelatedtotheSasaki–MokmetriconFM [18] thatextends theSasakimetriconTM.As for theaboveRiemannianmetriconFM, theSasaki–Mok metricallowspaths inFM tohavederivatives in thevertical spaceVFM.OnHFM, theRiemannian metric gR is here lifted to themetric gSM = (vu,wu) = gR(π∗(vu),π∗(wu)) (i.e., themetric is not anisotropicallyweighted). The lineelement is in thiscaseds2= gijdxidxj+XβαgijDαiD βj. GeodesicsforgSMareliftsofRiemanniangeodesicsforgRonM, incontrasttothesub-Riemannian normalgeodesics forgFMwhichwewill characterizebelow.Thefamilyofcurvesarisingasprojections to M of normal geodesics for gFM includesRiemanniangeodesics for gR (and thusprojections of geodesics forgSM),but the family is ingeneral larger thangeodesics forgR. 4.ConstrainedEvolutions Extremalpaths for (5) canbe interpretedasmostprobablepaths for thedrivingprocessWtwhen u0 definesananisotropicdiffusion. This iscaptured in the followingdefinition[3]: Definition1. Amostprobablepath for thedrivingprocess (MPP) fromx=π(u0)∈Mtoy∈Misasmooth pathxt : [0,1]→Mwithx0= xandx1= ysuch that its anti-developmentϕ−1u0 (xt) is amostprobablepath forWt; i.e., xt∈ argminσ,σ0=x,σ1=y ∫ 1 0 −L Rd(ϕ −1 u0 (σt), d dtϕ −1 u0 (σt))dt withL Rd being theOnsager–Machlup function for theprocessWt onR d [22]. The definition uses the one-to-one relation between W(Rd) and W(M) provided by ϕu0 to characterize the paths using the Rd Onsager–Machlup function L Rd. When M is Riemannian with metric gR, the Onsager–Machlup function for a g-Brownian motion on M is L(xt, x˙t) = −12‖x˙t‖2gR + 112SgR(xt)with SgR denoting the scalar curvature. This curvature term vanishes on Rd, andthereforeL Rd(γt,γ˙t)=−12‖γ˙t‖2 foracurveγt∈Rd. Bypullingxt∈Mback toRdusingϕ−1u0 , theconstructionremoves the 112SgR(xt) scalarcurvature correctiontermpresent inthenon-EuclideanOnsager–Machlupfunction. It therebyprovidesarelation betweengeodesicenergyandmostprobablepaths for thedrivingprocess. This is contained in the followingcharacterizationofmostprobablepaths for thedrivingprocess as extremalpathsof the sub-Riemanniandistance [3] that followsfromtheEuclideanspaceOnsager–Machluptheorem[22]. Theorem1 ([3]). LetQ(u0)denote theprincipal sub-bundleofFMofpointsz∈FMreachable fromu0∈FM byhorizontalpaths. Suppose theHörmander condition is satisfiedonQ(u0), and thatQ(u0)has compactfibers. Then,most probable paths fromx0 to y∈Mfor the driving process of Xt exist, and they are projections of sub-Riemanniangeodesics inFMminimizing the sub-Riemanniandistance fromu0 toπ−1(y). Below,wewillderiveevolutionequations for thesetofsuchextremalpaths thatcorrespondto normalsub-Riemanniangeodesics. 4.1.NormalGeodesics for gFM Connectedto themetricgFM is theHamiltonian H(z)= 1 2 (z|gFM(z)) (12) 412