This paper introduces a finite dimensional approximation method for the Wiener measure on symmetric spaces of non-compact type, aiding in rigorous interpretation of path integrals in mathematical physics.
Contribution
It develops a new approximation scheme for Brownian bridges on symmetric spaces, extending finite dimensional techniques to non-compact settings.
Findings
01
Provides a convergent approximation of Wiener measure on symmetric spaces
02
Enables rigorous analysis of path integrals in geometric contexts
03
Facilitates applications in quantum field theory and index theorems
Abstract
Path integrals developed by Richard Feynman have been an important tool in Physics in studying quantum field theory. In mathematics, it has also been widely used in providing formal proofs in the study of Index theorem and asymptotic behaviors of heat kernels. Finite dimensional approximations to path integral representations give a way to interpret path integrals and make the formal argument rigorous. The central idea is to restrict a path integral to smaller path spaces where everything is well defined and then to interpret the original path integral as a "limit" when smaller path spaces "exhaust" the full path space (Wiener space). In this paper I will present a finite dimensional approximation to Brownian bridge on a symmetric space of non---compact type using pinned piecewise geodesic space adapted to partitions of time.
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Taxonomy
Topicsadvanced mathematical theories · Algebraic and Geometric Analysis · Quantum Mechanics and Applications
Full text
A finite dimensional approximation to pinned Wiener measure on some symmetric spaces
Zhehua Li
Abstract
Let M be a Riemannian manifold, o∈M be a fixed base point, Wo(M) be the space of continuous paths from [0,1] to
M starting at o∈M, and let νx denote Wiener measure on Wo(M) conditioned to end at x∈M. The goal of this paper is to give
a rigorous interpretation of the informal path integral expression for
νx;
[TABLE]
In this expression E(σ) is the “energy” of the path σ,δx is the δ
– function based at x,Dσ is interpreted as an infinite
dimensional volume “measure” and Z is a
certain “normalization” constant. We will
interpret the above path integral expression as a limit of measures,
νP,x1, indexed by partitions, P of [0,1]. The measures
νP,x1 are constructed by restricting the above path
integral expression to the finite dimensional manifolds, HP,x(M), of piecewise geodesics in Wo(M) which
are allowed to have jumps in their derivatives at the partition points and end at x. The
informal volume measure, Dσ, is then taken to be a certain
Riemannian volume measure on HP,x(M). When M
is a symmetric space of non–compact type, we show how to
naturally interpret the pinning condition, i.e. the δ – function term,
in such a way that νP,x1, are in fact well defined finite
measures on HP,x(M). The main theorem of this paper then asserts that νP,x1→νx (in
a weak sense) as the mesh size of P tends to zero.
Let (∇) be the Levi-Civita covariant derivative on M which we add to the default set up (Md,g,∇,o).
The path space
[TABLE]
is known as the Wiener space on M and let ν be the Wiener measure on
Wo(M)—i.e. the law of M–valued Brownian motion
which starts at o∈M.
Consider the heat equation of the following form:
[TABLE]
where H=−21Δg+V is the Schrödinger operator, Δg is the Laplace-Beltrami operator on (M,g,o) and V:M→R is an external potential.
Let e−tH be the solution operator to (\refheateqn). Under modest regularity conditions, this operator admits an integral kernel ptH(⋅,⋅). In the physics literature one frequently finds Feynman type informal identities of the form,
[TABLE]
and
[TABLE]
Variants
of these informal path integrals are often used as the basis for “defining”and making computations in quantum-field theories. From a mathematical perspective,
making sense of such path integrals is thought to be a necessary step
to developing a rigorous definition of interacting quantum field theories,
(see for example; Glimm and Jaffe [13], Barry Simon [21],
the Clay Mathematics Institute’s Millennium problem involving Yang-Mills
and Mass Gap). In this paper we give an interpretation of the formal identity using a finite dimensional approximation scheme when the manifold is a symmetric space of non—compact type, for example, hyperbolic space Hd.
[TABLE]
where pt(x,y) is the heat kernel associated to 21Δg on M.
1.1 Finite Dimensional Approximation Scheme for Path Integrals
The central idea behind finite dimensional approximation scheme is to define a path integral as a limit of the same integrands restricted to “natural” approximate path spaces, for example, piecewise linear paths, broken lines, polygonal paths and so on. The ill–defined expression under these finite dimensional approximations usually becomes well–defined or has better interpretations, see ([12], [15]). For example, when M=Rd, it is known that Wiener measure on W0(Rd)
may be approximated by Gaussian measures on piecewise linear path
spaces. More specifically, Eq. (1.3) with V=0 and restricted
to a finite dimensional subspace of piecewise linear paths based on
a partition of [0,1] has a natural interpretation as
Gaussian probability measure. The interpretation results from the canonical isometry between the piecewise linear path space and Rdn, where n is the number of partition points. By combining Wiener’s theorem on the
existence of Wiener measure with the dominated convergence theorem,
one can see that these Gaussian measures converge weakly to ν
as the mesh of partition tends to zero, (see for example [9, Proposition 6.17] for details).
An analogous theory on general manifolds was also developed, see
for example Pinsky [20], Atiyah [2], Bismut
[3], Andersson and Driver [1]
and references therein. In [1], followed by [19] and [18], the finite dimensional approximation problem is viewed in its full
geometric form by restricting the expression in Eq. (1.3)
to finite dimensional sub-manifolds of piecewise geodesic paths on
M. Unlike the flat case (M=Rd) where the choice of translation invariant Riemannian metric on path spaces is irrelevant, various Riemannian metrics on approximate path spaces are explored. Based on these metrics, different approximate measures are constructed which lead to different limiting measures on Wo(M), see [1], [18], and [19]. In this paper we adopt a so–called GP1 metric on the piecewise geodesic space.
In the remainder of this section, we establish some necessary notations.
Definition 1.1** (Cameron-Martin space on (M,g,o))**
Let
[TABLE]
be the* Cameron-Martin space** on (M,g,o). (Here a.c. means absolutely continuous.)*
Notation 1.2
Let Γ(TM) be differentiable sections of TM and Γσ(TM) be differentiable sections of TM along σ∈H(M).
The space, H(M), is an infinite dimensional Hilbert manifold which is a central object in problems related to calculus of variations on Wo(M). Klingenberg [16] contains a good exposition of the manifold of paths. In particular, Theorem 1.2.9 in [16] presents its differentiable structure in terms of atlases. We will be interested in certain Riemannian metrics on H(M) and on certain finite dimensional submanifolds where the formal path integrals make sense.
Definition 1.3
For any σ∈H(M) and X,Y∈Γσa.c.(TM), We define a metric G1 as follows:
[TABLE]
where Γσa.c.(TM) is the set of absolutely continuous vector fields along σ with finite
energy, i.e. ∫01⟨ds∇X(s),ds∇X(s)⟩gds<∞.
Remark 1.4
To see that G1 is a metric on H(M), we identify the tangent space TσH(M) with Γσa.c.(TM). To motivate this identification, consider a differentiable one-parameter family of curves σt in H(M) such that σ0=σ. By definition of tangent vector, dtd∣0σt(s) should be viewed as a tangent vector at σ. This is actually the case, for detailed proof, see Theorem 1.3.1 in [16].
Definition 1.5** (Piecewise geodesic space)**
Given a partition
[TABLE]
define:
[TABLE]
The piecewise geodesic space HP(M) is a finite dimensional embedded submanifold of H(M). As for its tangent space, following the argument of Theorem 1.3.1 in [16], for any σ∈HP(M), the tangent space
TσHP(M) may be identified with
vector-fields along σ of the form X(s)∈Tσ(s)M
where s→X(s) is piecewise C2 and satisfies
Jacobi equation for s∈/P, i.e.
[TABLE]
where R is the curvature tensor. (See Theorem 2.29 below for a more detailed description of THP(M)). After specifying the tangent space of HP(M), we can define the GP1 metric as follows.
Definition 1.6
For any σ∈HP(M)
and X,Y∈TσHP(M), let
[TABLE]
where Δj=sj−sj−1 and ds∇Y(sj−1+)=lims↓sj−1ds∇Y(s).
Endowed with the Riemannian metric GP1, HP(M) becomes a finite dimensional Riemannian manifold and the right hand side of (\refeq:−20−1) is now well–defined on HP(M) if Dσ is interpreted as the volume measure induced from this Riemannian metric. This motivates the following approximate measure definition.
Definition 1.7** (Approximate measure on HP(M))**
Let
νP1 be the probability measure on HP(M)
defined by;
[TABLE]
where dvolGP1 is the volume measure on HP(M)
induced from the metric GP1 and ZP1
is the normalization constant.
1.2 Main Theorems
In this section we state the main results of this paper while avoiding many technical details.
We prove below in Proposition 3.9 that when M has non–positive sectional curvature, HP,x(M) is an embedded submanifold of HP(M).
Theorem 1.9
If M is a Hadamard manifold with
bounded sectional curvature and P={k/n}k=0n
are equally-spaced partitions, then there exists a finite measure νP,x1 supported
on HP,x(M), such that for any bounded continuous function f on HP(M),
[TABLE]
where δx(m) is an approximate sequence
of δx in C0∞(M).
Remark 1.10
The formula for dνP,x1 is explicitly given, see Definition 3.11.
The next theorem asserts, under additional geometric restrictions, that the
measure νP,x1 we obtained from Theorem 1.9
serves as a good approximation to pinned Wiener measure νx.
Theorem 1.11
If M is a symmetric space of non–compact type, i.e. it is a Hadamard manifold with parallel curvature tensor, then for any cylinder function f∈FCb1, see Definition 2.25,
[TABLE]
where νx is pinned Wiener measure, see Theorem 2.13 below.
1.3 Structure of the Paper
For the guidance to the reader, we give a brief summary of the contents of this paper.
In Section 2 we set up some notations and preliminaries
in probability and geometry. In particular we present the Eells-Elworthy-Malliavin construction of Brownian motion on manifolds.
In Section 3 we define explicitly the pinned approximate measure νP,x1 and study its properties. In Theorem 3.13, we prove that νP,x1 is a finite measure and that x→∫HP,x(M)fdνP,x1 is a continuous function on M provided f is bounded and continuous. This property is the key ingredient in proving Theorem 1.9. The proof of Theorem 1.9 is also given in this section.
In Section 4 we develop the so–called orthogonal lift of a vector field X on M to vector fields X~P on HP(M) and X~ on Wo(M). Integration by parts formulae for these two operators are presented which will serve as an important tool in the proof of Theorem 1.11.
In Section 5, (using the development maps introduced in Section 2), we view X~P as defined on all of Wo(M) and show that for any bounded cylinder function f (also introduced in Section 2), X~Pf−X~fLq(Wo(M))→0 as ∣P∣→0 for any q≥1 and more challengingly, we show the same result for X~tr,νf−X~Ptr,νP1f, where X~tr,ν is the adjoint of X~ with respect to ν and X~Ptr,νP1 is the adjoint of X~P with respect to νP1.
In Section 6, we combine all the tools that are developed from previous sections to prove the main Theorem 1.11 of this paper.
Acknowledgement 1.12
I want to thank my advisor Bruce Driver for many meaningful discussions and careful reading of my dissertation whose main part gives rise to this paper.
2 Preliminaries in Geometry and Probability
For the remainder of this paper, let u0:Rd→ToM be a fixed linear isometry which we add to the standard setup (M,g,o,u0,∇). We use u0 to identify ToM with Rd. Suggested references for this section are Chapter 2 of [14] and Sections 2, 3 of [7]. Some other references are [1], [10], [4] and [8] to name just a few.
For any x∈M, denote by O(M)x the
space of orthonormal frames on TxM, i.e. the space
of linear isometries from Rd to TxM. Denote O(M):=∪x∈MO(M)x
and let π:O(M)→M be the (fiber) projection
map, i.e. for each u∈O(M)x, π(u)=x.
The pair (O(M),π) is the orthonormal
frame bundle over M.
In this paper we use the connection on O(M) that are specified by the following connection form.
Definition 2.2** (Connection Form on O(M))**
We define a so(d)–valued connection form ω∇ on O(M) in the following way; for any u∈O(M) and X∈TuO(M),
[TABLE]
where u(⋅) is a differentiable curve on O(M) such that u(0)=u and dsdu(s)∣s=0=X. For any ξ∈Rd, ds∇u(s)∣s=0ξ:=ds∇u(s)ξ∣s=0 is the covariant derivative of u(⋅)ξ along π(u(⋅)) at π(u).
Definition 2.3** (Horizontal Bundle H)**
Given a connection form ω∇, the horizontal bundle H⊂TO(M) is defined to be the kernel of ω∇.
Definition 2.4
For any a∈Rd, define the horizontal lift Ba∈Γ(H) in the following way: for any u∈O(M), Ba(u)∈Hu⊂TuO(M) is uniquely determined by
[TABLE]
Definition 2.5** (Horizontal Lift of a Path)**
For any σ∈H(M), a curve
u:[0,1]→O(M) is said to be a
horizontal lift of σ if π∘u=σ and u′(s)∈Hu(s)∀s∈[0,1].
Remark 2.6
In this paper we only consider horizontal lift with fixed start point u0∈π−1(σ(0)). Under this assumption, given σ∈H(M), its horizontal lift u(σ,⋅) is unique.
We denote u by ψ(σ) and call ψ the horizontal lift map.
Definition 2.7** (Development Map)**
Given
w∈H(Rd), the solution to the ordinary
differential equation
[TABLE]
is defined to be the development of w and we will denote this map w→u by η, i.e. η(w)=u. Here {ei}i=1d is the standard basis of Rd.
Definition 2.8** (Rolling Map)**
ϕ=π∘η:H(Rd)→H(M)*
is said to be the rolling map to H(M).*
Definition 2.9** (Anti-rolling Map)**
Given σ∈H(M) with u=ψ(σ).
The anti-rolling of σ is a curve w∈H(Rd)
defined by:
[TABLE]
Remark 2.10
It is not hard to see w=ϕ−1(σ) and u(σ,s)u0−1 is the parallel translation along σ∈H(M).
The Eells-Elworthy-Malliavin construction of Brownian motion depends in essence on a stochastic version of the maps defined above. Since the
development maps on the smooth category are defined through ordinary
differential equations, a natural way to introduce probability is to replace
ODEs by (Stratonovich) stochastic differential equations.
First we set up some measure theoretic notations and conventions. Suppose
that (Ω,{Gs},G,P)
is a filtered measurable space with a finite measure P. For any
G—measurable function f, we use P(f)
and EP[f] (if P is a probability measure)
to denote the integral ∫ΩfdP. Given two filtered measurable spaces
(Ω,{Gs},G,P)
and (Ω′,{Gs′},G′,P′)
and a G/G′ measurable map f:Ω→Ω′,
the law of f under P is the push-forward measure f∗P(⋅):=P(f−1(⋅)).
We are mostly interested in the path spaces Wo(M),
W0(Rd) and Wu0(O(M)),
where the following notation is being used.
Notation 2.11
If (Y,y) is a pointed manifold, let W(Y):=C([0,1],Y) be the space
of all continuous paths in Y equipped with the uniform topology, Wy(Y):={w∈W(Y)∣w(0)=y}
be the subset of continuous paths that start at y.
Definition 2.12
For any s∈[0,1] let Σs:Wy(Y)→Y
be the coordinate functions given by Σs(σ)=σ(s).
We will often view Σ as a map from Wy(Y) to Wy(Y)
in the following way: for any σ∈Wy(Y) and
s∈[0,1], Σ(σ)(s)=Σs(σ).
Let Fso be the σ−algebra generated by {Στ:τ≤s}.
We use F1o as the raw σ−algebra and {Fso}0≤s≤1
as the filtration on Wy(Y). The next theorem defines
the Wiener measure ν and pinned Wiener measure νx on
(Wy(Y),F1o).
Theorem 2.13
Assume Y is a stochastically complete Riemannian manifold, then there exist two finite measures ν and νx on (Wy(Y),F1o)
which are uniquely determined by their finite dimensional distributions
as follows. For any partition 0=s0<s1<⋯<sn−1<sn=1
of [0,1] and bounded functions f:Yn→R;
[TABLE]
and
[TABLE]
where pt(⋅,⋅) is the heat kernel on Y associated to 21Δg,
Δi=si−si−1, x0≡o and xn≡x
in (\refeq:−37).
Definition 2.14** (Brownian motion)**
A stochastic process X:\left(\Omega,\mathcal{G}_{s},\left\{\mathcal{G}\right\},P\right)$$\to\left(W_{y}\left(Y\right),\nu\right)
is said to be a Brownian motion on Y if the law of X
is ν i.e. X∗P:=P∘X−1=ν.
Remark 2.15
From Theorem 2.13 it is clear that the law
of the adapted process Σ:Wy(Y)→Wy(Y)
is ν and Σ is a Brownian motion. We will call Σ the canonical Brownian motion on Y.
Remark 2.16
Using Theorem 2.13, we can construct Wiener
measure and pinned Wiener measure on W0(Rd),
Wo(M) and Wu0(O(M))
respectively. In order to avoid ambiguity from moving between W0(Rd)
and Wo(M), we fix the symbol μ(μx)
as the Wiener (pinned Wiener) measure on W0(Rd)
and reserve the symbol ν(νx) as the Wiener (pinned Wiener) measure on Wo(M). Meanwhile we reserve Σ
as the canonical Brownian motion on M.
Theorem 2.17** (Stochastic Horizontal Lift of Brownian Motion)**
If Σ is the canonical Brownian motion on M, then there exists a unique (up to ν−equivalence)u~∈Wu0(O(M)) such that
If Σ is the canonical Brownian motion on M, then the stochastic anti–rolling β of Σ is defined by,
[TABLE]
u~ and β defined above are linked through the (stochastic) development map.
Definition 2.19** (Stochastic Development Map)**
Let u~ and β be as defined in Theorem 2.17 and Definition 2.18, then u~ satisfies the following SDE driven by β,
[TABLE]
and u~ is said to be the development of β.
Fact 2.20
The following facts are well known, the proofs may be found in the references listed at the begining of this section, for example, Theorem 3.3 in [7].
•
ϕ* is a diffeomorphism from H(Rd) to
H(M),*
•
β* is a Brownian motion on (Wo(Rd),μ).*
From now on some notations are fixed for the convenience of consistency.
Notation 2.21
For any σ∈H(M), u(⋅)(σ)∈Hu0(O(M)) is its horizontal lift and b(⋅)(σ)∈H(Rd) is its anti-rolling. Recall that {Σs}0≤s≤1 is fixed to be the canonical Brownian motion on (Wo(M),ν). We also fix β(⋅) to be the stochastic anti-rolling of Σ, (which is a Brownian motion on Rd) and u~(⋅) to be the stochastic horizontal lift of Σ.
Notation 2.22
Given a partition P of [0,1], βP is the piecewise linear approximation to the Brownian motion β
on Rd given by:
[TABLE]
where Δiβ=β(si)−β(si−1)and
Δi=si−si−1.
Notation 2.23** (Geometric Notation)**
•
curvature tensor* For any X,Y,Z∈Γ(TM), define
the (Riemann) curvature tensor R:Γ(TM)×Γ(TM)→Γ(End(TM))
to be:*
[TABLE]
•
For any σ∈H(M), define Ru(σ,s)(⋅,⋅)⋅ to be a map from Rd⊗Rd to End(Rd) given by;
[TABLE]
where R is the curvature tensor of M.
Similarly we define Ru~(σ,s)(⋅,⋅)⋅ to be a random map (up to ν-equivalence) from Rd⊗Rd to Rd as follows:
[TABLE]
•
Ric(⋅):=∑i=1dR(vi,⋅)vi*
is the Ricci curvature tensor on M. Here {vi}i=1d
is an orthonormal basis of proper tangent space. Using u(σ,s) or u~(σ,s) to pull back R as in (\refn1) and (\refn2), we can define Ricu(σ,s) and Ricu~(σ,s) to be maps (random maps) from Rd to Rd.*
Convention 2.24
Since most of our results require a curvature bound, it would be convenient to fix a symbol N for it, i.e. ∥R∥≤N when it is viewed as a tensor of order 4. Following this manner, we have ∥Ric∥≤(d−1)N. A generic constant will be denoted by C, it can vary from line to line. Sometimes C(⋅) or C(⋅) are used to specify its dependence on some parameters.
Definition 2.25
f:Wo(M)↦R* is a cylinder function if there exists a partition*
[TABLE]
of [0,1] and a function F:Cm(Mn,R)
such that
[TABLE]
We denote this space by FCm.
Notation 2.26
Denote
[TABLE]
Remark 2.27
In this paper the partition P is always
equally spaced, so ∣P∣≡Δi≡n1
for i=1,...,n.
Definition 2.28** (Jacobi equation)**
For σ∈H(M), Y∈Γσ(TM),
we say Y(s)∈Tσ(s)M satisfies Jacobi
equation if:
[TABLE]
Further if the horizontal lift u(s) of σ is used,
we let y(s):=u−1(s)Y(s). It
then follows that y(s) satisfies the pulled back Jacobi
equation,
[TABLE]
where b′(s)=u(s)−1σ′(s).
Once we have Jacobi equation, we can describe the tangent space THP(M)
of HP(M).
We formalize the tangent space of HP(M) mentioned in Definition 1.5.
be a
partition of [0,1],Ki:=[si−1,si]
and Δi:=si−si−1 for 1≤i≤n, and say that
f(s) satisfies the i –Jacobi’s equation if
[TABLE]
where u−1σ′(s):=u(σ,s)−1σ′(s)∈Rd.
We now let CP,i(σ,s) and SP,i(σ,s)∈End(Rd)
denote the solution to Eq. (2.9) with initial conditions,
[TABLE]
and we further let
[TABLE]
Here we view CP,i(s) and SP,i(s)
as maps from HP(M) to End(Rd).
Definition 2.32
Define for all i=1,⋯,n,
[TABLE]
with the convention that SP,0≡∣P∣I and fP,0≡I.
Remark 2.33
SP,j(s), CP,j(s) may be expressed in terms of {fP,i}i=0n by
[TABLE]
[TABLE]
3 Approximate Pinned Measures
3.1 Representation of δ
– function
Let Y be a smooth Riemannian manifold, we will denote the distribution on Y by D′(Y) and, compactly supported distribution by E′(Y). For a matrix A, let eig(A) denote the set of eigenvalues of A.
For each x∈Y, let δx∈E′(Y) be the δ–function at x defined by
[TABLE]
Lemma 3.1** (Representation of δ0 on flat space)**
There exist functions {gi}i=0d with g0∈C0∞(Rd),{gj}j=1d⊂C∞(Rd/{0}) with supports contained in a compact subset K⊂Rd and satisfying
[TABLE]
such that
[TABLE]
In more detail, for any f∈C0∞(Rd),
[TABLE]
Proof. This lemma can be derived from Lemma 10.10 in [22].
Based on this representation we can get a representation of δp for any p∈M.
Theorem 3.2** (Representation of δ
– function on manifold)**
For
any p∈M, there exist functions {gj}j=0d⊂C∞(M/{p})∩Ld−1d(M) with supports in a compact subset K of M and smooth vector fields {Xj}j=1d⊂Γ∞(TM)
with compact support such that
[TABLE]
Proof. Pick a chart {U,x} near p∈M
such that x(p)=0. Since x(U)=Rd,
one can apply Lemma 3.1 on x(U)≃Rd
and get:
[TABLE]
where δ0 is the delta mass on x(U) supported
at the origin. So for any h∈C∞(U)
[TABLE]
where dλ is the Lebesgue measure on Rd. Consider
{detgg~j∘x}j=0d
where g=(gij)1≤i,j≤d is the metric matrix,
i.e. gij=⟨∂xi∂,∂xj∂⟩g.
From Lemma 3.1 we know that detgg~j∘x
has compact support in U and therefore K:=∪j=1dsupp(detgg~j∘x)
is compact in U. Using the smooth Urysohn lemma we can construct a
smooth function ϕ∈C∞(M→[0,1])
such that ϕ−1({0})=M/U and ϕ−1({1})=K.
Define
[TABLE]
and
[TABLE]
Then for any f∈C∞(M),
[TABLE]
where dvol is the volume measure on M.
Since ϕ⋅(x−1)∗∂xj∂ϕ≡0
and ϕ≡1 on K, we have:
[TABLE]
Therefore, by the Divergence Theorem, we can write down δp in distributional sense as
[TABLE]
where g0=g^0−∑j=1dg^j⋅divXj and for j=1,…,n, gj=−g^j.
From the construction one can see that Xj∈Γ∞(TM) with compact support
and {gj}j=0d⊂C∞(M/{p})∩Ld−1d(M)
with supports being a compact subset of M.
Remark 3.3
Since C0∞(M) is dense in Lq(M),∀q≥1, for any gj,j=1,⋯,d, we can find a sequence {gj(m)}m⊂C0∞(M) such that
[TABLE]
In particular, we can make ∪msupp(gj(m)) to be compact.
Corollary 3.4
Define
[TABLE]
Then {δx(m)}m is an approximating sequence of delta mass δx, i.e. δx(m)→δx in D′(M).
Proof. Using integration by parts, we have for any f∈C∞(M),
[TABLE]
Since K:=∪msupp(gj(m)) is compact, f⋅1K and Xj∗f⋅1K∈L∞−(M), then Corollary 3.4 easily follows from Holder’s inequality.
3.2 Definition of νP,x1
In this section we will give an explicit definition of νP,x1
proposed in Theorem 1.9.
Definition 3.5** (End point map)**
Define E1:H(M)→M to be E1(σ)=σ(1) and let E1P denote E1∣HP(M).
In general, it is not
guaranteed that E1P is a submersion, which would guarantee that HP,x(M) is an embedded
submanifold of HP(M). The following is an easy, yet illuminating, example
showing what can go wrong:
Example 3.6
If M=S2, o is the north pole and P:={0,1}, then dimHP(M)=2. Consider
[TABLE]
where
[TABLE]
An one parameter family realizing X(σ,s) would
be
[TABLE]
from which one can easily see that:
[TABLE]
So by Rank-Nullity theorem, E1∗σP is not surjective.
The problem comes from the conjugate points on M. Two points p and q are conjugate points along a geodesic σ if there exists non-zero Jacobi field (smooth vector field along σ satisfying Jacobi equation) vanishing at p and q. This fact will allow the kernel of E1∗P to be “overly large ”(more accurately dimension exceeds (n−1)d), so by Rank-nullity theorem, E1∗P can not be surjective. In this paper we consider manifolds with non–positive sectional curvature. These manifolds do not have conjugate points. From the next proposition we will see that E1P is a submersion on these manifolds.
Notation 3.7
We construct a GP1–orthonormal frame
[TABLE]
of HP(M)
as follows: for any σ∈HP(M), Xhα,i(σ,⋅)=u(⋅)(σ)hα,i(σ,⋅), where
[TABLE]
and the definition of HP,σ can be found in
Eq.(\refequ.6.2).
If M is complete with non-positive sectional
curvature, then for any x∈M, HP,x(M):=(E1P)−1({x})
is an embedded submanifold of HP(M).
Proof. It suffices to show E1P is a submersion. Since M is complete, for any y∈M, there exists a geodesic σ parametrized on [0,1] and connecting o and y. So E1P is surjective. To show E1∗P is surjective, we use a class of vector fields {Xhα,n}α=1d in Notation 3.7. Since
[TABLE]
where u(⋅)=u(σ,⋅) is the horizontal lift of σ∈HP(M). From Proposition A.3 we know SP,n is invertible, therefore {E1P∗(Xhα,n)}α=1d spans TE1P(σ)M. So E1P∗ is surjective.
Since HP,x(M) is an embedded submanifold of HP(M), we can restrict the Riemannian metric GP1
on THP(M) in Eq. (1.6) to
a Riemannian metric on THP,x(M).
Notation 3.10
Assuming M has non-positive
sectional curvature, for any x∈M, let GP,x1
be the restriction of GP1 to TσHP,x(M)⊂TσHP(M).
Further, let volGP,x1 be the
associated volume measure on HP,x(M).
Based on the volume measure volGP,x1
on HP,x(M), we can construct the pinned approximate
measure νP,x1:
Definition 3.11
Let νP,x1
be the measure on HP,x(M) defined by
[TABLE]
where JP(σ):=det(E1P∗σE1P∗σtr), ZP1:=(2π)2dn and E(σ)=∫01⟨σ′(s),σ′(s)⟩gds is the energy of σ.
3.3 Continuous Dependence of νP,x1 on x
Throughout this section we further assume M is simply connected, i.e. M is a Hadamard manifold, and the sectional curvature of M is bounded from below by −N. The following theorem illustrates that the measures, νP,x1, are finite and “continuously varying”with respect to x.
Notation 3.12
We will denote by Cb(Y) bounded continuous
functions on a topological space Y.
Theorem 3.13
For any x∈M, νP,x1 is a finite measure. Moreover, for any f∈Cb(HP,x(M)), define
[TABLE]
If the mesh size ∣P∣:=n1 of the partition P is small enough, i.e. n≥3dN, then hPf(x)∈C(M).
Before proving this theorem, we need to set up some notations and
auxiliary results.
Notation 3.14
We fix n∈N and let si:=ni and τ:=1−n1=sn−1. We further define K:=HP([0,τ],M)
be the space of piecewise geodesic paths, σ:[0,τ]→M
such that σ(0)=o∈M.
Lemma 3.15
For x,y∈M, we can choose an unique element logx(y)∈TxM so that
[TABLE]
is the unique minimal-lengh-geodesic connecting x to y such that γy,x(τ)=x
and γy,x(1)=y.
Proof. Since M is a Hadamard manifold, by the Theorem of Hadamard (See Theorem 3.1 in [5]), expx:TxM→M is a diffeomorphism. Therefore we can see that logx(y)=expx−1(y) is unique and it follows that the geodesic γy,x is unique.
Definition 3.16
For any given y∈M, let ψy:K→HP,y(M):=(E1P)−1({y}) be defined by
[TABLE]
where
[TABLE]
Notation 3.17
For any σ∈HP,y(M), let ξy,σ:=u(σ,τ)−1logσ(τ)(y)∈Rd and Aξy(σ,s)=Ru(σ,1−s)(ξy,σ,⋅)ξy,σ and 0≤s≤n1. Denote by Cy(σ,s),Sy(σ,s) the solutions to y′′(σ,s)=Aξy(σ,s)y(σ,s) with initial values Cy(σ,0)=I,Cy′(σ,0)=0,Sy(σ,0)=0,Sy′(σ,0)=I.
The next lemma characterizes the differential of ψy:
Lemma 3.18
For any σ∈K and Xh(σ,⋅):=u(σ,⋅)h(σ,⋅)∈TσK,
[TABLE]
where
[TABLE]
Proof. From now on we will suppress the path argument ψy(σ) in h^. Suppose that t→σt∈K
is an one-parameter family of curves in K
such that σ0=σ and dtd∣0σt=Xh(σ).
Then we have
[TABLE]
If s∈[0,τ], then
[TABLE]
While if s∈[τ,1] we have
[TABLE]
where Xsh^ is uniquely determined by,
h^ satisfies Jacobi’s equation,
2. 2.
h^(τ)=h(τ) and h^(1)=0.
Denote h^(s) by g(1−s) for s∈[τ,1],
the above conditions are equivalent to g being the solution to
the following boundary value problem:
[TABLE]
Then we use Sy(⋅) to express the solution. Here we have used Proposition A.3 to see that Sy(s) is invertible for s∈[0,n1], therefore
[TABLE]
and thus
[TABLE]
Corollary 3.19
For any y∈M, ψy is a diffeomorphism.
Proof. From Lemma 3.18 it is easy to see that the
push forward (ψy)∗ of ψy is one to
one and thus an isomorphism since dim(K)=dim(HP,y(M)).
Therefore the inverse function theorem implies that ψy is
a local diffeomorphism. Furthermore, M being a Hadamard manifold
implies that ψy is bijective, so ψy is actually
a diffeomorphism.
Remark 3.20
An orthonormal frame {Xhα,i:1≤α≤d,1≤i≤n−1} of K can be constructed similarly to Notation 3.7,
[TABLE]
In this section we will use the same notation for both these two sets
of orthonormal frames as the meaning should be clear from the context.
Definition 3.21
f:M→N* is a differentiable map between two Riemannian manifolds M,N. The Normal Jacobian of f is defined to be det(f∗f∗tr).*
We will use the orthonormal frame {Xhα,i:1≤α≤d,1≤i≤n−1} of
K to estimate the Normal Jacobian JP of E1
in Lemma 3.22 and the “volume change
”Vx (See precise definition in Lemma 3.24) brought by
the diffeomorphism ψx in Lemma 3.24 and 3.25.
Lemma 3.22
Let JP:=detE1P∗(E1P∗)tr be the Normal Jacobian of E1P, then
[TABLE]
Proof. Note that
[TABLE]
so if v∈TE1P(σ)M, then
[TABLE]
Therefore, using the orthonormal frame of THP(M) given by
[TABLE]
we find
[TABLE]
Let {eα}α=1d be the standard basis of Rd, since u(1) is an isometry, {u(1)eα}α=1d is an O.N. basis of TE1P(σ)M. Using
[TABLE]
we can compute:
[TABLE]
Using the expression of JP in Lemma 3.22, we can easily derive the following estimate.
Corollary 3.23
Let JP be defined as above, then for any σ∈HP(M), JP(σ)≥1.
Proof. For any v∈Cd, using Proposition A.3, we
have:
[TABLE]
So by Min-max theorem, eig(n1∑i=1nfP,i(σ,1)fP,itr(σ,1))⊂[1,+∞)
and therefore:
[TABLE]
Lemma 3.24
For any σ∈K, let Vx:K→R+ be the normal Jacobian of ψx:K→HP,x(M), i.e. Vx:=det((ψx∗)trψx∗), then
[TABLE]
where
[TABLE]
Proof. Using (3.9) and differentiating h^
with respect to s, we get:
For any X, Y∈TK(the tangent bundle of K), define two metrics GP,τ0,
GP,τ1 to be:
[TABLE]
and
[TABLE]
Lemma 3.27
GP,τ0* is a metric on K.*
Proof. The only non–trivial part is to check ⟨X,X⟩GP,τ0=0⟹X=0.
Since M has non–positive curvature, there are no conjugate points. For each 0≤i≤n−1, there is a unique piece of Jacobi field X along σ∣[si,si+1] with specified boundary values X(si) and X(si+1). ⟨X,X⟩GP,τ0=0⟹X(si)=0 for any 1≤i≤n. By the uniqueness of Jacobi field, X≡0.
Based on the metric GP,τ0
and GP,τ1, we define measures νP,τ0
and νP,τ1 on K as follows.
Definition 3.28
Let
[TABLE]
and
[TABLE]
Lemma 3.29
If
[TABLE]
then dνP,τ0=ρPdνP,τ1 and moreover, ρP(σ)≥1∀σ∈K.
Proof. The argument to show ρP is the density
of νP,τ0 with respect to νP,τ1
is almost exactly the same as Theorem 5.9 in [1]
with a slight change of ending point from 1 to τ. Here we
focus on the lower bound estimate of ρP(σ).
Since for any v∈Cd,
[TABLE]
we know from Proposition A.3 that for any λ∈eig(nSP,i),
[TABLE]
and from which we know:
[TABLE]
Proof of Theorem 3.13. Since ψx
is a diffeomorphism,
[TABLE]
Notice that
[TABLE]
so
[TABLE]
Combining (\refeq:−13), (\refeq:−20) we know that:
[TABLE]
So
[TABLE]
When n is large enough, i.e. n>Nk, e−2n−Nkd2(σ(τ),x)≤1 and thus it suffices to show
[TABLE]
For each k≤d we have:
[TABLE]
Using Lemma A.2 in Appendix A we obtain a bound of the right–hand side of Eq. (3.18) (the bound here depends on n).
Since for any σ∈K, JPf∘ψx(σ)e−2nd2(σ(τ),x)Vx(σ) is continuous with respect to x∈M, so by dominated convergence theorem, hPf(x)∈C(M).
Not only can we show that hPf(x) is a continuous function, it is bounded uniformly in x∈M and partition P, as is shown in the following proposition.
Now define the projection map πP:K→Mn−1,
for any σ∈K,
[TABLE]
Since M is a Hadamard manifold, πP is
a diffeomorphism. From there one can get:
[TABLE]
Corollary 4.2 in [23] gives a lower bound of the heat kernel of manifold M, provided Ric≥(1−d)N:
[TABLE]
where N is the curvature bound and C is some constant depending
only on d and N and ρ=d(x,y). Using the fact
that
[TABLE]
it follows that
[TABLE]
Let t=n−N11, where N1=2Nd+2N(d−1), we have, for any j∈{0,…,n−1}:
[TABLE]
So
[TABLE]
Since the heat kernel is continuous with respect to time t , combining (\refeq:−48) ,(\refeq:−49) and (\refeq:−17) we get
[TABLE]
and hence
[TABLE]
where C is a constant depending only on d and N.
Theorem 3.13 shows that the class of approximate pinned
measures {νP,x1} are finite measures
and using the continuity result for hP(x),
one can see that νP,x1 is deserved to be formally
expressed as δx(σ(1))νP1
and it should be interpreted in the sense of Corollary 3.32 below. First we state a co–area formula.
Let H and M be two Riemannian manifolds with volume measures dvolH and dvolM respectively. If p:H→M is a smooth submersion, g:H→[0,∞) is a density function, for each x∈M, let dvolHx be the volume measure on Hx:=p−1({x}) and J(y):=det(p∗yp∗ytr) on y∈Hx, then for any non–negative measurable function f:M→[0,∞),
[TABLE]
Corollary 3.32
Denote by δx∈E′(M)
the delta–function at x∈M and {δx(m)}⊂C0∞(M) is an approximate sequence to δx(δx(m)→δx in E′(M)) i.e. for any h∈C∞(M), we have:
From Theorem 3.13 we know hPf(x)∈C(M),
therefore:
[TABLE]
4 Orthogonal Lifting Technique
4.1 The Orthogonal Lift X~P on HP(M)
As a remainder, unless mentioned separately, M is a complete Riemannian manifold with non–positive sectional curvature bounded below by −N. In this subsection we focus on the unpinned piecewise geodesic space HP(M).
4.1.1 A Parametrization of TσHP(M)
Recall from Theorem 2.29 that Y∈Γ(THP(M)) iff for each σ∈HP(M), J(σ,s):=u(σ,s)−1Y(σ,s) satisfies (in the following equation we suppress σ)
[TABLE]
where b=ϕ(σ)∈H(Rd) is the anti–rolling of σ.
From above we observe that J can be parametrized by
[TABLE]
where (k0,k1,…,kn−1) is an arbitrary element
of (Rd)n. Proposition 4.1 explains
this parametrization in more detail.
Proposition 4.1
If (k0,k1,…,kn−1)∈(Rd)n, then the unique J\left(\cdot\right)\in$$C\left(\left[0,1\right],\mathbb{R}^{d}\right) satisfying (\refA) and (\refB) above is given by
[TABLE]
Proof. From the definition of fP,i+1 (see Definition 2.32), J in Eq. (4.3) may be written as
[TABLE]
To finish the proof, we need only verify that J is continuous, J′(si+)=ki for 0≤i≤n−1 and J solves the Jacobi equation (4.1).
Since CP,l(s) and SP,l(s) satisfies Jacobi equation for s∈[sl−1,sl), J satisfies (4.1) and is continuous at s∈/P. For each sl, 1≤l≤n−1, since CP,l+1(sl)=I, SP,l+1(sl)=0 and J is right continuous on [0,1],
[TABLE]
So J is also continuous at partition points. Then since
[TABLE]
J satisfies (4.2). The uniqueness of J is easily seen from the uniqueness of solutions to linear ODE with initial values.
Definition 4.2
For each s∈[0,1], define Ls:(Rd)n→Rd as follows: for s∈[sl−1,sl],
[TABLE]
We now compute the adjoint of L1.
Lemma 4.3
For any v∈Rd, let L1∗:Rd→(Rd)n be the adjoint of L1, then
[TABLE]
where fP,i∗(1) is the matrix adjoint of fP,i(1).
Proof. Equation (4.5) immediately follows from the identity;
[TABLE]
Definition 4.4
We now define
[TABLE]
In particular,
[TABLE]
Recall that given a matrix A, eig(A) denotes the
eigenvalues of A.
Lemma 4.5** (Invertibility of KP(1))**
If
M has non-positive sectional curvature, then
[TABLE]
and thus KP(1)
is invertible.
Proof. Denote Rus(b′(si−1+),⋅)b′(si−1+) by AP,i(s):HP(M)→End(Rd). Notice that M having non-positive
sectional curvature guarantees AP,i(s)
is positive semi-definite. Then apply Proposition A.3 to find, for
any i=1,⋯,n and v∈Cd,
[TABLE]
From these inequalities it follows that
[TABLE]
So fP,i(1) is invertible and fP,i∗(1)−1=fP,i(1)−1≤1. Therefore for any v∈Cd,
[TABLE]
now replace v by fP,i∗(1)v, we get fP,i∗(1)v≥∥v∥ and thus
[TABLE]
This implies that
[TABLE]
In particular, KP(1) is invertible.
4.1.2 Orthogonal Lift on HP(M)
In this subsection we use the least square method to lift a vector field X∈Γ(TM) to a vector field X~P∈Γ(THP(M)), here lift means X~P satisfies Eq. (\refequ.6.5).
Theorem 4.6** (Orthogonal lift)**
For
all X∈Γ(TM), there exists a unique vector field X~P∈Γ(THP(M)) satisfying;
For all h∈C1(M),
[TABLE]
2. 2.
For all σ∈HP(M),
[TABLE]
In this paper X~P is referred to as the orthogonal lift of X to (HP(M),GP1).
First we use the parametrization in Subsection 4.1.1 to characterize {Nul(E1∗,σ)}⊥.
Lemma 4.7
Suppose Y∈Γ(THP(M)) with k(⋅):=u(⋅)−1Y(⋅):HP(M)→H(Rd). Then Y∈{Nul(E1∗)}⊥
iff
[TABLE]
Proof. Given Y(⋅):=u(⋅)k(⋅) and Z(⋅):=u(⋅)J(⋅)∈Γ(THP(M)), then
[TABLE]
and
[TABLE]
Recall from Proposition 4.1 and Definition 4.2 that (here we suppress σ)
[TABLE]
so
[TABLE]
Since
[TABLE]
so Y∈{Nul(E1∗)}⊥
iff
[TABLE]
Remark 4.8
According to (\refequ.6.4), it is immediate that
[TABLE]
Definition 4.9
Given X∈Γ(TM), define X~P∈Γ(THP(M)) to be X~P(⋅)=u(⋅)JP(⋅)
where
[TABLE]
Proof of Theorem 4.6.
We will show X~P is the unique orthogonal lift of X. Since TσHP(M)=Nul(E1∗,σ)⊕GP1{Nul(E1∗,σ)}⊥, given a lift Z∈Γ(THP(M)) of X∈Γ(TM), its orthogonal projection to {Nul(E1∗,σ)}⊥ is also a lift but with smaller GP1 norm. So if Z is an orthogonal lift, then Z∈{Nul(E1∗)}⊥. From Lemma 4.7 and Remark 4.8 it follows that if k(⋅):=u−1(⋅)Z(⋅), then
[TABLE]
for some v∈Rd. Then using Definition 4.4 and Proposition 4.1, k must have the following form,
[TABLE]
for some v∈Rd to be determined.
To specify v, we use condition (4.10)
[TABLE]
This implies KP(1)v=u1−1X∘E1. Since
KP(1) is invertible, we can just
pick v to be KP(1)−1u1−1X∘E1.
Definition 4.10
We will view X~P as a differential operator with domain,
[TABLE]
Here
[TABLE]
Since Cb1(HP(M)) is dense in L2(HP(M),νP1), we can also view X~P as a densely defined operator on L2(HP(M),νP1).
4.1.3 Restricted Adjoint X~Ptr,νP1
In this subsection we study X~Ptr,νP1—the adjoint of X~P with respect to νP1 restricted to D(X~P), i.e. we require D(X~Ptr,νP1)=D(X~P).
Lemma 4.11
Given X∈Γ(TM), if X~P is the orthogonal lift of X, then
[TABLE]
where M⋅ is the multiplication operator, b is the anti-rolling of σ and divX~P
is the divergence of X~P with respect to volGP1.
Proof. In this proof we identify the measure νP1
with the associated nd—form. So by “Cartan’s magic formula”, first assume f∈Cb1(HP(M)) with compact support,
[TABLE]
Since fνP1 is a top degree form, d(fνP1)=0.
By Stokes’ theorem,
Combining (\refequ.8.2), (\refequ.8.3) and (\refequ.8.4) we get, if f∈Cb1(HP(M)) with compact support, then
[TABLE]
where X~Ptr,νP1 is defined in Eq. (\refequ.8.1).
For the general case choose a cut–off function ϕ∈C0∞(R,[0,1])
such that ϕ≡1 on [−1,1] and ϕ≡0
on R/[−2,2]. Let fn:=f⋅ϕ(nE), observe, using product rule, that
[TABLE]
so
X~Pfn→X~Pf as n→∞νP1 a.s.
Using Proposition 5.20 and Lemma 6.1 we have for any q≥1, there exists M=M(q)>0 such that ∀P with ∣P∣≤M1,
[TABLE]
Since f has bounded differential, from Definition 4.9 and 4.4 we have
[TABLE]
Lemma 4.5 states KP(1)−1 is bounded. Then utilizing Lemma 5.6 we have for any q≥1, there exists M=M(q)>0 such that ∀P with ∣P∣≤M1,
[TABLE]
Lemma 6.2 shows that X~Ptr,νP11∈Lq(HP(M),dνP1) provided ∣P∣≤M1 for some M=M(q), so applying DCT to both sides of Eq.(4.17) with fn→f gives Eq. (4.13).
Summing Eq.(4.26) on α and j while making use of (4.20) gives (4.19).
4.2 The Orthogonal Lift X~ on Wo(M)
Definition 4.13** (Cameron-Martin vector field)**
A Cameron-Martin process, h, is an Rd-valued process on Wo(M) such that s→h(s) is in H(Rd)ν−a.s. and a TM-valued process Xh on (Wo(M),ν) is called a Cameron-Martin vector field (denote this space by X) if π(Xs)=Σsν−a.s., h(s):=u~s−1Xsh is a Cameron-Martin process and
[TABLE]
Cameron-Martin vector field is the key concept in path space analysis. In this section we are going to introduce a non-adapted Cameron-Martin vector field (see Definition 4.21) which “lift”a vector field on a manifold M to a “vector field”on the corresponding path space Wo(M).
Definition 4.14
Define T~(⋅):[0,1]×Wo(M)→End(Rd) to be
the solution to the following initial value problem:
[TABLE]
Definition 4.15
Using T~s, we define K~:[0,1]×Wo(M)→End(Rd):
[TABLE]
Remark 4.16
Both T~ and K~ are defined up to ν−equivalence. We can pick a version at first place in order to avoid stating ν−a.s. in the following results.
Lemma 4.17
For all s∈[0,1], T~s
is invertible. Further both 0≤s≤1supT~s and
0≤s≤1supT~s−1
are bounded by e21(d−1)N, where (d−1)N
is a bound of ∥Ric∥.
Proof. Denote by Us∈End(Rd) the solution to the following initial value problem:
[TABLE]
then direct computation shows that Ys:=T~sUs∈End(Rd) satisfies
[TABLE]
By the uniqueness of solutions for linear ODE, we get Ys≡I,
and this shows that Us is a left inverse to T~s.
As we are in finite dimensions it follows that T~s−1
exists and is equal to Us. The stated bounds now follow
by Gronwall’s inequality.
Lemma 4.18
K~1* is invertible
and K~1−1≤e(d−1)N, provided ∥Ric∥≤(d−1)N.*
Proof. Since
[TABLE]
is a symmetric positive semi-definite operator such that
from which it follows that eig(K~1)⊂[e−(d−1)N,∞) and K~1−1=min{λ:λ∈eig(K~1)}1≤e(d−1)N.
Definition 4.19
For each X∈Γ(T~M)
define two ν−equivalent maps H~:Wo(M)→Rd and J~:Wo(M)→H(Rd) by
[TABLE]
and
[TABLE]
Notation 4.20
Given a measurable function h:Wo(M)→H(Rd), let Zh:Wo(M)→H(Rd) be the solution to the following initial value problem:
[TABLE]
Definition 4.21** (Orthogonal Lift on Wo(M))**
For any X∈Γ(TM),
define X~∈X as follows.
[TABLE]
where
[TABLE]
Given f∈FCb1, define the gradient operatorDf∈X as follows,
[TABLE]
where F(Σs1,⋯,Σsn) is a representation of f and gradiF is the differential of F with respect to the ith variable.
Then we define X~f:=⟨Df,X~⟩G1.
Since FCb1 is dense in L2(Wo(M),ν), X~ can be viewed as a densely defined operator on L2(Wo(M),ν) which admits an integration by parts formula as below.
Theorem 4.22
For any f,g∈FCb1, we have
[TABLE]
where
[TABLE]
and
[TABLE]
Proof. See Lemma 4.23 of IVP.
Remark 4.23
The orthogonal lift X~ on Wo(M) can be viewed as a stochastic extension of the orthogonal lift in the sense of Theorem 4.6 where the path space is the curved Cameron-Martin space H(M) and the Riemannian metric is a damped metric related to Ricci curvature. Interested readers may refer to IVP for more details in this topic.
5 Convergence Result
In this section M is a complete Riemannian manifold with non–positive and bounded sectional curvature. Other conditions will be mentioned specifically in theorems if needed.
First we modify and abuse a few notations we have defined before in order to avoid messy arguments.
Notation 5.1
Recall that β:Wo(M)→W0(Rd) is the Brownian motion on Rd defined in Definition 2.18. We have also defined βP:Wo(M)→HP(Rd) to be the linear approximation to Brownian motion on Rd as in Notation 2.22. Now denote by uP:=η∘βP the development map of βP. Notice that ϕ∘βP∈HP(M)–ν a.s, here ϕ is the rolling map onto H(M). So after identifying CP,i, SP,i
and hence fP,i with CP,i∘ϕ∘βP,
SP,i∘ϕ∘βP
and fP,i∘ϕ∘βP,
we can view them as maps from Wo(M) to End(Rd). The point here is to make the notations short and it should not cause confusions after this explanation.
Convention 5.2
We use C to denote a generic constant. It can vary from line to line. In this section it depends only on an upper bound of the mesh size ∣P∣:=n1 of the partition P (We may allow C to depend on some other factors as well, but this is good enough for our purpose of taking the limit as ∣P∣→0. )
5.1 Convergence of X~P to X~
5.1.1 Some Useful Estimates for {CP,i}i=1n and
{SP,i}i=1n
We apply Proposition A.5 to get some commonly used estimates listed as Lemmas 5.3.
Lemma 5.3
For any i∈{1,...,n} and s∈[si−1,si], we have
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Lemma 5.4
For all γ∈(0,21),
define Kγ:=s,t∈[0,1],s=tsup{∣t−s∣γ∣βt−βs∣},
then there exists an ϵγ>0 such that E[eϵKγ2]<∞.
Proof. See Fernique’s Theorem (Theorem 3.2) in [17].
Remark 5.5
From Lemma 5.4, it is easy to see any polynomial
of ϵKγ has finite moments of all orders.
5.1.2 Size Estimates of fP,i(s)
Recall from Definition 2.32 that fP,i:Wo(M)×[0,1]→End(Rd)0≤i≤n is given by
[TABLE]
with the convention that SP,0≡∣P∣I and fP,0≡I.
Using the estimates in Subsection 5.1.1,
it is easy to get an estimate of fP,i(s).
Lemma 5.6
Recall from the begining of this section that n:=∣P∣1 and N is the sectional curvature bound. For each q≥1, we have
[TABLE]
Proof. For all i,j∈{0,⋯,n}, if j<i, fP,i(sj)≡0. So we only
need to consider the case when j≥i.
Since
Since the right–hand side is independent of i, we proved (\refequ.7.22).
Secondly, define
[TABLE]
We will show, for each q≥1,γ∈(0,21),
there exists a constant C>0 such that for all n>5qN, we have
[TABLE]
For each j∈{1,⋯,n},
[TABLE]
Since ∣fP,j+1(1)∣≤eN∑k=1n∣Δkβ∣2 by (5.6), ans also T~1T~sj+1−1≤1, we have
[TABLE]
Thus for all i∈{1,⋯,n},
[TABLE]
Since (eN∑k=1n∣Δkβ∣2+1)q≤eqN∑k=1n∣Δkβ∣2, using Holder’s inequality and Theorem 5.9 we get
[TABLE]
Lastly, we estimate K^P(si)−KP(si). Using (\refequ.7.21) we have
[TABLE]
Since
[TABLE]
and from Proposition A.3, we know fP,i−1(1)−1≤1,
and
[TABLE]
So
[TABLE]
Then using Theorem 5.9, Lemma 5.6 and Holder’s
inequality we have
[TABLE]
Finally Lemma 5.13 is proved by combining (\refequ.7.22),(\refequ.7.23) and (\refequ.7.24).
Lemma 5.14
For each q≥1, there exists a constant C>0 such that
[TABLE]
Proof. By the fundamental theorem of calculus, we have
[TABLE]
Using Lemma 4.17, note that Ric is bounded by (d−1)N, we have
[TABLE]
where C and (d−1)N are two constants independent of s. Then using Gronwall’s inequality we get
[TABLE]
so s∈[0,1]supK~s
is bounded. Then using the fundamental theorem of calculus again from
s to s we have
[TABLE]
Therefore
[TABLE]
By Gronwall’s inequality again we have
[TABLE]
and thus
[TABLE]
The next theorem is a generalization to Proposition 5.13 in the sense that s now can be taken to be arbitrary between 0 and 1.
Theorem 5.15
For each q≥1 and γ∈(0,21), there exists a constant C>0 such that for all n>5qN,
[TABLE]
Proof. For any s∈[0,1], s∈[si−1,si]
for some i∈{1,⋯,n}. So
[TABLE]
Then using Lemma 5.12, Proposition 5.13 and 5.14 we prove this theorem.
5.1.4 Convergence of JP(s) to J~s
Recall from Definition 4.9 that JP(s):=KP(s)KP(1)−1HP, where HP:Wo(M)→Rd is given by HP=uP(1)−1X(π∘uP(1)) and uP is interpreted in Notation 5.1.
Proposition 5.16
Let J~s
be as in Definition 4.19 and X∈Γ(TM) with compact support, then for any q≥1,
[TABLE]
Proof.
[TABLE]
where
[TABLE]
For IP(s), since X has compact support, ∣HP(σ)∣
is bounded. By Lemma 4.5KP(1)−1≤1.
Then using Theorem 5.15 we have
[TABLE]
For IIP(s): since
[TABLE]
so
[TABLE]
Recall from (\refequ.7.25) that s∈[0,1]supK~s is bounded (the bound is deterministic), using Theorem 5.15 again we have
[TABLE]
For IIIP(s): Since F:O(M)→Rd given by F(y)=y−1X∘π(y) is bounded and continuous, and by the Wong–Zakai approximation Theorem (for example, see Theorem 10 in [10]), uP(1)→u~1 in probability as ∣P∣→0, by DCT,
[TABLE]
Also since s∈[0,1]supK~s and K~1−1 are bounded, we have
[TABLE]
Combining Eq.(\refequ.7.26), (\refequ.7.27) and (\refequ.7.28) we prove
this proposition.
If M has parallel curvature tensor, i.e. ∇R≡0, then for any f∈FCb1 and q≥1,
[TABLE]
where according to Notation 5.1, X~Ptr,νP1f is interpreted as (X~Ptr,νP1(f∣HP(M)))∘ϕ∘βP.
Proof. In correspondence with the three–term formulae (5.40) and (5.41), this theorem is decomposed as three propositions: Proposition 5.19 states that
Thus the proof will be complete once the stated propositions are proved.
Remark 5.18
For Proposition 5.19 and 5.20 we assume the assumption of bounded sectional curvature as is mentioned in the beginning of this section. For Proposition 5.21 we further require the curvature tensor to be covariantly constant.
Proposition 5.19
If X∈Γ(TM) with compact support and f∈FC1, then for any q≥1,
[TABLE]
Proposition 5.20
Keeping the notation above, we have for any q≥1,
[TABLE]
Proposition 5.21
Continuing the notation above, if we further assume ∇R≡0, then for any q≥1,
[TABLE]
Proof of Proposition 5.19.
Using Eq.(4.31) and the fact that π∘uP=ϕ∘βP, we have
[TABLE]
and
[TABLE]
where F is a representation of f as in Definition 2.25.
Since W(O(M))∋y→ysi−1gradiF(π∘y)∈Rd is continuous and bounded, using Theorem 10 in [10] and DCT, we know
[TABLE]
The proof is then completed by making use of (5.42) and Proposition 5.16.
Note that gi(si)=SP,i−CP,iSP,i−1, so by Eq.(5.1),
[TABLE]
and thus
[TABLE]
Picking γ∈(21,31) we know for any q≥1,
[TABLE]
Since fP,i−1(1)=fP,0(1)fP,0−1(si−1)Δi−1SP,i−1,
so
[TABLE]
Using Proposition A.3 we have fP,0−1(si−1)≤1. Then using Eq.(5.3) we obtain
[TABLE]
Therefore for each q≥1,
[TABLE]
Picking γ∈(31,21), we have
[TABLE]
Rewrite ∑i=1nfP,0−1(si−1)Δiβ as ∫01fP(s)dβs, where fP(s):=∑i=1nfP,0−1(si−1)1[si−1,si)(s). Define a martingale
Mr:=∫0rfP(s)dβs−∫0rT~s−1dβs. Then by Burkholder-Davis-Gundy inequality, for each q≥1,
The proof is completed by combining Eq.(5.48) and (5.49).
We state two supplementary lemmas.
Lemma 5.22
If the curvature tensor is parallel, i.e. ∇R≡0, then
[TABLE]
where As⟨Zα⟩=∫0sRu~r(Zα(r),δβr).
Proof. See Lemma 4.25 in IVP.
Lemma 5.23
Fix s∈[0,1],
consider an one parameter family of paths {σt}⊂HP(M)
and denote by ut(⋅) the horizontal lift of σt. For simplicity, we will denote ut(s) by ut, σ0 by σ, the derivative with respect to t by ⋅ and the derivative with respect to s by ′. For any X∈Γ(TM), define fX:O(M)↦Rd≃ToM
by
[TABLE]
Then:
[TABLE]
Proof. The connection on O(M) defined in Definition 2.2 gives the following decomposition:
[TABLE]
where a=u0−1dtd∣0σt(s)=u0−1σ˙(s)∈ToM
and A~(u0)=dtd∣0u0etA for some A=u_{0}^{-1}\frac{\triangledown u_{t}}{dt}\left(0\right)\in\text{\mathfrak{so}}(d)
and Ba(u0)=dtd∣0//t(γ)u0
where γ satisfies γ˙(0)=u0a and γ(0)=σ(s).
In this example, we can choose γ(⋅) to be σ(⋅)(s). So
[TABLE]
and
[TABLE]
Following the computation in Theorem 3.3 in [1], we know that
[TABLE]
Proof of Proposition 5.21. Because of Lemma
5.22, it suffices to prove
[TABLE]
From Definition 4.9 we get, for each α∈{1,…,d} and j∈{1,…,n}, that
We first compute IVP. After viewing L(⋅)=uP(1)−1∇uP(1)(⋅)X
as a linear functional on Rd we have
[TABLE]
where in Eq. (5.53) we use identity (4.8) ∑j=1nΔjfP,j(1)fP,j∗(1)=KP(1) and given A∈Md×d, Tr(A):=∑α=1d⟨Aeα,eα⟩ is the trace of the matrix A.
The proof of the lemma will be completed by Lemma 5.25 below which shows term VP converges to the right side of Eq. (5.52).
Lemma 5.25
Let VP be defined as in Lemma 5.24 and ∇R≡0, then for any q≥1,
[TABLE]
Proof. Recall that
[TABLE]
For each α∈{1,…,d} and j∈{1,…,n}, since hα,j(r)=ΔjfP,j(r)eα, we have
[TABLE]
where e0:=e0,1+e0,2
[TABLE]
and
[TABLE]
Since ∇R≡0, dRus=∇dbsR≡0. So Rus≡Ru0 is independent
of u, therefore e0,1=0.
As for e0,2, since
[TABLE]
using (\refequ.7.3) we have
[TABLE]
and from which it follows
[TABLE]
Picking γ>31 such that q(3γ−1)>0 for any q≥1, so E[∣e0,2∣q]→0 as ∣P∣→0.
Next we analyze
[TABLE]
where
[TABLE]
Define
[TABLE]
For each i=1,2,3,4,
denote eP,i(r)=∫0rgi(s)dβs−∫0rgi+1(s)dβs, then
[TABLE]
Notice that {gi}i=15 are all adapted, so based on the same computation as in Lemma 5.10 (mainly Burkholder-Davis-Gundy inequality), we can show for each i∈{1,2,3,4} and for any q≥1,
, where B1(u,v):=⟨T~1∗K1−1∫01Ru~r(dβr,T~ru)H~,∫0rT~s−1(T~s−1)∗vds⟩. Define
[TABLE]
and
[TABLE]
Then
[TABLE]
Since
[TABLE]
and g(0)=0=f(0), Eq. (5.63) is proved by observing that V^P=f1=g1=∑α=1dB1(eα,eα).
Lastly, notice that
[TABLE]
and T~r∫0rT~s−1(T~s−1)∗dseα=Zα(r),
we combine Eq. (5.59), (5.61) and (5.63) to prove Eq.(5.54).
Lemma 5.26
If ∇R≡0, then for any q≥1,
[TABLE]
Proof. Define g~j(s):=Xhα,jfP,j(s)
and gj(s):=g~j(s)−g~j(s).
Then we know that gj(s) satisfies the following ODE: for k=j,⋯,n
[TABLE]
where A˙P,k(s)=dtd∣0(RuP(t,s)(βP′(t,s),⋅)βP′(t,s)), β(t,⋅):=ϕ~−1(E(tXhα,j)) is the stochastic anti-rolling of the approximate flow E(tXhα,j)(See Corollary 4.6 in \cite[cite][\@@bibrefDriver1999]), βP(t,⋅) is the piecewise linear approximation of β(t,⋅) and uP(t,⋅)=η∘βP(t,⋅) is the horizontal lift of βP(t,⋅).
Since ∇R≡0, AP,k(s) is a constant operator for s∈[sk−1,sk], therefore by DuHammel’s principle,
[TABLE]
Using Eq. 5.3 and 5.8 we obtain the following estimate,
[TABLE]
Therefore
[TABLE]
Then we analyze k∈{1,…,n}supA˙P,k. Since ∇R≡0,
[TABLE]
Notice that βP′(t,s)=uP−1(s)σP′(t,s), where σP(t,⋅)=ϕ∘βP(t,⋅) is the rolling of βP(t,⋅), so we can use Lemma 5.23 to get
[TABLE]
Therefore
[TABLE]
where f(Kγ) is some random variable in L1(Wo(M)),
so
[TABLE]
From above one can see
[TABLE]
From (\refequ.8.19) we know that ∑j=1n(g~j∗(1)∣P∣)→0 in L∞−(W), also notice that for any q≥1,
[TABLE]
So
[TABLE]
Lemma 5.27
If ∇R≡0,
then for any q≥1,
[TABLE]
Proof. Since
[TABLE]
so
[TABLE]
Then let g~j(s):=Xhα,j(KP(s))
and this lemma follows from a Lemma 5.26-type argument.
By Lemma 6.1, X~tr,νf∈L∞−(Wo(M)). Therefore Lemma 6.2 follows from triangle inequality and the fact that the law of ϕ(βP) under ν is νP1.
Notation 6.3
Denote by g any one of {gi}i=0d as in Theorem 3.2 and {g(m)}m⊂C0∞(M) be the approximate sequence in Ld−1d(M) as defined in Remark 3.3.
Lemma 6.4
Define g~(σ)=g(σ(1)) and g~(m)(σ)=g(m)(σ(1)), then for any f∈FCb1,
[TABLE]
and
[TABLE]
Proof. Since ν{σ:σ(1)=x}=0, so g~ is ν−a.s. well-defined. In particular, for any p>0,
[TABLE]
where λ is the volume measure on M.
Since g has compact support and p1(0,⋅)∈C∞(M),
[TABLE]
Combining Eq.(6.3) and (6.4) and letting p=d−1d we get
[TABLE]
Since X~tr,νf∈L∞−(Wo(M)) by Lemma 6.1, using Holder’s inequality we prove Eq.(6.1).
Since ∪msupp(g(m)) is contained in a compact set, Eq.(6.2) can be proved similarly with g replaced by g(m)−g.
Lemma 6.5
Define g~:HP(M)→R to be g~(σ)=g(σ(1)), then g~∈Ld−1d(HP(M),νP1).
Proof. Set
[TABLE]
Applying the co-area formula Eq.(3.22) to f(y)=1{y=x}, we have
[TABLE]
So g~ is νP1−a.s. well-defined. Then applying the co-area formula Eq.(3.22) again to ∣g~∣d−1d,
we have
[TABLE]
where hP(x)∈C(M)
is defined in Theorem 3.13 with f≡1. Since g has compact support,
[TABLE]
Therefore
g~∈Ld−1d(HP(M),νP1).
Lemma 6.6
Define g~(σ)=g(σ(1)) and g~(m)(σ)=g(m)(σ(1)), then there exists a constant M such that for any f∈FCb1 and P with ∣P∣<M1,
[TABLE]
and
[TABLE]
Proof. Using Lemma 6.2, Lemma 6.5
and Holder’s inequality, we can easily see Eq.(6.7). Then applying the co-area formula (3.22) with
[TABLE]
we have:
[TABLE]
Since ∪msupp(g(m)) is contained in a compact set, Eq.(6.8) can be proved using exactly the same argument as in Lemma 6.6 with g replaced by g(m)−g and letting m→∞.
Lemma 6.7
For any p≤d−1d, supPE[∣g~(ϕ∘βP)∣p]<∞.
Proof. Since the law of ϕ∘βP under ν is νP1, we have
[TABLE]
Then applying co-area formula (3.22) exactly as Eq. (6.6) we get
[TABLE]
Using Proposition 3.30, note that g has compact support, we have
[TABLE]
Theorem 6.8
For any f∈FCb1,
[TABLE]
Proof. Since
[TABLE]
and
[TABLE]
we have
[TABLE]
Choosing p<d−1d and using Holder’s inequality, we have
Then because of Lemma 6.1, it suffices to find a p≤d−1d such that
[TABLE]
Since for any ϵ>0, there exists a constant Cp,ϵ such that
[TABLE]
We choose p and ϵ such that p(1+ϵ)<d−1d. From Eq. (6.5) we know E[∣g~∣p(1+ϵ)]<∞. Then using Lemma 6.7 we have
[TABLE]
So supPEν[∣g~(ϕ∘βP)−g~∣p(1+ϵ)]<∞ and thus
[TABLE]
Then we want to show ∣g~(ϕ∘βP)−g~∣p→0 in probability.
Let UP:={σ∈Wo(M):E1∘Φ−1∘βP(σ)=x}, since the law of Φ−1∘βP under ν is νP1, recall from Lemma 6.5 that
VP:={σ∈HP(M):E1P(σ)=x} and νP1(VP)=0, so
Notice that g∈C∞(M/{x}) and π:O(M)→M is smooth, so excluding N, we have
[TABLE]
Combining this with uniformly integrability we proved Eq. (6.12).
Proposition 6.9
Let f∈FCb1, then
[TABLE]
where
Σr(σ)=σ(r) is the canonical
process on Wo(M).
Proof. Since f=F(Σs1,…,Σsn), we have by Markov property,
[TABLE]
Viewing ∫Mn−1F(x1,…,xn)Πj=1npn1(xj−1,xj)dx1⋯dxn−1 as a function of xn, observe that it is uniformly integrable with respect to xn, therefore it is a continuous function of xn. Thus
[TABLE]
Proof of Theorem 1.11.
Recall from Remark 3.3 that we can construct an approximate sequence of the delta mass δx on M:
[TABLE]
where {gj(m):0≤j≤d,m≥1}⊂C0∞(M) and {Xj:1≤j≤d}⊂Γ(TM) with compact supports.
Consider their orthogonal lift on HP(M) (referring to Theorem 4.6) as follows
[TABLE]
where g~(σ)=g∘E1P(σ) for any g∈C(M) and XP,i is the orthogonal lift of Xi into Γ(THP(M)).
For any 0≤j≤d (with the convention that XP,0=I), using integration by parts, we get:
If X is a normal random variable with mean 0 and variance t, then
[TABLE]
Proof. The result follows from the direct computation below.
[TABLE]
If k≥2t1, then
[TABLE]
If k<2t1, then
[TABLE]
Lemma A.2
Let
β be a standard Brownian motion on Rd, {si=ni}i=0n be an equally spaced partition of [0,1] and Δiβ be βsi−βsi−1, then for any q>0, we have
[TABLE]
Proof. Since for each j, ∣Δjβ∣2=∑l=1d(Δjβ)l2, where {(Δjβ)l}l=1d are coordinates of Δjβ, i.e. Δjβ=((Δjβ)1,…,(Δjβ)d). Since β is a Brownian motion on Rd, {(Δjβ)l}l=1d are i.i.d with Gaussian distribution of mean 0 and variance n1. Using Lemma A.2 we have
[TABLE]
Then Eq.(\refGs) follows since (1−n2q)−2nd converges as n→∞.
Proposition A.3
Consider an ODE:
[TABLE]
where Y(s),A(s)∈Mn×n(R) are real n×n matrices
and A(s) is positive semi-definite.
Denote by {C(s),S(s)}the solutions
to this ODE with initial values:
[TABLE]
Recall that in this paper we use eig(X) to denote the set of eigenvalues of matrix X.
Then
•
If λ∈eig(C(s)), then ∣λ∣≥1.
•
If λ∈eig(S(s)), then ∣λ∣≥s.
Proof. For all v∈Cd, define v(s):=C(s)v,
then:
[TABLE]
Therefore,
[TABLE]
Since ⟨v′(0),v(0)⟩=0,
so ⟨v′(s),v(s)⟩≥0.
Therefore
[TABLE]
Notice that ∥v(0)∥2=∥v∥2,
so
[TABLE]
Therefore if λ∈eig(C(s)), choose v∈Cd to be an eigenvector associated to λ, then
[TABLE]
So
[TABLE]
Therefore C(s) is invertible and
[TABLE]
A lower bound result for ∥S(s)v∥ can be found
in [18, Appendix E]:
[TABLE]
From there it follows
[TABLE]
and S(s) is invertible with
[TABLE]
Definition A.4
Fix ξ∈Rd, σ∈H(M), denote Ru(s)(ξ,⋅)ξ
by Aξ(s), and let Cξ(s),Sξ(s)∈Md×d be the solutions to
V′′(s)=Aξx(s)V(s) with initial values Cξ(0)=I,Cξ′(0)=0 and Sξ(0)=0,Sξ′(0)=I.
Proposition A.5
If R is bounded by a constant
N, i.e. ∣R(ξ,⋅)ξ∣≤N∣ξ∣2,
then
[TABLE]
[TABLE]
[TABLE]
and
[TABLE]
Proof.A.2 and A.3 are quite elementary,
so here we only present the proof of A.4 and A.5.
By Taylor’s expansion,
[TABLE]
[TABLE]
Define f(s):=∣Sξ(s)−sI∣,
then we have:
[TABLE]
By Gronwall’s inequality:
[TABLE]
Then we consider Cξ(s):
[TABLE]
So
[TABLE]
Define f(s):=∣Cξ(s)−I∣,
then we have:
[TABLE]
By Gronwall’s inequality:
[TABLE]
Appendix B Matrix Analysis
Theorem B.1
Suppose that V is a finite
dimensional inner product space, A:Vn→V is a linear
map, and
[TABLE]
Then
[TABLE]
Proof. First observe that
[TABLE]
We denote dim(V)=d and let {uj}j=1d⊂V be an orthonormal
basis of eigenvectors for AAtr:V→V
so that AAtruj=λjuj and then let
vj:=Atruj. Then it follows that
[TABLE]
Now extend {vj}j=1d to a basis for all
Vn. From this we will find that StrS has
eigenvalues {1}∪{1+λj}j=1d
and therefore
[TABLE]
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