Fisher Information Geometry of Poisson Kernels and Heat Kernel on Riemannian Manifolds

Fisher Information Geometry of Poisson Kernels and Heat Kernel on Riemannian Manifolds

Mitsuhiro Itoh

Institute of Mathematics, University of Tsukuba, Ten-odai, 1-1-1, Tsukuba, 305- 8751, Ibaraki, Japan

e-mail : itohm@sakura.cc.tsukuba.ac.jp Hiroyasu Satoh

Institute of Mathematics, University of Tsukuba, Ten-odai, 1-1-1, Tsukuba, 305- 8751, Ibaraki, Japan

e-mail : hiroyasu@math.tsukuba.ac.jp

(2000 Mathematics Subject Classification : 53C35, 53C42, 58B20.)

Abstract. Information geometry of Poisson kernels and heat kernel on an Hadamard manifoldX which is harmonic is discussed in terms of the Fisher information metric. The Poisson kernel map and the heat kernel map, both, turn out to be a homothetic immersion fromXinto the space of probability measures. Certain geometric properties of Shannon’s entropy for the Poisson kernels and the heat kernel are shown. It is verified that there exists a crucial relation associated with the heat kernel between the homothety constant and the Shannon’s entropy.

0 Introduction and Main Theorems

The classical Dirichl´et problem for the Laplace-Beltrami operator ∆ =−∇ⁱ∇i

on then-dimensional unit ball inRⁿ is solved by the integration as u(x) =

Z

∂Bⁿ

Φ(x, θ)ψ(θ)dθ by using the Poisson kernel.

Now, let (X, h) be a connected, simply connected, completen-dimensional Rie- mannian manifold with curvature strictly bounded as −b² ≤KX ≤ −a² <0. We can consider in this case the Dirichl´et problem with respect to the ideal bound- ary ∂X. (X, h) admits, then, Poisson kernels Φ(x, θ) such that any solution to the Dirichl´et problem with boundary condition at infinity is written as the Poisson kernel integration, similarly as on the Euclidean unit ball. The detailed argument is referred to [Sch-Y]. Thus, any Poisson kernel together with the standard vol- ume form gives rise to a probability measure on the ideal boundary so that we can consider a map from X into P(∂X), the space of probability measures on∂X, by assigning to any pointxin X a probability measure ρ= Φ(x, θ)dθon∂X.

1

On the other hand, consider a heat flow equation onRⁿ;

∂u

∂t(t, x) + ∆xu(t, x) = 0, u(0, x) =f(x)

Then, it is well known that there exists a fundamental solution, called heat kernel k(x, y;t), a real valued function defined on Rⁿ×Rⁿ×R⁺ such that any solution of the above equation can be written in heat kernel integration

u(t, x) = Z

k(x, y;t)f(y)dy.

We observe that the heat kernel together with the standard volume formdv yields probability measures on Rⁿ, parametrized with (x, t) ∈ Rⁿ ×R⁺. So, we can consider a map fromRⁿ into P(X),X =Rⁿ, same as in the Poisson kernel case.

LetM be a connected, oriented complete Riemannian manifold, not necessarily compact. Consider the spaceP(M) consisting of probability measures with positive density function defined onM and provide a topology on it in terms of a (weighted) Sobolev norm. This space is equiped with an infinite dimensional manifold structure with a Riemannian metric, called Fisher information metric. This naturally defined metric is a generalization of Fisher information matrix, introduced in the theory of statistical influence([A-N]). The Fisher information metricGcarries geometrically significant properties. For examples, any orientation preserving diffeomorphism of M induces an isometric action on (P(M), G) and furthermore (P(M), G) is a manifold of constant sectional curvature equal to 1/4. The details will be given in

§.1.2.

We are now at a position to consider the map Φ, called the Poisson kernel map, from a connected, simply connected, complete n-dimensional Riemannian manifold (X, h) with negative curvature, strictly bounded from below and above, into (P(∂X), G), by the aid of the Poisson kernels, and also the map ϕt, called the heat kernel map, from a Hadamard manifold (X, h) into (P(X), G) by the heat kernel, respectively. Further we have another very important concept in the information theory, the Shannon’s entropy, which measures information uncertainty of any probability measure.

The aim of this paper is to investigate information geometry of the map Φ and ϕtin terms of the Fisher information metricGand the manifold metrich, and also from the viewpoint of Shannon’s entropy.

We summarize the main results as that these kernel maps are homothetic, as shown in Theorems 1 and 5, provided (X, h) is a manifold with strictly negative curvature, which is Riemannian homogeneous (in case of Poisson kernel map) or, a Hadamard manifold (in case of heat kernel map).

In Theorems 2 and 6 we will state that the Shannon’s entropyI associated with the Poisson kernel map, viewed as a function onX, must satisfiy ∆(I(ϕ)) =λ²for the volume entropy λof X, and that the entropy associated with the heat kernel map must satisfy, for its time derivative, _dt^dIt=nCt>0, whereCtis the homothety constant of the heat kernel map.

In Part I we introduce the space of probability measures with positive density function and give definition of Fisher information metric together with its basic geometrical properties and also definition of the Shannon’s entropy.

Part II deals with the main results. Basic treatment of the ideal boundary of a simply-connected, complete, Riemannian manifold of strictly negative curvature together with normalized Poisson kernels is given in§2.1 and 2.2. The Poisson kernel map is defined at §2.3. Also the Busemann functions, normalized at a reference point, are introduced in §2.4, where we give a complete proof to Proposition 2, a key for Theorem 1. Notice that the proof of the cocyle formula and the Jacobian formula with respect to an isometric action are given. These formulas are a key of our Theorem 1. Definition and basic properties of heat kernels and the heat kernel map are given at§2.6. We deal briefly with harmonic manifolds in order to exhibit Theorem 4, valid for harmonic Hadamard manifolds.

Acknowledgements. The first author would like to thank Professor Young Jin Suh to invite him and to deliver an opportunity of giving talks on our research. He thanks awfully the hospitality during the 12th workshop.

1 Part 1. Geometry of Probability Measures 1.1 Space of probability measures

Let (M, h) be a compact, connected, orientedn-dimensional smooth manifold with a smooth Riemannian metric h=P

h_ijdxⁱdx^j with unit volume form dv = (dethij)^1/2 dx¹∧ · · · ∧dxⁿ.

A probability measureρ=p dv overM is ann-form defined overM satisfying R

Mρ=R

x∈Mp(x)dv(x) = 1. Each probability measure is assumed to have density functionp,everywhere positive. We define the space of probability measures having positive density function as:

P(M) ={ρ=p(x)dv(x)|p∈L²_k(M), p(x)>0, Z

M

ρ= 1},

where k is an integer satisfying k > n/2. The Sobolev space L²_k(M) is needed to assure that Sobolev inequality argument works. Any probability measure with positive density function is considered as a point of the spaceP(M).

Note. P(M)⊂Ωⁿ(M), where Ωⁿ(M) = Γ(M; Λⁿ(M)) is the space of n-forms.

By using Sobolev space argument, this space is an infinite dimensional manifold with tangent space TρP(M) at a pointρ:

{τ=q(x)dv(x)|q(x)∈L²_k(M), Z

M

τ= 0}

Take a pointρ∈ P(M) and a tangent vector τ at ρ. Then, it is easily shown

that in the spaceP(M) there exists a parametrized curveρ(s) =ρ+sτins∈(−², ²) for a sufficiently small²such thatρ(0) =ρand the velocity vector _ds^d|s=0ρ=τ.

1.2 Fisher Information Metric

On the space P(M) we introduce the Fisher information metric G, a general- ization of the Fisher information matices (Gij);

Definition1. An inner productGρ is defined overTρ=TρP(M) as Gρ(τ1, τ2) =

Z

M

dτ1

dρ dτ2

dρ ρ, (1)

whereτi=qi(x)dv(x)∈TρP(M),i= 1,2 are tangent vectors atρand dτi

dρ = qi(x) p(x),

are the Radon-Nikodym derivative ofτi with respect toρ,i= 1,2. So Gρ(τ1, τ2) =

Z

M

q1(x) p(x)

q2(x)

p(x)p(x)dv(x) =Eρ(dτ1

dρ dτ2

dρ), (2)

whereEρ(·) denotes the expectation of·with respect to the probability measureρ.

It is not hard to see that theGρ is positive definite.

Example 1. LetN(µ, σ²) be normal distributions of expectationµ, varianceσ²; p(x) = 1

(2πσ²)^1/2 exp µ

−(x−µ)² 2σ²

¶ ,

a family inP(R) parametrized with (ξ1, ξ2) = (µ, σ). The Fisher information matrix is then

µ ₁

σ² 0 0 _σ²2

¶ , that is,

Xgijdξidξj = 1

σ²(dξ₁²+ 2dξ²₂).

The parameter space

Ξ ={(ξ1, ξ2) = (µ, σ)| − ∞< µ <∞, 0< σ}

is then considered as the hyperbolic plane of constant Gaussian curvature−2.

Theorem ([F]). The inner products G={Gρ, ρ∈ P(M)} yields a Riemannian structure onP(M), called Fisher information metric, which enjoys the following

(i) the Levi-Civita connection∇ atρis

∇τ1τ= 1 2

µdτ dρ

dτ1

dρ − Z

M

dτ dρ

dτ1

dρρ

¶

ρ, τ1, τ ∈TρP(M), (3)

where τi,i= 1,2 are considered as a vector field extended parallelly, (ii) it has constant sectional curvature equal to 1/4,

(iii) (Dif f)⁺_k+1(M) acts onP(M) transitively and isometrically so that P(M) = (Dif f)⁺_k+1(M)/K,

where (Dif f)⁺_k+1(M) is the group of all orientation preserving diffeomorphisms of M whose Jacobian matrices are in L²_k+1, and K is the isotropy subgroup of (Dif f)⁺_k+1(M) fixing some ρ∈ P(M).

To explain the formula in (i) it suffices to see the connection defined above fulfills ∇G= 0 and is torsion-free, i.e., ∇τ1τ2− ∇τ2τ1−[τ1, τ2] = 0 for parallelly extended vector fieldsτ_i,i= 1,2 so that the torsion field is trivial. While we omit a detail, a direct substitution of (3) into∇Gis readily checked to be zero. See [F]

or [Shi].

(ii) is a straightforward computation. (iii) stems from well known Moser’s theo- rem on volume forms. Letµandθbe smooth volume forms onM withR

Mµ=R

Mθ.

Then, there exists a smooth, orientation preserving diffeomorphism ψ such that µ=ψ^∗θ

Example 2(discrete case). Consider the space of discrete probabilities:

P={µ= (p1, p2, p3)∈R³|pi>0,X pi= 1}

with the Fisher information metricGlooks just as a portion of 2-sphere of radius 2 so that Gaussian curvature is equal to 1/4.

In fact, P is an open 2-simplex ∆⊂ {R³|x+y+z= 1}, whose tangent space is TµP ={τ = (v1, v2, v3)∈R³|X

vi= 0}, with a basis

τ1= (1,0,−1), τ2= (0,1,−1)∈TµP Then the Fisher metricGhas components atµ= (p1, p2, p3):

µ G11 G12

G21 G22

¶

=

µ 1/p1+ 1/p3 1/p3

1/p3 1/p2+ 1/p3

¶

for which a suitable variable change indicates that the Gaussian curvature is con- stant 1/4.

Remark. The Fisher metric G, which turns out to be not geodesically com- plete(see [F]), is defined statistically. However, it is also very natural from a view- point of geometry. Define the square-root map

ψ: (P(M), G) −→ S₊^∞(M)⊂(L²_k(M, dv),h·,·iL²) : ρ = p(x)dv7→p

p(x), where

S₊^∞(M) ={f ∈C⁰(M)|f(x)>0, Z

M

f²(x)dv= 1}

which is appeared in [B-C-G] in order to discuss their rigidity argument.

Proposition1 ([I-S-Shi]). ψis homothetic;

ψ^∗(h·,·iL² |S₊^∞) =1 4 G Proof. We have forτ =q(x)dv atρ=p(x)dv

ψ∗(τ) = d

dt(p+tq)^1/2|t=0=1 2

q(x) p(x)^1/2 so that together withτ1=q1(x)dv

hψ∗(τ), ψ∗(τ1)i= 1 4

Z

M

q(x) p(x)^1/2

q1(x)

p(x)^1/2dv=1 4

Z

M

q(x) p(x)

q1(x)

p(x)p(x)dv=1

4Gρ(τ, τ1)

1.3 Shannon’s Entropy

Definition 2 ([Shan]). TheShannon’s entropy I(ρ) for aρ=p(x)dx ∈ P(M) is defined

I(ρ) =− Z

X

p(x) logp(x)dv(x) =Eρ(−logp), that is, the expectation of a probability variable−logp(x), x∈M.

I(ρ) measures information uncertainty of the probability measureρ.

2 Part 2. Information Geometry of Poisson Kernels and Heat Kernels 2.1 Negatively curved manifold and the ideal boundary

Let (X, h) be a connected, simply connected, completen-dimensional Rieman- nian manifold. Assume, here and in what follows, the negative curvature condition:

−b²≤KY ≤ −a²<0,

for any plane section Y. This is required to guarantee existence and uniqueness of Poisson kernel. Cartan-Hadamard theorem implies that X is diffeomorphic toRⁿ. Example3. Typical examples are rank one symmetric spaces of non-compact type ([H]): real hyperbolic space H_Rⁿ, complex hyperbolic spaceH_Cⁿ, quaternionic hy- perbolic space H_Hⁿ, octonionic hyperbolic spaceH_Q¹. For instance,

Dⁿ={x∈Rⁿ| |x|<1}, ho= 4 (1− |x|²)²

Xdx²_i

gives a unit ball model of the space H_Rⁿ. Similarly Dⁿ_C = {z∈Cⁿ| |z|<1},

ho = 4

1− |z|²

X|dzi|²+X

i,j

4zizjdzidzj

(1− |z|²)² is a complex ball model of the space, H_Cⁿ.

To define an ideal boundary of (X, h) we take the setRX of all (geodesic) rays in X. Here a curve c: [0,∞)−→X is called a ray, ifc is a geodesic of unit speed.

The negative (rather, nonpositive) curvature condition implies that any segment of c attains the distance of the endpoints asd(c(t), c(t⁰)) = |t−t⁰| for any t, t⁰. We then define asymptotical equivalence between them as

Definition3. Raysc1andc2inX are equivalent: c1∼c2⇐⇒if there is a positive constant a >0 such thatd(c1(t), c2(t))< afor allt∈[0,∞).

Definition 4 (ideal boundary). ∂X =RX/∼ is called the ideal boundary of (X, h).

Remark. The ideal boundary ∂X is identified with the (n−1)-dim unit sphere Sⁿ⁻¹via the exponential map exp_xoat a fixed reference pointxo. In fact, any point θ of∂X can be realized uniquely as [c], where cis a ray issuing fromx_o so that

UxoX ={v∈TxoX||v|= 1} ∼=∂X;

by identifyingc for whichθ= [c]∈∂X withc⁰(0) =v∈Tx_oX

Note. (i) ForH_Rⁿ the unit sphereSⁿ⁻¹={x∈Rⁿ| |x|= 1}, the real boundary of the open unit ballDⁿ, is actually the ideal boundary via the open unit ball Poincare model.

(ii) X∪∂X is equiped with the cone topology, a compactification ofX ([B-G- Schr],[E]).

(iii) The ideal boundary can be defined also for a Hadamard manifold, a simply connected, complete Riemannian manifold of non-positive curvature. Even the Eu- clideann-space admits the ideal boundary, homeomorphic to the (n−1)-dimensional sphere.

2.2 Poisson Kernel

The definition of Poisson kernel requires the Laplace-Beltrami operator ∆ =

−P

h^ij∇i∇j operating on smooth functionsC^∞(X).

Consider the Dirichl´et problem at infinity: Givenψ∈C⁰(∂X), solve

∆u= 0 on X, u|∂X=ψ.

The solutionuis given in the Poisson integral formula u(x) =

Z

∂X

Φ(x, θ)ψ(θ)dθ, x∈X.

Definition5 (normalized Poisson kernel) ([Sch-Y]).

Given a pointx_o∈Xand a boundary pointθ∈∂X. Then, a function Φ_θ(x) = Φ(x, θ)∈C⁰(X∪∂X\{θ}), associated withθ, is called a normalized Poisson kernel, if it satisfies the following conditions;

(i) (harmonicity) Φ is harmonic inx, i.e., ∆xΦ = 0, (ii) (positivity) Φ(x, θ)>0 for allx∈X,

(iii) (normalization) Φ(xo, θ) = 1 at xo,

(iv) (approaching the delta measure) if an interior pointxofX goes to θ⁰ ∈∂X in the cone topology andθ⁰6=θ, then

x→θlim⁰Φ(x, θ) = 0.

Thus defined normalized Poisson kernel depends on the choice of reference point.

Example 4. On (Dⁿ, h), the real hyperbolic space, the Poisson kernel is Φ(x, θ) =

µ1− |x|²

|x−θ|²

¶_n−1

, x∈Dⁿ, θ∈∂Dⁿ, (4)

where the originois the reference point. It is easily checked that the Φ(x, θ) satisfies (ii),(iii),(iv) of the definition, except (i). (i) is easily checked by using the formula;

∆f =− 1

√h

∂

∂xⁱ(√ hh^ij ∂f

∂x^j), wherehij=F δij,h^ij =F⁻¹δ^ij and√

h=F^n/2,F =_(1−|x|⁴ 2)².

The existence theorem of Poisson kernels is assured for complete Riemannian manifolds of strictly negative curvature.

Theorem([A], [Sul], [Sch-Y]). Let (X, h) be a connected, simply connected, com- pleten-dimensional Riemannian manifold.

Under the negative curvature condition;−b² ≤K_Y ≤ −a²<0, (X, h) admits Poisson kernels Φ(x, θ), normalized at any reference point.

Moreover, the uniqueness of normalized Poisson kernels is guaranteed.

Remark. The Poisson kernel integration formula is a generalization of the classical Poisson integral over the unit circle in the plane.

2.3 Poisson kernel map

Definition 6. Define the map, called the Poisson kernel map by the aid of the normalized Poisson kernels

Ψ :X −→ P(∂X);x7→Ψ(x) = Φ(x, θ)dθ,

wheredθdenotes the standard unit measure onUxoX ∼=∂X such thatR

∂Xdθ= 1.

In fact, Φ(x, θ) dθ gives a probability measure on ∂X with positive density function, parametrized with a pointxofX, since by solving the Dirichl´et problem at infinity withψ= 1 identically overP(∂X) , one has

1 = Z

∂X

Φ(x, θ)dθ, ∀x∈X

Theorem 1 ([I-Shi]).

Let (X, h) be an n-dimensional rank one symmetric space of noncompact type and Ψ : (X, h)−→(P(∂X), G) the Poisson kernel map. Then,

(i) it is homothetic, i.e.,

Ψ^∗G=λ² nh (ii) and it is a minimal embedding.

Hereλis the growth rate of volume of the geodesic ballB_x(r) atx∈X, called thevolume entropyof (X, h):

λ(x) = lim

r→∞

1

rlog vol(Bx(r)), x∈X.

(see [B-C-G]).

Remark. (i) The volume entropy λ is in general a function of X. However, it is an isometry invariant so it is constant on a Riemannian homogeneous manifold.

The above definition of λis equivalent to saying that for a sufficiently large r the volume of a geodesic ball of radius r is given as vol(By(r))∼cexp (λr). Note λ coincides with the topological entropy when X is a compact Riemannian manifold of nonpositive curvature.

(ii) For rank one symmetric spaces H_Rⁿ, H_Cⁿ,H_Hⁿ andH_Q¹: λ=n+ dim_RF−2, for F=R,C,H or Q (see p. 740, [B-C-G]).

The proof of Theorem 1 will be given in§2.5 via the Busemann function argu- ment and homogeneity of (X, h).

2.4 Poisson kernels and Busemann functions

Definition 7 (Busemann function). Let (X, h) be as above and xo a reference point. For anyθ∈∂X the function defined in the following is called a Busemann functionB(x, θ), normalized atxo:

B(x, θ) = lim

t−→∞(d(x, c(t))−t), wherec is a ray starting atxo and approachingθ.

It is known that for a simply connected, connected, complete Riemanninian manifold (X, h) satisfying the negative curvature condition there exist Busemann functions B(·, θ), associated to θ∈ ∂(X) which are at least C²-class. See for this [B-G-Schr], [E] or [A-A].

The normalized Busemann functions satisfy the following:

(i) |∇B(x, θ)|= 1 for anyxand θ, (ii) B(xo, θ) = 0 for anyθ,

(iii) limx→θB(x, θ) =−∞ for allθ, (iv) limx→θ⁰B(x, θ) = +∞,θ⁰6=θ.

There exists a similarlity ! between the Poisson kernels and the Busemann functions. In fact, we have

Proposition 2. Let (X, h) be a simply connected, connected, complete Rieman- ninian manifold satisfying the negative curvature condition. Assume that (X, h) is harmonic. Then

Φ(x, θ) = exp(−λB(x, θ)), whereλis the volume entropy of (X, h).

We will give a proof, although the formula is appeared in p.740, [B-C-G] without a proof.

Before proving this, we will equip the notion of horosphere. Here, a horoshere Σ = Σx,θ, centered at θ ∈ ∂X through a point x ∈ X, is the level set of the Busemann functionB(·, θ), that is,

Σx,θ={y∈X|B(y, θ) =B(x, θ)}.

It is known that the horosphere Σ_x,θ is a limit of geodesic spheres(see [A-A]). In fact, putB(x, θ) =sand take the geodesicc(t), −∞< t <∞issuing fromxoand c(+∞) =θ ( hence, B(c(−s), θ) = s=B(x, θ), i.e., c(−s)∈Σx,θ ) and consider the geodesic spheres S_c(s+t)(t), t > 0, centered at the pointc(s+t), situated on the geodesic c, and of radius t. Noticing that all such geodesic spheres pass the common pointc(−s), we have the expression of the horosphere as

Σx,θ = lim

t→∞Sc(s+t)(t).

Let Φ(x, θ) be a normalized Poisson kernel, whose existence is guaranteed by Theorem([A], [Sul], [Sch-Y]). In order to show the proposition, it suffices to assert that every horosphere has constant mean curvature whose value is equal to −λ. In fact, if so, then, as we set Ψ(x, θ) = exp(−λB(x, θ)), then we see the harmonicity of Ψ as

∆Ψ(x, θ) = (−λ)Ψ(x, θ){∆B(x, θ)−λ|B(x, θ)|²}= 0,

where the mean curvature of a horosphere is given ∆B(·, θ) which must equal λ.

Further, it is easily seen that the Ψ is Poisson kernel, normalized at xo from the properties of Busemann function. Since (X, h) is harmonic, whose definition will be given in §.2.7, (X, h) is analytic so that the Busemann functions and hence horospheres are analytic. Then, the mean curvature of Σx,θ at a point y is a limit of mean curvature η =η(r) of the geodesic spheres. However, η is the minus sign of the trace of the Hessian Ddr of the distance functionr from its center so that η(r) =−∆rfor which we have the formula(see 4.16,[G-H-L])

∆r=−n−1

r − 1

ω(r, u)

∂ω(r, u)

∂r ,

where ω(r, u) is the volume density function in a geodesic ballB centerd aty and (r, u) denoting the spherical coordinate inB. So, the mean curvature of Σx,θ at a pointy is then given as

− lim

r→∞

1 ω(r, u)

∂ω(r, u)

∂r . (5)

From the harmonicity of X, ω(r, u) is independent of the spherical coordinates u.

So the Riemannian volume form of X is dv = ω(r, u)rⁿ⁻¹drdu = ω(r)rⁿ⁻¹drdu.

Then, the volume of the ball By(r) of radius r is vol(Sⁿ⁻¹)×R_r

0 tⁿ⁻¹ω(r)dr. On the other hand,

vol(By(r))∼const.exp(λr), r−→ ∞

from the volume entropy formula. Thus, we can conclude from (5) that the mean curvature of Σx,θ aty is equal toλ.

2.5 Isometrical property of Poisson kernel map

Let I_o(X, h) be the group of orientation preserving isometries of (X, h). Then we have

Proposition 3([I-Shi]). The Poisson kernel map Ψ commutes with the isometric actionγ;

Ψ(γ(x)) =γ(Ψ(x)), x∈X, namely

Φ(γ(x), θ)dθ=γ(Φ(x, θ)dθ), x∈X,

X −→ P(∂X)

↓ γ ↓ γ

X −→ P(∂X)

Proof. Letγ∈Io(X, h). Thenγinduces an action on the ideal boundary ∂X:

γ:∂X −→∂X; θ= [c]7→ [γ◦c],

because γ maps any ray c = c(t) to another ray γ◦c(t) so that if c ∼ c⁰, then γ◦c∼γ◦c⁰.

A transformation γ : ∂X −→ ∂X, thus defined, induces further an action of P(∂X) as

γ(ρ) = (γ⁻¹)^∗(ρ), ρ∈ P(∂X)⊂Ωⁿ⁻¹(∂X),

Here (γ⁻¹)^∗is the pull-back by γ⁻¹. So we have the left action ofIo(X, h);

(γ◦γ1)(ρ) =γ(γ1(ρ)), γ, γ1∈Io(X, h).

Proposition4 (Cocycle formula)([B-C-G]). The normalized Poisson kernels obey the following transformation formula; for anyx∈X and anyθ∈∂X it holds

Φ(γ(x), θ) = Φ(x, γ⁻¹(θ)) Φ(γ(x_o), θ),

This is a consequence of the uniqueness of normalized Poisson kernels, if we define a function Σ onXas Σ(x) = _Φ(γ(x^Φ(γ(x),θ)

o),θ). In fact Σ(x) satisfies all the conditions of normalized Poisson kernel except for (iv). However, if we setθ⁰ =γ⁻¹θ, and let xapproach any point different fromθ⁰. This meansγ(x) approaches another point than θ so that Σ(x) = _Φ(γ(x^Φ(γ(x),θ)

o),θ) tends to zero. Hence, from the uniqueness Σ(x) must be Φ(x, γ⁻¹(θ)). Then, Proposition 3 follows from the above proposition together with

Proposition 5 (Jacobian formula)([B-C-G]).

γ^∗dθ= Φ(γ⁻¹(xo, θ)dθ,

The proof is obtained from the following consideration.

Let u be a unique solution to the Dirichl´et problem with a given boundary condition asu(θ) =ψ(θ),θ∈∂X. Then,uis givenu(x) =R

θΦ(x, θ)ψ(θ)dθ.

Further, let γ ∈ I0(X, h) and set v = γ^∗u = u ◦ γ. Then v fullfils v(x) = R

θΦ(γx, θ)ψ(θ)dθ, which reduces from the cocyle formula to v(x) = R

θΦ(x, γ⁻¹θ)Φ(γxo, θ)ψ(θ)dθ. On the other hand, since v satisfies the Laplace equation and also the boundary condition v(θ) = u(γθ) = ψ(γθ) so that v can be written as v(x) = R

θΦ(x, θ)ψ(γθ)dθ. Changing the variable yields v(x) = R

θΦ(x, γ⁻¹θ)ψ(θ)(γ⁻¹)^∗dθ.

So, we getR

θΦ(x, γ⁻¹θ)ψ(θ){Φ(γx_o, θ)dθ−(γ⁻¹)^∗dθ}= 0 which holds for any ψ ∈ C⁰(∂X). Write (γ⁻¹)^∗dθ=f(θ)dθ and set ψ(θ) = Φ(γxo, θ)−f(θ), θ ∈∂X so that we haveR

θΦ(x, γ⁻¹θ){Φ(γxo, θ)−f(θ)}²dθ= 0 which implies Φ(γxo, θ)− f(θ) = 0 and hence (γ⁻¹)^∗dθ= Φ(γxo, θ)dθ. So, we get the above Jacobian formula.

Proof of Theorem 1. (i): We first give basic formula.

tr(Ψ^∗G) = Xn

i

G(dΨ(ei), dΨ(ei)) = Z

|∇log Φ(x, θ)|²Φ(x, θ)dθ (6)

This is immediate from the definition.

Put

c= (Ψ^∗G)(u, u) for u∈TxX,|u|= 1.

Then, c is independent of choice of uand x ∈ X, because (X, h) is Riemannian homogeneous and any isometry of X commutes with the Poisson kernel map, as Proposition 3 indicates. Moreover, its value is

c= λ² n,

since Φ(x, θ) is written as Φ(x, θ) = exp(−λB(x, θ)) from Proposition 2, so log Φ(x, θ) =−λB(x, θ) and then, by using the above basic formula (6)

nc=tr(Ψ^∗G) = (−λ)² Z

∂X

|∇B(·, θ)|²Φ(x, θ)dθ=λ². Here we used the property (i) of Busemann functions.

(ii): To see the map Ψ is injective, it suffices to show that, ifB(x, θ) =B(y, θ) for allθ∈∂(X), thenxcoincides with yas points.

Assume x6=y and take a geodesic ray c starting atxand passing through y.

Thenc induces the pointθ0 =c(+∞) in ∂X. Now we take another geodesic ray ˜c issuing from the xo and approaching the same pointθ0. Then we have two Buse- mann functions, theBc(·, θ0) with respect toc and the B(·, θ0), normalized at xo. However, the differenceBc(·, θ0)−B(·, θ0) is just a constant, since two Busemann functions are same up to additive constant, if they are associated with the same boundary point, as shown in 3.5 Cor., [G-S-B]. From the definitionBc(x, θ0) = 0, whereasBc(y, θ0) =−d(x, y)<0 so that B(x, θ0)6=B(y, θ0). This is a contradic- tion.

We will see the embedding Ψ is minimal. For this we make use of the Levi-Civita connection’s formula of the Fisher information metricG.

X

i

∇τiτi= 1 2

X

i

µdτi

dρ dτi

dρ − Z

M

dτi

dρ dτi

dρρ

¶ ρ,

whereρ= Ψ(x) = Φ(x, θ)dθandτi= (eilog Φ(x, θ))ρwith respect to an orthonor- mal basis{ei} at a point ofX. So ^dτ_dρⁱ =eilog Φ(x, θ) and hence,

X

i

dτi

dρ dτi

dρ =X

i

(e_ilog Φ)²=|∇log Φ(·, θ)|²=λ² is constant. So

X

i

(dτi

dρ dτi

dρ − Z

M

dτi

dρ dτi

dρρ) =λ²−λ² Z

dρ=λ²−λ²= 0.

Hence the normal component of the above also vanishes and the embedding must be minimal.

Theorem 2 ([I-Shi]). Let (X, h) be a connected, simply connected, complete n- dimensional Riemannian manifold satisfying the negative curvature condition (n≥ 3). Suppose that the Poisson kernels have the following form

Φ(x, θ) = exp{−λB(x, θ)}, x∈X

and, moreover (X, h) admits a compact smooth quotient. Then (X, h) must be a rank one symmetric space of non-compact type.

This follows from the following and Cor. 9.18 in [B-C-G].

Theorem 3 ([I-Shi]). Let (X, h) be as above. The Poisson kernels admit the exponential representation of Busemann functions as above if and only if (X, h) is asymptotically harmonic. Here, we say that ((X, h) is asymptotically harmonic, when there exists a constant c such that every horosphere has mean curvature constant equal toc.

Notice that the mean curvature c turns out to be minus sign of the volume entropy of (X, h), as discussed in Proposition 2 in§.2.4.

Remark. Assume that a function Φ is represented Φ(x) = exp(cB(x)),

with respect to a Busemann function B and a constantc. Then it holds (∆Φ)(x) =cΦ(x){∆B−c|∇B|²}(x).

So, Φ is harmonic iff trHess of B is constant −c, i.e., every horosphere Σ has constant mean curvature c, and the latter condition means that the Busemann functionB is an isoparametric function, as seen in [B-T-V], since∇B andHess(B) give rise to the unit normal and second fundamental form of the hypersurface Σ.

Question. Characterize complete manifolds (X, h) for which the Poisson kernel map Ψ :X−→ P(∂X) is minimal homothetic immersion.

Theorem 4. Let (X, h) be a simply connected, complete, n-dimensional Rie- mannian manifold satisfying the negative curvature condition. Let ρ = Φ(x, θ)dθ be a probability measure on ∂X induced by the Poisson kernel. Then,

(i) its Shannon’s entropyI(ρ) is a function onX satisfying the following

∆xI(ρ) =tr(Ψ^∗G) = Z

|∇log Φ(x, θ)|²Φ(x, θ)dθ and

(ii) if the Poisson kernel is given by Φ(x, θ) = exp(−λB(x, θ)) in terms of the Busemann function, then

∆xI(ρ) =λ²

Proof. (i) Setϕ(x) = log Φ(x, θ). Then the LHS of the equation is

∆xI(ρ) = ∆x(−

Z

θ

Φϕdθ)

= −

Z

{∆xΦ·ϕ−2h∇Φ,∇ϕi+ Φ∆xϕ}dθ= Z

|∇ϕ|²Φ(x, θ)dθ, since ∆Φ = Φ(∆ϕ− |∇ϕ|²) = 0.

(ii) follows fromϕ(x) =−λB(x, θ).

2.6 Heat kernels

The heat kernel k(x, y;t) on a complete Riemannian manifold (X, h) is the fundamental solution of the heat equation onX:

∂u

∂t + ∆u= 0,

namely, the heat kernelk=k(x, y;t)∈C^∞(X×X×R⁺) satisfies the following;

(i) symmetry and positivity;

k(x, y;t) =k(y, x;t)>0, (ii) solution to the heat equation;

(∂

∂t+ ∆_x)k(x, y;t) = (∂

∂t+ ∆_y)k(x, y;t) = 0,

(iii) for anyxofX, the measurek(x, y;t)dv(y) converges toδx(y) atxast−→+0 (iv) the semi-group property;

k(x, y;t+s) = Z

z∈X

k(x, z;t)k(z, y;s)dv(z), (v) isometry invariance;

k(ψx, ψy;t) =k(x, y;t), ψ∈Io(X, h) Then,

u(x, t) = Z

X

k(x, y;t)f(y)dv(y)

is a solution of the heat equation with initial condition u(x,0) = f(x) for a given f ∈C^∞(X).

In what follows, we restrict ourselves to connected, simply connected, com- plete Riemannian manifolds of nonpositive curvature and whose Ricci curvature is bounded below. The boundedness of the Ricci curvature ensures the uniqueness of heat kernels(see, for this, [D]). We can also consider compact manifold case in which ∆ has discrete spectrum. An investigation of compact case will be given somewhere.

Definition 8. Define theheat kernel map,

ϕt:X −→ P(X); x7→ϕt(x) =k(x, y;t)dv(y), parametrized with timet.

Example5. (Euclidean space ) k(x, y;t) = 1

(4πt)^n/2exp µ

−|x−y|² 4t

¶

Observation The heat kernel map for the Euclidean space enjoys ϕ^∗_tG= (1/2t)ho.

(7)

This is straightforward. Take{e_i=_∂x^∂i}a basis ofT_xRⁿ. Thendϕ_t(e_i), which we denote byτi, is

∂

∂xⁱk(x, y;t)dv=k(x, y;t)× µ

−2(xⁱ−yⁱ) 4t

¶ dv

The Radon-Nikodym derivative ^dτ_dρⁱ, atρ=ϕt(x), is−^2(xi_4t^−yi). So, G(dϕt(ei), dϕt(ej)) =

Z

Rⁿ

(−xⁱ−yⁱ

2t )(−x^j−y^j

2t )k(x, y;t)dv(y) which reduces to

= 1

4t²(4πt)^n/2 Z

z∈Rⁿ

zⁱz^je^−|z|2/4tdz¹∧ · · · ∧dzⁿ

= δij

4t²(4πt)^n/2 µZ _∞

−∞

r²e^−r2/4tdr

¶ µZ _∞

−∞

e^−r2/4tdr

¶_n−1 . The formula is derived then from the following special integrals:

Z _∞

−∞

e^−r2/4tdr= (4πt)^1/2, Z _∞

−∞

r²e^−r2/4tdr= 2t (4πt)^1/2.

Here, we comment on the topology ofP(X), becauseX is a noncompact mani- fold. To define suitable topology onP(X) we prepare the spaceC_c^∞(X) of smooth functions of compact support and take its completion with respect to the (weighted) SobolevL²_k-norm in terms of the metrich, the Levi-Civita connection and the Rie- mannian volume form dv= det(hij)^1/2dx¹· · ·dxⁿ. See [Gilk], [D], for this.

2.7 Information geometry of harmonic spaces Like the Euclidean space, we have

Theorem 5 ([I-S-Shi]). Let (X, h) be a simply-connected, noncompact complete Riemannian manifold of nonpositive curvature, which is harmonic.

Then, it holdsϕ^∗_tG=Cth, whereCt is a positive constant depending only on t.

Definition9 ([Besse], [Sz]). A complete Riemannian manifold (X, h) isharmonic if it satisfies one of the following conditions equivalent each other;

(i) for any x of X the Laplacian ∆y r²(x, y) of r²(x, y) is again a function of r²(x, y), wherer(x, y) is the distance ofxandy,

(ii) the volume density functionωx=ωx(r, u) in a normal coordinates is a radial function around any point x ∈ X, where r and u denote the distance from the center xand the spherical coordinates, respectively.

(iii) (mean value property) for any harmonic function f locally defined at any x∈X the average functionAx(f) over a geodesic sphereSx(r) aroundxof radius rhas

Ax(f)(z) =f(x), ∀ z∈Sx(r),

(iv) every sufficiently small geodesic sphere has constant mean curvature, when dimX >2

Moreover, we have another notion:

Definition 10. A complete Riemannian manifold (X, h) is called strongly har- monic, if the heat kernelk(x, y;t) is a function of the distancer(x, y) andt.

Theorem ([Sz]). A simply connected complete Riemannian manifold is strongly harmonic iff it is harmonic.

Remark. A harmonic manifold is Einstein and hence analytic. See [Besse].

Lichnerowicz conjecture. Let (X, h) be a complete harmonic manifold. Then its universal covering must be either a rank one symmetric space or the Euclidean space.

Theorem ([Sz],[B-C-G]). Let (X, h) be a compact harmonic manifold. Then it or its universal covering, if necessary, is rank one symmetric, provided it either has finite fundamental group (Szab´o), or is of negative sectional curvature (B-C-G).

Non-compact version of Lichnerowicz conjecture is, however, still open. Damek- Ricci spaces, i.e., solvable Lie groups with left invariant metric, exhibit harmonic Hadamard manifolds, giving counterexamples of the conjecture. See [B-T-V], for the details.

Theorem 5 can be verified by applying the argument in p.14, [Sz]. Actually, set ϕt(x) =k(x, y, t)dv(y) =k(r, t)dv(y), r=r(x, y)

and take a geodesicγ(s) inX,γ(0) =x,γ⁰(0) =u. Then dϕt(u) =

µ

−∂

∂r logk(r, t)

¶

k(r, t)×cosα dv(y),

whereα=∠(γ⁰(0), v), and vis the tangent vector of the geodesiccjoiningx=γ(0) andy. So,

G(dϕ(u), dϕ(u)) = Z

X

µ

−∂

∂rlogk(r, t)

¶₂

hu, vi²k(r, t)dv(y)

= Z _∞

0

µ

−∂

∂rlogk(r, t)

¶₂

k(r, t)rⁿ⁻¹ω(r)dr× Z

|v|=1

hu, vi²dvSⁿ⁻¹

which does not depend on any choice of uandx.

2.8 Information geometry of homothety constants

We will show a crucial relation between the homothety constants and the so- called C. Shannon’s information entropyI. We summarize the argument in [I-S-Shi]

as

Theorem 6. Let (X, h) be an n-dimensional harmonic Hadamard manifold. Sup- pose that each probability measure ρt = k(x, y;t)dv(y) admits finite Shannon’s entropy It=I(ρt).

Then the homothety constantCt, appeared inϕ^∗_tG=Cth, represents the time gradient ofIt;

Ct= 1 n

d dtIt>0.

Therefore,Itis an increasing function in t, independent of any choice of point x.

Remark. In the euclidean case, a direct calculation yields the value of the entropy as

It= n

2 (logt+ log 4π+ 1) and Ct= 1 2t. (8)

The proof of the theorem follows from the following lemma, since the log heat kernelK(x, y;t) = logk(x, y;t) satisfies

∆xK− |∇xK|²+ ∂

∂tK= 0 (9)

Lemma1([I-S-Shi]).

traceh(ϕ^∗_tG) = Z

X

k(x, y;t)|∇xK(x, y;t)|²dv(y) (10)

= −∂

∂t Z

X

k(x, y;t)K(x, y;t)dv(y).

See, for the proof, [I-S-Shi].

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