getdoc4ea6. 224KB Jun 04 2011 12:04:22 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y}

Vol. 16 (2011), Paper no. 9, pages 271–292. Journal URL

http://www.math.washington.edu/~ejpecp/

A new probability measure-valued stochastic process with

Ferguson-Dirichlet process as reversible measure

∗

Jinghai Shao

†

School of Mathematical Sciences, Beijing Normal University, 100875, Beijing, China Institut für Angewandte Mathematik, Universität Bonn, DE-53115 Bonn, Germany

Abstract

A new diffusion process taking values in the space of all probability measures over [0, 1] is constructed through Dirichlet form theory in this paper. This process is reversible with respect to the Ferguson-Dirichlet process (also called Poisson Dirichlet process), which is the reversible measure of the Fleming-Viot process with parent independent mutation. The intrinsic distance of this process is in the class of Wasserstein distances, so it’s also a kind of Wasserstein diffusion. Moreover, this process satisfies the Log-Sobolev inequality.

Key words:Wasserstein diffusion; Logarithmic Sobolev inequalities; Ferguson-Dirichlet process; Fleming-Viot process.

AMS 2000 Subject Classification:Primary 60J68; Secondary: 60J35; 28A33; 58J65; 47D07. Submitted to EJP on August 26, 2010, final version accepted December 11, 2010.

1

Introduction

This work is motivated by Von Renesse, M-K. and Sturm, K.T.[24]about Wasserstein diffusion on one dimensional space. There they constructed a probability measure-valued stochastic process, which is reversible with respect to (for short, w.r.t.) an “entropy measure". Let P([0, 1])denote the collection of all probability measures on[0, 1]. Then under the entropy measure, almost surely the measureµ∈ P([0, 1])has no absolutely continuous part and no discrete part. The topological support ofµis negligible w.r.t. Lebesgue measure.

In this paper we are interested in the Ferguson-Dirichlet process, which is also called Poisson Dirich-let process. It is a probability measure on the space of all probability measures over a measurable space. The concrete definition will be given in next section. Ferguson-Dirichlet process is an im-portant probability measure on the space of probability measures. It plays imim-portant roles in many research fields such as population genetics and Bayesian statistics. As proved in [7], it is the re-versible measure of the Fleming-Viot processes with parent independent mutation. It was shown in

[7]and[8]that under the Ferguson-Dirichlet process, almost surely the measureµ∈ P([0, 1]) is discrete and has full topological support. This is very different to the “entropy measure". However,

[24, section 3] has pointed out some connection between the “entropy measure" and Ferguson-Dirichlet process. Moreover, Stannat, W.[18]showed that the Fleming-Viot process doesn’t satisfies the Log-Sobolev inequality unless the cardinality of the support ofν0in the Ferguson-Dirichlet

pro-cess is finite (see the definition of Ferguson-Dirichlet propro-cess in (2.4)). Döring, M. and Stannat, W.[3]showed that the Wasserstein diffusion satisfies the Log-Sobolev inequality. So based on the works[24]and[3], it’s of great interest to ask whether there exists a probability measure-valued stochastic process, which is reversible w.r.t. Ferguson-Dirichlet process and satisfies the Log-Sobolev inequality at the same time. If it does, then it provides us a process which not only converges expo-nentially inL2-norm to the Ferguson-Dirichlet process, but also converges exponentially in entropy to the Ferguson-Dirichlet process. To be more precise, due to general results of functional inequali-ties, for the semigroup(P_t)_t≥0 determined by a Dirichlet form, which admits invariant measureπ,

it’s known that the Poincaré inequality is equivalent to the L2-exponential convergence:

kP_tf −π(f)k2_L2_(π)≤Var(f)exp[−λt]

for some positive constantλ, where Var(f) =π(f2)−π(f)2. However, the Log-Sobolev inequality can yield the Poincaré inequality and

Ent(P_tf)≤Ent(f)exp[−σt]

for some positive constantσ, where Ent(f) = π(f logf)−π(f)logπ(f) for positive function f. The well known exponential ergodicity for Markov process means that

kP_t(x,·)−πk_var≤C(x)e−αt for someα >0,

wherek · k_varis the total variation norm. According to Chen[2, Theorem 1.9], for reversible Markov process theL2-exponential convergence is equivalent toπ-a.e. exponential ergodicity. Refer to Chen

the rate of convergence in exponential ergodicity based on the explicit formula of the transition semigroup.

In this paper, we consider the space of all probability measures on the interval [0, 1]. On it, we will construct a new probability measure-valued process, whose reversible measure is the Ferguson-Dirichlet process. We shall show our new process satisfies the Log-Sobolev inequality, so, by the theory of functional inequalities, its associated semigroup will converge in entropy to its equilibrium with exponential rate. Our process is constructed through classical Dirichlet form theory. Moreover, we show that the intrinsic metric of this Dirichlet form is in the class of Wasserstein distances. This means that our new probability measure-valued process is also a kind of Wasserstein diffusion.

The structure of this paper is as follows: in next section we recall some facts about the Fleming-Viot process, and via comparing with the Fleming-Viot process we sketch the idea of our construction of the new process. In the third section we are concerned with constructing the Dirichlet form. In order to prove the pre-defined bilinear form to be closable, we consider the quasi-invariance property of Ferguson-Dirichlet measure (see Theorem 3.4) and establish the integration by parts formula (see Theorem 3.6). Then we constructed a regular Dirichlet form with Ferguson-Dirichlet process to be the reversible measure in Theorem 3.8. Furthermore, we discuss the intrinsic metric of this Dirichlet form (see Theorem 3.11). The last section is devoted to establish the Log-Sobolev inequality for our new probability measure-valued process. This is proved by the method of finite dimensional approximation based on our construction of Dirichlet form and Döring, M. and Stannat, W.’s work

[3].

2

Comparison with Fleming-Viot process

Let’s first introduce the Fleming-Viot process. Let E be a Polish space, i.e. a complete separable metric space. P(E)denotes the set of all probability measures onE, andP(P(E))denotes the set of all probability measures onP(E). The Fleming-Viot process is initially established as the scaling limit of the measure-valued Moran model. The Fleming-Viot operator is defined as

Lφ(µ) =1

2 Z

E

Z

E

µ(dx) δx(dy)−µ(dy)

δ2_φ(µ)

δµ(x)δµ(y)

+

Z

E

µ(dx)Aδφ(µ)

δµ(·)

(x), φ∈Cyl(D(A)),

(2.1)

where δφ(µ)

δµ(x) =

d dt

t=0φ(µ+tδx),δx denotes the Dirac measure at x andAis the generator of a

Feller semigroup onC(E)with domainD(A), which represents the random mutation. Here

Cyl(D(A)) ={φ:φ(µ) =F(〈f1,µ〉, . . . ,〈fn,µ〉),F ∈C∞(Rn),fi∈ D(A),n∈N}, (2.2)

where〈f,µ〉=R_E fdµforf ∈ B(E). WhenAis the parent independent mutation, i.e.

Af(x) = θ

2 Z

E

f(y)−f(x)ν₀(dy), (2.3)

where θ > 0 and ν0 ∈ P(E), Shiga, T. [17] showed that there is a unique reversible measure

Π_θ,ν

partition ofEinto disjoint sets, the mappingµ7→ µ(Λ₁), . . . ,µ(Λ_k)under Π_θ,ν the Ferguson-Dirichlet process or Poisson Dirichlet process. Ferguson[8]and Ethier and Kurtz[5, 7]

showed that

almost everywhereµ∈ P(E)is a purely atomic probability measure.

With the Fleming-Viot operatorL and Poisson Dirichlet processΠ_θ,ν0, one can associate a symmetric Dirichlet form(E,D(E))as the closure of

In the sequel of this paper, we only consider the Fleming-Viot process with parent independent mutation Aas in (2.3). Then the Fleming-Viot operator acting on the cylindrical function u(µ) =

F(〈f1,µ〉, . . . ,〈fn,µ〉)has the representation functionals depending only on its values onP(E), it’s more appropriate to define

It’s easy to check through direct calculation that the Fleming-Viot operatorL can also be represented by

Lu(µ) = 1

2 Z

E

∇2_xu(µ)µ(dx) +θ

2 Z

E

µ(dx)

Z

E

∇_yu(µ)− ∇_xu(µ)ν0(dy)

foru∈Cyl. Moreover, the associated Dirichlet form has the form

E(u,v) =−

Z

P(E)

uLvΠ_θ,ν

0(dµ) =

Z

P(E)

〈∇_·u(µ),∇_·v(µ)〉_µΠ_θ,ν

0(dµ),

where〈f,g〉_µ=R

E f(x)g(x)µ(dx)for f, g∈C(E).

Recently there are many works looking onP(E)as an infinite dimensional Riemannian manifold. For instance, Jordan, R. et al. [10] defined a Riemannian manifold structure on the space of all probability measures onRd and constructed the solution of Fokker-Planck equation in Rd through establishing gradient flow of relative entropy functional. Otto, F. and Villani, C. [13] used it as a guideline to find the interrelation between the transportation cost inequality and the Log-Sobolev inequality. This kind of viewpoint also stimulate them to find the so called “HWI" inequality, which includes three important quantities: “H" entropy, “W" Wasserstein distance, and “I" Fisher informa-tion into one inequality. Sturm, K.T.[20, 21] and Lott, J. and Villani, C. [11] further established the concept of lower bound of Ricci curvature on metric measure space E. Particularly, when E is a Riemannian manifold, their lower bound of Ricci curvature coincides with the geometric lower bound of Ricci curvature. Taking this viewpoint, as done in Schied [15], one can look on L2(µ)

as the tangent space at each pointµ ∈ P(E), i.e. T_µP(E) = L2(µ) and for any f, g ∈ T_µP(E), define its inner product by〈f,g〉_µ. Then the function x 7→ ∇_xu(µ) for some good functionaluon

P(E)is a tangent vector atµand the Carré du champs operatorΓ(u,v) =〈∇·u(µ),∇·v(µ)〉µ asso-ciated to the Dirichlet form(E,D(E))is just the inner product of tangent vectors. Define a mapping Sf :P(E)→ P(E)for f ∈ Bb(E)by

d Sfµ=

ef

〈ef_,µ〉dµ. (2.7)

This mapping is called the exponential map of the “Riemannian manifold" P(E) since the map t7→S_{t f}µfromR_{toP(E)generates a continuous curve inP(E)and}

d dt

_t=0u S_{t f}µ=〈∇_·u(µ),f〉_µ. (2.8)

Indeed, foru∈Cyl, for each f ∈T_µP(E) =L2(µ)we have

d dt

_t=0u S_{t f}µ=〈∇·u(µ),f〉µ=

n

X

i=1

∂_iF(〈~f,µ〉) 〈f_if,µ〉 − 〈f_i,µ〉〈f,µ〉.

In this paper, we mainly consider a special case of E, that is, E = [0, 1]. Compared with the Fleming-Viot process onP([0, 1]), we define the tangent space atµas a subspace ofL2((g_µ)_∗Le b), whereg_µ denotes the cumulative distribution function ofµ, Le bdenotes the Lebesgue measure and

(g_µ)_∗Le b:=Le b◦g_µ−1denotes the push forward measure ofLe bunder the mapg_µ:[0, 1]→[0, 1].

We will introduce a new exponential mapSef :P([0, 1])→ P([0, 1])forf ∈TµP([0, 1]). Precisely, letG₀denote the space of all right continuous nondecreasing mapsg:[0, 1]→[0, 1]withg(0) =0, g(1) =1. A C2-isomorphism h∈ G₀ means thath:[0, 1]→[0, 1] is increasing homeomorphism such thathandh−1 are bounded inC2([0, 1]). For eachC2-isomorphismh∈ G₀, define a mapping ˜

τ_h : P([0, 1]) → P([0, 1]) as: for each µ∈ P([0, 1]) with cumulative distribution function g_µ, ˜

τh(µ) is defined to be an absolutely continuous probability measure w.r.t. µwith density function

¯_hg

µ, namely,

d˜τh(µ)

dµ (x) =¯hgµ(x), (2.9)

where

¯_h

g(x) =

Z 1

0

h′(r g(x+) + (1−r)g(x−))dr, g(x+) = lim

y→x+g(y), g(x

−_{) = lim}

y→x−g(y).

When g is a continuous function, ¯hg(x) = h′(g(x)). When g is of bounded variation, dh(g(x)) =

¯_h

g(x)dg(x)(refer to[1, Section 3.10]for chain rule of bounded variation functions).

Given a function φ ∈ C∞([0, 1],R) satisfying φ(0) = φ(1) = 0, let t 7→ X(t,x) be the unique solution of the following ODE:

dX_t

dt =φ(Xt), X0=x. (2.10) Put etφ(x) = X(t,x), then etφ(x) = eφ(t,x) = etφ(1,x) and e(t+s)φ = etφ◦esφ for t,s ∈Rand x ∈[0, 1]. The assumptionφ(0) =φ(1) =0 yields thate_tφ(0) =0,e_tφ(1) =1 for allt∈R_{. Hence,}

e_tφ is aC2-isomorphism inG₀. Set

H₀=φ∈C∞([0, 1],R); φ(0) =φ(1) =0 . (2.11)

Now we defineeS_φ:P([0, 1])→ P([0, 1])by

dSe_φ(µ) =d˜τ_eφ(µ), whereφ∈ H₀, (2.12)

then it holdseS(t+s)φ=Setφ◦Sesφfort,s∈R. For a functionu:P([0, 1])→R, define itsdirectional derivativealongφ∈ H₀by

D_φu(µ) =lim

t→0

1 t

u(τ˜etφ(µ))−u(µ)

(2.13)

provided the limit exists. The tangent space of P([0, 1]) at a point µ is now defined to be the closure of H₀ in the norm of L2((g_µ)_∗Le b), denoted by T_µP. We say that a function u has a gradient at µ if there exists a function x 7→ ∇_xu(µ) such that D_φu(µ) = 〈∇·u(µ),φ〉TµP :=

R

[0,1]∇gµ(x)u(µ)φ(gµ(x))dx for eachφ∈ H0. Define a symmetric bilinear form by

E(u,v) =

Z

P([0,1])

〈∇_·u(µ),∇_·v(µ)〉_TµP Π_θ,ν

foru, v ∈Cyl, where Cyl is defined in (3.4) below. We shall prove that(E, Cyl)is closable and by classical Dirichlet form theory, we can obtain a probability measure-valued process. This process is reversible w.r.t. the Ferguson-Dirichlet processΠ_θ,ν

0. Moreover, we will show this process satisfies

the Log-Sobolev inequality.

3

Construction of the Dirichlet form

3.1

Quasi-invariance property

In order to prove the symmetric bilinear formE defined in (2.14) is closable, we need to consider first the quasi-invariance of Π_θ,ν

0 under the map eSf for f ∈ H0. From now on for simplicity of

notation, we setP =P([0, 1])andΠ_θ = Π_{θ,Le b}(that is, the Ferguson-Dirichlet measureΠ_θ,ν

0 with

ν0=Lebesgue measure). It’s known thatP is compact and complete under the weak topology, and

its weak topology coincides with the topology determined by Lp-Wasserstein distancedw,p, p ≥1,

where

dw,p(µ,ν) = inf

π∈C(µ,ν) Z

[0,1]2

|x−y|pπ(dx, dy)1/p.

Here and in the sequel,C(µ,ν)stands for the collection of all probability measures on[0, 1]×[0, 1]

with marginalsµandν respectively. Set d_w =d_w,2. Refer to[23]for these fundamental results on probability measure space.

Recall thatG₀denotes the space of all right continuous nondecreasing mapsg:[0, 1]→[0, 1]. Each g∈ G₀can be extended to the full interval by setting g(1) =1.G₀ is equipped with L2-distance

kg1−g2kL2=

Z 1

0

|g1(t)−g2(t)|2dt

1/2

.

Definition 3.1 Forθ >0, there exists a unique probability measure Qθ₀ onG₀, called Dirichlet process, with the property that for each n∈N, and each family0=t₀<t₁<. . .<t_n<t_n+1=1,

Qθ₀(gt1∈dx1, . . . ,gtn∈dxn) =

Γ(θ)

Πn_i=1Γ(θ(t_i+1−t_i)) n

Y

i=1

(xi+1−xi)θ(ti+1−ti)−1dx1· · ·dxn,

with xn+1=1.

The measure Qθ₀ is sometimes called entropy measure, but as in [24], in this paper we use the entropy measure to only denote the push forward measure ofQθ₀ under the map g7→g∗Le b. Define the map ζ : G₀ → P, g 7→ dg. It’s easy to see that (ζ)∗Qθ₀ := Qθ₀ ◦ζ−1 = Πθ. Its inverse ζ−1 assigns to each probability measure its distribution function. In the following we will study the quasi-invariance property ofΠ_θ,ν

0 throughQ

θ

0 andζ. Von Renesse, M-K. and Sturm, K.T.[24]has

Theorem 3.2 ([24]Theorem 4.3 ) Each C2-isomorphism h∈ G₀ induces a bijection mapτh:G0→

G₀, g7→h◦g, which leaves Qθ₀ quasi-invariant:

dQθ₀(h◦g) =Y_hθ_,0(g)dQθ₀(g),

and Y_hθ_,0is bounded from above and below. Here

Y_hθ_,0(g) =X_hθ(g)Y_h,0(g), (3.1)

where

Yh,0(g) =

1 p

h′(0)h′(1)

Y

a∈Jg

p

h′_(g(a−_))−h′_(g(a+₎₎

δ(h◦g)

δg (a)

,

X_h(g) =expθ

Z 1

0

logh′(g(s))ds,

Jg=x∈[0, 1]; g(x+)6=g(x−) ,

δ(h◦g)

δg (a) =

h(g(a+))−h(g(a−))

g(a+_)−g(a−₎ .

Due to the compactness of the interval[0, 1]andP, several well known topologies onG₀ andP

coincide. More precisely, for each sequence (g_n) ⊂ G₀, and each g ∈ G₀, the following types of convergence are equivalent:

• g_n(t)→g(t)for eacht ∈[0, 1]in which gis continuous;

• gn→gin Lp([0, 1])for eachp≥1;

• µ_gn→µ_g weakly;

• µgn→µt in theL

p_{-Wasserstein distance for each p≥1.}

Refer to[24]for a sketch of the idea of the argument.

Lemma 3.3 For each C2-isomorphism h∈ G₀,τ˜h:P → P defined by (2.9) is continuous.

Proof. Let µ_n ∈ P, n ≥ 1 and µ_n converges to µ as n → ∞. g_µn and g_µ denote the corre-sponding cumulative distribution functions of µn and µ. Then gµn converges to gµ in L

p_{([0, 1])}

for each p ≥ 1. Set νn = τ˜h(µn), ν = τ˜h(µ). Then the cumulative function of νn and ν

are h◦ g_µn and h◦ g_µ respectively. We denote by d_w,1 the L1-Wasserstein distance, that is,

d_w,1(µ,ν) = inf_{π∈C(µ,ν)}n R

[0,1]2|x − y|π(dx, dy)

o

. Then according to [22, Theorem 2.18]about optimal transport onR, we have

d_w,1(νn,ν) =

Z 1

0

|g_ν−1

n(t)−g

−1

ν (t)|dt= Z 1

0

|g_νn(t)−g_ν(t)|dt

=

Z 1

0

|h◦g_µ

n(t)−h◦gµ(t)|dt≤_smax_∈[0,1]|h

′_(s)| Z 1

0

|g_µ

n(t)−gµ(t)|dt,

which yields that asµnweakly converges toµ,νnconverges toν as well. This is the desired result.

Theorem 3.4 (Quasi-invariance) Letν0be the Lebesgue measure on[0, 1]. For each C2-isomorphism

h∈ G₀, the Ferguson-Dirichlet measureΠ_θ,ν

0is quasi-invariant under the transformationτ˜h, and

dΠ_θ,ν

0 τ˜h(µ)

₌

Y_hθ_,0(g_µ)dΠ_θ,ν

0(µ), (3.2)

where Y_hθ_,0(g_µ)is defined as (3.1), and g_µ denotes the cumulative distribution function ofµ.

Proof. For any bounded measurable functionuonP, it can induce a bounded measurable function ¯

uonG₀by

¯

u(g):=u(ζ(g)).

Note that forC2-isomorphismh∈ G₀,τ−_h1=τh−1 (see Theorem 3.2 for the definition), and

ζ◦τ_h◦ζ−1=τ˜_h. (3.3)

So ˜τh is a bijection map and ˜τ−_h1 = τ˜h−1. To see this, noting that g_µ = ζ−1(µ), we have for any

f ∈ B([0, 1]), for anyµ∈ P, Z 1

0

f(x)d ζ◦τh(gµ) = Z 1

0

f(x)dτh(gµ) = Z 1

0

¯_h(g

µ(x))dgµ(x) = Z 1

0

f(x)d˜τh(µ).

According to Theorem 3.2, we obtain Z

P

u(µ)dΠ_θ,ν

0(τ˜h(µ)) =

Z

P

u(τ˜−_h1(µ))dΠ_θ,ν

0(µ) =

Z

G0

¯

u(τ−_h1(g))dQθ₀(g)

=

Z

G0

¯

u(g)dQθ₀(τ_h(g)) =

Z

G0

¯

u(g)Y_hθ_,0(g)dQθ₀(g)

=

Z

P

u(µ)Y_hθ_,0(g_µ)dΠ_θ,ν0(µ),

which concludes the proof.

3.2

Integration by parts formula

In section 2, we have defined the mapetφ:[0, 1]→[0, 1]and the derivative of a functionu:P →R by

D_φu(µ) =lim

t→0

1 t

u(τ˜etφ)(µ)−u(µ)

alongφ∈ H₀, provided the limit exists. Let Cyl be the set of all functions onP in the form

u(µ) =F(〈f₁,µ〉, . . . ,〈f_n,µ〉), (3.4)

where F∈C1(Rn_{), fi ∈C}1_{([0, 1])fori=1, . . . ,nandn∈}N_{. Let Cyl(G0)be the set of all functions}

w:G₀→Rin the form

w(g) =F Z f1dg, . . . ,

Z

fndg

where F ∈ C1(Rn), f_i ∈ C1([0, 1]) and n∈N. For a function w : G₀ → R, define its directional

Proof. (i) By virtue of integration by parts formula on[0, 1], we have

Z 1

Invoking (3.3),

Before stating the integration by parts formula, we recall a result on the derivative of g7→Y_hθ_,0(g)

appeared in the quasi-invariance formula. According to[24, Lemma 5.7], forφ∈C2([0, 1]) with

Proof. We shall only prove (i), and (ii) can be proved by the similar method used in the previous lemma. By the quasi-invariance ofQθ₀, one has

which concludes (i).

3.3

Tangent space and Dirichlet form

The goal of this subsection is to obtain our stochastic process through establishing a Dirichlet form on P. The key point is how to specify a suitable pre-Hilbert norm k · k_µ on T_µP such that the direction derivative D_φu(µ) of a nice functionuon P could determine a bounded linear function

φ7→D_φu(µ)on H₀. Then Riesz representation theorem yields that there exists a unique element Du(µ)∈T_µP with

D_φu(µ) =〈Du(µ),φ〉_TµP, ∀φ∈T_µP. HereT_µP is the completion ofH₀w.r.t. the pre-Hilbert normk · k_µ.

Naturally, the nice functions are cylindrical functions. Letu∈Cyl in the form (3.4). Forφ∈ H₀,

D_φu(µ) = n

X

i=1

∂_iF(〈~f,µ〉)·

Z 1

0

f_i(x)φ¯_gµ(x)dg_µ(x)

=−

n

X

i=1

∂iF(〈~f,µ〉)·

Z 1

0

f_i′(x)φ(g_µ(x))dx

=−

n

X

i=1

∂iF(〈~f,µ〉)·

Z 1

0

f_i′(g−_µ1(x))φ(x)d(g_µ)_∗Le b

=

Z 1

0

−

n

X

i=1

∂_iF(〈~f,µ〉)· f_i′(g_µ−1(x))φ(x)d(g_µ)∗Le b,

(3.14)

where 〈~f,µ〉 = (〈f₁,µ〉, . . . ,〈f_n,µ〉), R₀1f(x)d(g_µ)∗Le b = R1

0 f(gµ(x))dx for f ∈ Bb([0, 1]).

Ac-cording to this expression, we choose the pre-Hilbert norm to be the norm of L2((g_µ)∗Le b) at

µ ∈ P. Then φ 7→ D_φu(µ) is a bounded linear function on H₀. So it can be extended to be a bounded linear functional on the completion ofH₀w.r.t. the norm of L2((g_µ)_∗Le b). Therefore, the tangent space T_µP is defined to be the completion ofH₀w.r.t. the norm of L2((g_µ)∗Le b).

Definition 3.7 The gradient of function u:P →Ris said to exist atµ∈ P, if there exists a function

ψ:[0, 1]→R_{such that for anyφ}∈ H₀,

D_φu(µ) =

Z 1

0

ψ(x)φ(x)d(g_µ)_∗Le b=

Z 1

0

ψ(g_µ(x))φ(g_µ(x))dx.

Then the gradientψ(·)atµis denoted by∇_·u(µ).

Similarly, for a function u:G₀→R_{, if there exists a functionψ:}[0, 1]→R_{such that}

D_φu(g) =

Z 1

0

φ(x)ψ(x)dg_∗Le b, ∀φ∈ H₀,

Due to the calculation (3.14), we know that for a cylindrical functionuin the form (3.4), its gradient

Next, we define a symmetric bilinear form by

E(u,v) = reversible measureΠ_θ =Ferguson-Dirichlet measure.

By the integration by parts formula (3.12), previous equality

=

Z

P

n

X

j=1

D∗_g′

j◦gµ−1

∂_jG(〈~g,µ〉)u(µ) Π_θ(dµ)

=−

Z

P f

Lv(µ)·u(µ) Π_θ(dµ).

This proves that(Lf, Cyl₀)is a symmetric operator, and(E, Cyl₀)is closable, its generator coincides with the Friedrichs extension ofLf.

b) Now let’s prove that Cyl₀is dense in Cyl. For simplicity, assume thatuis of formu(µ) =F(〈f,µ〉), F ∈ C1(R_{), and f ∈C}1_{([0, 1]), that is, for simplicity, consider n= 1. Let Fǫ ∈C}∞₍R_{), for ǫ >0}

be smooth approximation of F withkF−F_ǫk_∞+kF′−F_ǫ′k_∞ → 0, as ǫ →0. Let f_ǫ ∈ C∞ with f_ǫ′(0) = f_ǫ′(1) = 0 be smooth approximation of f withkf − f_ǫk_∞ →0 and f_ǫ′(t) → f′(t) for all t ∈ (0, 1) as ǫ →0. Moreover, assume that sup_ǫkf_ǫ′k∞ < ∞. Define uǫ(µ) = Fǫ(〈fǫ,µ〉). Then u_ǫ∈Cyl₀ andu_ǫ→uin L2(Π_θ)asǫ→0 by dominated convergence theorem. Since

sup ǫ

sup µ∈P

Z 1

0

F_ǫ′(〈f_ǫ,µ〉)2f_ǫ′(x)2dx ≤C, Z 1

0

F_ǫ′(〈f_ǫ,µ〉)2f_ǫ′(x)2dx→

Z 1

0

F′(〈f,µ〉)2f′(x)2dx,

we obtain

E(u_ǫ,u_ǫ) =

Z

P Z 1

0

F_ǫ′(〈f_ǫ,µ〉)2f_ǫ′(x)2dxΠ_θ(dµ)

−→

Z

P Z 1

0

F′(〈f,µ〉)2f′(x)2dxΠ_θ(dµ)

asǫ→0 by dominated convergence theorem inL2(Π_θ). This implies that(u_ǫ)_ǫconstitutes a Cauchy sequence relative to the norm

kvk2_E,1:=kvk2_L2+E(v,v).

In fact, since(u_ǫ)_ǫ is uniformly bounded w.r.t. k · k_E,1, by weak compactness there exists a subse-quence converging weakly in(D(E),k · k_E,1). Since the associated norm converges,

ku_ǫ−uk2_E,1=ku_ǫk2_E,1−2〈u_ǫ,u〉_L2−2E(u_ǫ,u) +kuk2

E,1−→0.

Moreover, asu_ǫ →u in L2(Π_θ), the limit is unique. Hence the entire sequence converges tou ∈

(D(E),k · kE,1). Particularly, E(uǫ,uǫ)→ E(u,u). This proves Cyl0 is dense in Cyl. Combining with

(E, Cyl₀) is closable proved in a), we get(E, Cyl) is closable and that the closures of Cyl₀ and Cyl coincide.

According to the theory of Dirichlet form (refer to[12]), any regular Dirichlet form possessing local property, i.e.

E(u,v) =0, for allu,v∈ D(E)with supp[u]∩supp[v]=;,

where supp[u]denotes the support ofu, admits a diffusion process, that is, a strong Markov process with continuous sample paths with probability 1. It’s clear that Dirichlet form(E,D(E))possesses the local property, so it admits a diffusion process. Note that here sample paths are continuous in the weak topology ofP. Is this process still continuous in the topology determined by the total variation norm? We tend to believe the negative answer, but we can’t prove it now.

Remark 3.9 In the argument of Theorem 3.8, we know that operatorLfis symmetric w.r.t.Π_θ, i.e. Z

P

u(µ) Lfv(µ)dΠ_θ(µ) =

Z

P

v(µ) Lfu(µ)dΠ_θ(µ), u,v∈ D(Lf).

So the diffusion process associated(E,D(E))is reversible w.r.t.Π_θ.

Our next goal is to give a description of the intrinsic metric associated to our Dirichlet form. In[24], they constructed a Dirichlet form onP whose intrinsic metric is theL2-Wasserstein distance onP. This result is obtained based on Rademacher theorem ([24, Theorem 7.9, Theorem 7.11]) on the spaceG₀. There they considered different kinds of cylindrical functions. Using their idea, we can establish Rademacher theorem in our setting onG₀.

Proposition 3.10 It holds that

kg1−g0kL2=supu(g₁)−u(g₀); u∈Cyl(G₀),k∇u(g)k_T

gG0≤1, Q

θ

0−a.e.onG0 (3.18)

for all g₀,g₁∈ G₀, wherekg₁−g₀k2

L2=

R1

0 |g1(x)−g0(x)| 2_dx.

Sketch of the proof. Note Cyl(G₀) can be proved to be a core of a Dirichlet form on G₀ in the same method to construct(E,D(E)) onP, so (3.18) equals to the intrinsic metric of this Dirichlet form onG₀. Almost following the same argument of[24]Theorems 7.9 and 7.11, we can establish Rademacher theorem, then this proposition corresponds to[24, Corollary 7.14].

Theorem 3.11 Let dess denote the intrinsic metric associated with the Dirichlet form(E,D(E))onP,

that is,

d_ess(µ1,µ0) =sup

u(µ1)−u(µ0); u∈Cyl,k∇u(µ)kTµP ≤1, Πθ-a.e. onP

for allµ0,µ1∈ P. Then dess(µ1,µ0) =kgµ1−gµ0kL2, and

dw,1(µ1,µ0)≤dess(µ1,µ0)≤

p

dw,1(µ1,µ0), (3.19)

where d_w,1(µ1,µ0)denotes the L1-Wasserstein distance onP.

Proof. Letu∈Cyl such thatk∇u(µ)k_TµP ≤1,Π_θ-a.e. onP. Set ¯u(g) =u(ζ(g)), then ¯u∈Cyl(G₀). By (3.3), we get

which yields∇¯u(g) =∇u(ζ(g))andk∇u¯(g)k_T

gG0=k∇u(ζ(g))kTζ(g)P. It follows that

k∇u¯(g)k_T

gG0≤1, Q

θ

0−a.e. inP.

As ζ is reversible, for each ¯u ∈ Cyl(G₀) with k∇u¯(g)k_T

gG0 ≤ 1, Q

θ

0-a.e. in G0, define u(µ) =

¯

u(ζ−1_{(µ)). After similar deduction as above, we obtain u ∈ Cyl and k∇u(µ)k}

TµP ≤ 1, Πθ-a.e.

onP. Due to Proposition 3.10, forµ0,µ1∈ P,

kg_µ

1−gµ0kL2

=sup{u¯(g_µ1)−u¯(g_µ0); ¯u∈Cyl(G₀),k∇u¯(g)k_T

gG0≤1, Q

θ

0-a.e. onP }

=sup{u(µ1)−u(µ0); u∈Cyl,k∇u(µ)kT_µP ≤1, Πθ-a.e. onP }.

Hence,

d_ess(µ₁,µ₀) =kg_µ1−g_µ0k_L2. (3.20)

It holds

kg_µ

1−gµ0kL1=

Z 1

0

|g_µ

1(x)−gµ0(x)|dx =

Z 1

0

|g_µ−1

1(x)−g

−1

µ0(x)|dx=dw,1(µ1,µ0),

where the second equality is easily verified by seeing the area between graphs and the last inequality follows from[22, Theorem 2.18]. By Hölder’s inequality,

d_w,1(µ1,µ0) =kgµ1−gµ0kL1≤ kgµ1−gµ0kL2=dess(µ1,µ0).

On the other hand, as|g_µ1(x)−g_µ0(x)| ≤1 at each x ∈[0, 1],

kg_µ

1−gµ0kL2≤

Z 1

0

|g_µ

1(x)−gµ0(x)|dx

1/2

= kg_µ

1−gµ0kL1

1/2

.

Therefore,d_ess(µ1,µ0)≤

p

d_w,1(µ1,µ0), and we get the desired results.

4

Log-Sobolev inequalities for the process

In this section, we shall establish the Log-Sobolev inequality for(E,D(E)). Döring, M. and Stannat, W.[3]established the Log-Sobolev inequality for the Wasserstein diffusion constructed by[24]on

P. There they have done lots of calculation on finite dimensional approximation, especially they established the Log-Sobolev inequality for the Dirichlet distribution on finite simplex. We shall take advantage of their work and to establish the Log-Sobolev inequality for(E,D(E)) on P again by finite dimensional approximation. But to deal with our Dirichlet form, we should take different approximation sequence.

First let’s recall the definition of Dirichlet processQθ₀ on G₀forθ >0. Qθ₀ is the unique probability measure onG₀such that for anyn∈N_{, 0}=t0<t1<. . .<tn−1<tn=1,

Qθ₀(gt1∈dx1, . . . ,gtn−1∈dxn−1)

where forq= (q₁, . . . ,q_n)∈Rn₊

We now include a result proved in[3, Proposition 2.3]as follows:

Proposition 4.1 Letq∈Rn

satisfies the Log-Sobolev inequality with constant less than4c1/q∗, that is,

Z

Theorem 4.2 (Log-Sobolev inequality) There exists a universal constant C>0such that(E,D(E))

onP satisfies the Log-Sobolev inequality with constant less than C/θ, i.e.

Z

Proof.We will use the idea of[3]to establish the Log-Sobolev inequality through finite dimensional approximation. But compared with[3], we use different type of approximation sequence. In order to use the calculation results of[3], we first make our calculations onG₀, then obtain the desired results under the help of mapζ.

Hence we choose the approximation sequence by

Note that this is different to Döring, M. and Stannat, W. [3]. There they used the approximation induced by

When f, gare both continuous or f is continuous, gis of bounded variation, one has

lim

Formula (4.3) can be written more explicitly and take projection onto finite dimensional space. Put

Remark 4.3 It’s well known that inequality (4.2) implies that the semigroup (etLf)_t≥0 is hyper-contractive, i.e. ketLfk_2→4 ≤ 1 for some t > 0, where k · k_2→4 denotes the operator norm from L2(Π_θ)to L4(Π_θ). The contraction properties of Markov semigroups including hypercontractivity, supercontractivity and ultracontractivity are closely related to various functional inequalities. To be more precise, given a complete, connected, noncompact Riemannian manifoldM, let(Pt)t≥0be the

semigroup of a diffusion process generated by L= ∆ +Z for someC1-vector fieldZsatisfying

Ric(X,X)− 〈∇_XZ,Z〉 ≥ −KZ|X|, X ∈T M

for some K_Z ∈ R, where Ric denotes the Ricci curvature of M. Assume P_t admits an invariant measure µ which is positive and Radon. Then, according to [25, Theorem 5.7.1], if there exist C, t>0, andq>p>1 such thatkP_tk_p→q≤C, then

µ(f2logf2)≤βKZ,p,qµ(|∇f|

2_{) +} pq

q−plogC, ∀f ∈C

∞

0 (M), µ(f 2_{) =1,}

for some positive constantβ_KZ,p,q. Conversely, if there existC1,C2>0 such that

µ(f2logf2)≤C1µ(|∇f|2) +C2, f ∈C0∞(M), µ(f 2_{) =1,}

then

kP_tk_p→q≤exph4C₂ 1 p−

1 q

i

for t >0 and q > p > 1 satisfying exp(4t/C1)≥ (q−1)/(p−1). Refer to Wang’s book [25] for

more properties associated with Log-Sobolev inequalities and general discussion about functional inequalities including F-Sobolev inequalities, Harnack inequalities, and super and weak Poincaré inequalities.

Remark 4.4 In [19], the same Dirichlet form and Log-Sobolev inequality as this work have been obtained, but the generator and the intrinsic distance of the Dirichlet form haven’t been given there. Moreover, the explicit form of the intrinsic metric enable us to establish the transportation cost inequalities for Wasserstein diffusions, which will be done in our forthcoming work[16].

Acknowledgement: The author would like to express his thankfulness to the referees for pointing out to him Stannat, W.’s work[19].

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