数理解析研究所講究録
第 1837 巻 2013 年 87-101

87

ON NON-LINEAR SPECTRAL GAP FOR SYMMETRICMARKOV CHAINS WITH COARSE RICCICURVATURESEIKI KOKUBO AND KAZUHIRO KUWAE

ABSTRACT. In this note, we report the summary of [12] for the casethat the target space is a complete separable CAT(0)-space. We
prove an upper estimate of spectral radius for (non-linear) transition operator $P$ over-maps in the framework of symmetricMarkov chains on a Polish space with positive lower bound of $n-$
step coarse Ricci curvatures without its proof. As consequences,strong -Liouville property for -harmonic maps, a global Poincar\’einequality (spectral gaps) for energy functional over -maps (orfunctions), and spectral bounds of -generator of Markov chains$L^{p}$$L^{p}$$P$$L^{2}$$L^{2}$

are presented.

1. COARSE RICCI CURVATURE

Throughout this note, let $(E, d)$ be a Polish space with complete distance and. Denote by, the family of probability
measures on $(E, d)$ with finite p-th moment. We consider a conservative
Markov chain
with state space $(E, d)$ .
Then the transition kernel $P(x, dy)$ (or $P_{x}(dy)$ in short) of X defined
by $P(x, dy)$ $:=P_{x}(X_{1}\in dy),$ $x\in E$ satisfies(Pl) for each $x\in E,$ $\mathcal{B}(E)\ni A\mapsto P(x, A)$ is a probability measureon.
(P2) for each$E\ni x\mapsto P(x, A)$ is-measurable.
Conversely, for $P(x, dy)$ satisfying (Pl) and (P2), we can constructa conservative Markov chain X such that $P(x, dy)=P_{x}(X_{1}\in dy)$ ,$x\in E$ . We set $Pf(x)$ $:= \int_{X}f(y)P(x, dy)=E_{x}[f(X_{1})]$ for any nonnegative or bounded-measurable function on . For, if
we set $P^{n}f(x)$ $:=P(P^{n-1}f)(x)$ inductively, then $P^{n}f(x)=E_{x}[f(X_{n})]$$d$$\mathbb{N}_{0}$$:=\mathbb{N}\cup\{0\}$$\mathcal{P}^{p}(E)$$X=(\Omega, X_{k}, \theta_{k}, \mathcal{F}_{k}, \mathcal{F}_{\infty}, P_{x})_{x\in E}$$(E, \mathcal{B}(E))$$A\in \mathcal{B}(E),$$\mathcal{B}(E)$$f$$\mathcal{B}(E)$$E$$n\in \mathbb{N}$

Date: December 22, 2012.
2000 Mathematics Subject Classification. Primary $53C20$ ; Secondary $53C21,$$53C23.$

Key words and phrases. CAT(0)-space, barycenter, Jensen’s inequality, optimalmass transportation, Wasserstein distance, symmetric Markov chain, coarse Ricci
curvature, -step coarse Ricci curvature, -harmonic map, spectral radius, spectral
$P$

$n$

gap, eigenvalue, strong -Liouville property.
The second author is partially supported by a Grant-in-Aid for Scientific Research No. 22340036 from the Japan Society for the Promotion of Science.
$L^{p}$

88
and $P^{n}(x, A)$ $:=(P^{n}1_{A})(x)=P_{x}(X_{n}\in A)$ . For any non-negativeby, we define a measureandmeasure on.
$:= \int_{E}P^{n}(x, A)v(dx)=P_{\nu}(X_{n}\in A),$$vP^{n}(A)$$x\in E.$
Note that
We further assume the following condition to X:.
(P3) for each $x\in E,$, a Markov
For the given Markov chain X as above and a fixed
chainwith state space $(E, d)$ defined$P^{n}(x,dy)$
is called an -step Markov chain.
by the transition kernel
does so.
Note that if X satisfies (P3), then$L^{1}-Wasserstein/Kantorovich$
For
, the
-Rubinstein distanceis defined by$:=\langle\nu,$$\nu P^{n}$$n\in \mathbb{N}$$(E, \mathcal{B}(E))$$v$$A\in \mathcal{B}(E)$$P^{n}1_{A}\rangle$$\delta_{x}P^{n}=P_{x}^{n},$$P_{x}\in \mathcal{P}^{1}(E)$$n\in \mathbb{N}$$X^{n}=(\Omega, X_{k}^{n}, \theta_{k}^{n}, \Psi_{k}, y_{\infty}^{n}, P_{x}^{n})_{x\in E}$$n$$X^{n}$$\mu,$$v\in \mathcal{P}^{1}(E)$$d_{W_{1}}(\mu, \nu)$$d_{W_{1}}( \mu, \nu) :=\inf\{\int_{ExE}d(x, y)\pi(dxdy) \pi\in\Pi(\mu, \nu)\},$$:=\{\pi\in \mathcal{P}(E\cross E)|\pi(A\cross E)=\mu(A),$ $\pi(E\cross B)=$
where$\nu(B)$ for any $A,$$\Pi(\mu, \nu)$$B\in \mathcal{B}(E)\}.$

Definition 1.1 (Coarse Ricci Curvature, [22]). For a pair of distinct
points $y\in E$ , the coarse Ricci curvatureof X along $(xy)$ isdefined to be$\kappa(x, y)$$x,$$\kappa(x, y)$$:=1- \frac{d_{W_{1}}(P_{x},P_{y})}{d(x,y)}(>-\infty)$,$(x, y)\in E\cross E\backslash$

diag

and
is said to be
the lower bound of the coarse Ricci curvature. The -step coarse Ricci
curvatureof X along $(xy)$ is defined to be$\kappa$$:= \inf\{\kappa(x, y)|(x, y)\in E\cross E\backslash d\dot{\ovalbox{\tt\small REJECT}}ag\}\in[-\infty, 1]$$n$$\kappa_{n}(x, y)$$\kappa_{n}(x, y)$$:=1- \frac{d_{W_{1}}(P_{x}^{n},P_{y}^{n})}{d(x,y)}(\geq-\infty)$,$(x, y)\in E\cross E\backslash$diag

and $\kappa_{n}:=\inf\{\kappa_{n}(x, y)|(x, y)\in E\cross E\backslash diag\}\in[-\infty, 1]$ is said to be
the lower bound of the -step coarse Ricci curvature.
$n$Remark 1.2. We denote the family of Lipschitz functions on $E$ by
Lip $(E)$ .
(1) If, then $P^{n}f\in$ Lip $(E)$ for any $f\in$ Lip $(E)$ and Lip $(P^{n}f)\leq$$(1-\kappa)^{n}Lip(f)$ by [22, Proposition 20], which implies that (P3)
provided (P3) holds for X and, in parholds for all$\kappa_{n}(x,y)>-\infty$$x\neqy$ticular,for allandunder(2) The -step coarse Ricci curvature
nothingisbut thecoarse Ricci curvature forand $\kappa_{1}(x, y)=\kappa(x, y)$ for $(x_{\}}y)\in$
diag. In general, we have$\kappa\in \mathbb{R}$$X^{n}$$\kappa\in \mathbb{R}$$n\in \mathbb{N}$$\kappa\in \mathbb{R}.$$\kappa_{n}(x, y)$$n$$X^{n}$$E\cross E\backslash$$(1-\kappa_{k+\ell})\leq(1-\kappa_{k})(1-\kappa_{\ell}) , k, \ell\in \mathbb{N},$which implies
In particular,$\lim_{\ellarrow\infty}(1-\kappa_{\ell})^{1/\ell}=\inf_{n\in N}(1-\kappa_{n})^{1/n}\in[0, +\infty].$$\kappa\in \mathbb{R}$implies$\kappa_{n}\in \mathbb{R}$for all$n\in \mathbb{N}.$89
The following example due to [4] shows that the lower bound forthe coarse Ricci curvature does $not$ necessarily mean the same boundfor the -step coarse Ricci curvature.
$0$

$n$

Example 1.3 (Simple Random Walk on Cycle Graph, see [4]). Let$G=(V, E)$ be a cycle graph of size $N$ , that is,is an unweighted$E:=\{x_{i}x_{i+1}\}_{i=1}^{N}$byand edges
finite graph with vertices$x\inV$$d_{x}(G)$$x_{N+i}=x_{i}(i\in\mathbb{N})$is the numberfor. The degree
regardingof edges starting from is given by $d_{x}(G)=2$ for this cycle graph $G.$$N.$
The weight
for $xy\in E$ is given by $w_{x_{i}x_{i+1}}=1$ for $i=1,2,$
Consider a symmetric Markov chain X defined by the transition kernel. As for the simple random walk on
on X satisfies $\kappa(x, y)=0$ for, the coarse Ricci curvature
diag (by [8, Theorems 2,3,4 and 5], [4, Theorems 6 andsatisfies
7 , hence the -step coarse Ricci curvatureby [22, Proposition 25]. $X$ (hence thefor-step Markov chain) is $m$-symmetric with respect to $m(dy):=$. For simplicity, hereafter, we assume $N=5.3$-step$:=(V^{3}, E^{3})$Markov chain
is associated with the Cayley graph$:=\{x_{i}x_{j}|i,$ $j=1,2,3,4,5$ with $i\neq j\}.$$:=V$ anddefined byofis a weighted complete graph. The transition kernel
$G$

$V$

$:=\{x_{i}\}_{i=1}^{N}$$x$$\cdots,$$w_{xy}$$P_{x_{i}}(dy)$$:= \frac{1}{2}\delta_{x_{i-1}}(dy)+\frac{1}{2}\delta_{x_{i+1}}(dy)$$\kappa(x, y)$$\mathbb{Z}^{1}$$(x, y)\in V\cross V\backslash$
$])$

$0$

$\kappa_{n}(x, y)\geq$$\kappa_{n}(x, y)$$n$$(x, y)\in V\cross V\backslash d\dot{\ovalbox{\tt\small REJECT}}ag$$X^{n}$$n$$\frac{1}{N}\sum_{i=1}^{N}\delta_{x}.(dy)$$G^{3}$$X^{3}$$E^{3}$$V^{3}$$P_{x}^{3}(dy)$$G^{3}$$X^{3}$.
is given by
for $x\in V$ isis a weighted graph with no loop and the degree
for $xy\in E^{3}$ is given by
given by $d_{x}(G^{3})=4$ . The weight. Note here that our degree $d_{x}(G^{3})=4$ and theway for weighting on edges are different from those used in Section 6of [4], but the conclusion is the same as we calculate below. The 3-stepfor $xy\in E^{3}$ can be estimated by use ofcoarse Ricci curvature[4, Theorems 6 and 7]:$P_{x_{i}}^{3}(dy)= \frac{1}{8}\delta_{x_{i-2}}(dy)+\frac{3}{8}\delta_{x_{i-1}}(dy)+\frac{3}{8}\delta_{x_{i+1}}(dy)+\frac{1}{8}\delta_{x_{i+2}}(dy)$$G^{3}$$d_{x}(G^{3})$$w_{x_{i}x_{i}}= \frac{3}{2},$$w_{xy}$$w_{x_{i-2}x_{t}}=w_{x_{i}x_{i+2}}= \frac{3}{4}$$\kappa_{3}(x, y)$$\kappa_{3}(x_{i}, x_{i+1})=\frac{3}{8}, \frac{5}{8}\leq\kappa_{3}(x_{i}, x_{i+2})\leq\frac{7}{8}.$Therefore,$\kappa_{3}(x, y)\geq\frac{3}{8}$for all $(x, y)\in V\cross V\backslash$ diag.

Remark 1.4. For a continuous time parameter Markov process $M$ , thefor $M$ is discussed in [22], [28]:notion of coarse Ricci curvature$\kappa(x, y)$$\kappa(x, y)$$:= \varliminf_{tarrow 0}\frac{1}{t}(1-\frac{d_{W_{1}}(P_{t}(x,\cdot),P_{t}(y,\cdot))}{d(x,y)})$for $(x, y)\in E\cross E\backslash diag.$

We can also define the -step coarse Ricci curvature$n$$\kappa_{n}(x, y)$$:= \varliminf_{tarrow 0}\frac{1}{t}(1-\frac{d_{W_{1}}(P_{nt}(x,\cdot),P_{nt}(y,\cdot))}{d(x,y)})$$\kappa_{n}(x, y)$:for $(x, y)\in E\cross E\backslash diag,$

which is nothing but the coarse Ricci curvature for the time changed
process $M^{n}=(\Omega, X_{nt}, P_{x})$ . Then, we easily see $\kappa_{n}(x, y)=n\kappa(x, y)$for $(x, y)\in E\cross E\backslash$ diag, in particular, the positivity of lower bound90
for curvature is equivalent to each other between both coarse Ricci
curvatures. In our discrete setting, we have no such a relation.
Example 1.5 (Riemannian manifold). Let $(M, g)$ be a -dimensional
complete smooth Riemannian manifold whose Ricci curvature is boundbelow by $\kappa>0$ . In view of Bonnet-Myers theorem, $M$ is compact.
$d$Consider a Brownian motion $M=(\Omega, X_{t}, P_{x})$ on $M$ associated withthe following Dirichlet energy form on $L^{2}(M;m)$ ;$\{\begin{array}{l}D(\mathcal{E}) :=\{u\in L^{2}(M;m)|\int_{M}g(\nabla f, \nabla f)dm<\infty\}\mathcal{E}(f, g) ;=\int_{M}g(\nabla f, \nabla g)dm, f, g\in D(\mathcal{E})\end{array}$where $m$ is the volume element of $(M, g)$ . Let $P_{t}(x, dy)$ be the transition
kernel of M. Under the Ricci curvature lower bound, $M$ is a conservative process, that is,for any $t>0$ . Moreover, wesee
for any $t>0$ . We set $P(x, dy)$ $:=P_{1}(x, dy)$ andconsider an $m$-symmetric Markov chain X associated with $P(x, dy)$ . It
is proved in [30] that
$P_{t}(x, \cdot)\in \mathcal{P}(M)$$P_{t}(x, \cdot)\in \mathcal{P}^{1}(M)$$d_{W_{1}}(P_{t}(x, \cdot), P_{t}(y, \cdot))\leq e^{-\kappa t}d(x, y) , x, y\in M.$

So the coarse Ricci curvature$\kappa_{M}(x, y)$of $M$ has the lower estimate$\kappa_{M}(x, y):=\varliminf_{tarrow 0}\frac{1}{t}(1-\frac{d_{W_{1}}(P_{t}(x,\cdot),P_{t}(y,\cdot))}{d(x,y)})$$\geq\frac{d}{dt}(1-e^{-\kappa t})|_{t=0}=\kappa>0,$$(x, y)\in M\cross M\backslash$

On the other hand, the -step coarse Ricci curvature
the lower estimate$n$$\kappa_{n}(x, y)=1-\frac{d_{W_{1}}(P_{x}^{n},P_{y}^{n})}{d(x,y)}\geq 1-e^{-\kappa n}>0,$$\kappa_{n}(x, y)$diag.
of X has$(x, y)\in M\cross M\backslash$diag.

Note that the same conclusion also holds for a Markov process whose
coarse Ricci curvature is bounded below by$\kappa>0.$

2. CAT$(O)$-SPACESIn this section, we summarize the notions of CAT(0)-space and its
properties.

Definition 2.1 CAT $(O)-$ space) .
metric space $(Y, d)$ is called the$CAT(O)$ -space Hadamard space, $or$ global $NPC$ space) if for any pairof points
and any $t\in[0,1]$ there exists a point
suchthat for any $z\in Y$
$($

$A$

$($

$\gamma_{0},$(2.1)$\gamma_{1}\in Y$$\gamma_{t}\in Y$$d_{Y}^{2}(z, \gamma_{t})\leq(1-t)d_{Y}^{2}(z, \gamma_{0})+td_{Y}^{2}(z, \gamma_{1})-t(1-t)d_{Y}^{2}(\gamma_{0}, \gamma_{1})$.

By definition,is the minimal geodesic joining
and. Any CAT(0)-space is simply connected. Hadamard manifolds,
Euclidean Bruhat-Tits buildings (e.g. metric tree), spiders, booklets
and Hilbert spaces are typical examples of CAT(0)-spaces (cf. [25]).
$\gamma$$\gamma_{1}$$:=(\gamma_{t})_{t\in[0,1]}$$\gamma_{0}$91
Let $(Y, d_{Y})$ be a CAT(0)-space. Then the distance function:$Yarrow[O,$is convex (Corollary 2.5 in [25]) and Jensen’s inequality(Theorem 6.3 in [25]) can be applied to the convex function $Y\ni w\mapsto$$d_{Y}(w, z)$ for each $z\in Y.$$d_{Y}$$Y\cross$$\infty[$

The inequality (2.1) yields the (strict) convexity of $Y\ni x\mapsto d_{Y}^{2}(z, x)$
for a fixed $z\in Y$ . Any closed convex subset of a CAT(0)-space is again
a CAT(0)-space.
The unique existence of projection (or foot-point) to closed convex
set of CAT(0)-space is proved in [14] in more general setting.

Lemma 2.2 (Projection Map to Convex Set, see [25]). Let $(Y, d_{Y})$ be

a complete $CAT(O)$ -space. The following hold:(1) Letbe a closed convex subset of $(Y, d_{Y})$ . Then, for each$x\in Y_{f}$ there exists a unique element $\pi_{F}(x)\in F$ such that$d_{Y}(x, F)=d_{Y}(\pi_{F}(x), x)$ holds.
: $Yarrow F$ the proWe call$F.$jection map to(2) Letbe as above. Thensatisfies$F$$\pi_{F}$

$F$

(2.2)

$\pi_{F}$$d_{Y}^{2}(z, \pi_{F}(z))+d_{Y}^{2}(\pi_{F}(z), w)\leq d_{Y}^{2}(z, w) , forz\inY, w\in F,$

in particular, $d_{Y}(\pi_{F}(z), w)\leq d_{Y}(z, w)$ for $z\in Y,$ $w\in F.$

Let $(Y, d_{Y})$ be a metric space andmeasures on $Y$ . For $p\geq 1$ , we set$\mathcal{P}^{p}(Y)$$\mathcal{P}(Y)$a family of Borel probability$:= \{\mu\in \mathcal{P}(Y)|\int_{Y}d_{Y}^{p}(x, y)\mu(dy)<\infty$

Each elementment.

$\mu\in \mathcal{P}^{p}(Y)$for any/some $x\in Y\}.$is called a probability measure with p-th mo-

Definition 2.3 (Barycenter or Center of Mass, see [25]). For, if
has a minimizer $b(\mu)\in Y$ , then we callthe barycenter, or center of mass of. For
and $w\in Y$ , we consider the following function:$\mu\in$$\mathcal{P}^{2}(Y)$$z \mapsto\int_{Y}d_{Y}^{2}(z, x)\mu(dx)$$b(\mu)$$\mu\in \mathcal{P}^{2}(Y)$$\mu\in \mathcal{P}^{1}(Y)$$F_{w}$(2.3)$F_{w}(z) := \int_{Y}(d_{Y}^{2}(z, x)-d_{Y}^{2}(w, x))\mu(dx)$.

We easily see$|F_{w}(z)| \leq 2d_{Y}(z, w)\int_{Y}(d_{Y}(z, x)+d_{Y}(w, x))\mu(dx)<\infty.$

If $Y\ni z\mapsto F_{w}(z)$ admits a minimizer
independent of in thesense that $F_{w}(z)\geq F_{w}(b(\mu))$ if and only if $F_{v}(z)\geq F_{v}(b(\mu))$ for all$w,$ $v\in Y$ , we call it barycenter, or center. If theof mass ofbarycenter ofexists, then it is a barycenter of.
$b(\mu)$

$z,$

$w$

$\mu\in \mathcal{P}^{1}(Y)$$\mu\in \mathcal{P}^{2}(Y)$$\mu\in \mathcal{P}^{1}(Y)$Assume that $(Y, d_{Y})$ is a geodesic space. For a subset of , denote
by $C(F)$ the closed convex hull of . That is, $C(F)$ is the smallest
closed convex subset of containing $F.$
$F$

$F$

$Y$

$Y$

92
If $(Y, d_{Y})$ is a complete CAT(0)-space, we can obtain the uniqueproved in [25].
existence of barycenter of$\mu\in \mathcal{P}^{1}(Y)$Lemma 2.4 ([25], cf. [16],[21]). Let $(Y, d_{Y})$ be a complete $CAT(O)-$space. Thenadmits a unique barycenter.
$\mu\in \mathcal{P}^{1}(Y)$For any metric space $(Y, d_{Y})$ , we easily see $b(\delta_{x})=x$ for $x\in Y.$
The following proposition is proved in Proposition 5.5 in [25].
Theorem 2.5 (Jensen’s Inequality, see [25, Theorem 6.3]). Let $(Y, d_{Y})$be a complete $CAT(O)$ -space. Let be a lower semi-continuous convex. Then we have
. Supposefunction on and$\varphi$$Y$$\varphi\in L^{1}(Y;\mu)$$\mu\in \mathcal{P}^{1}(Y)$(2.4)$\varphi(b(\mu))\leq\int_{Y}\varphi(x)\mu(dx)$.

Corollary 2.6 (Fundamental Contraction Property, see [25]). Let $(Y, d_{Y})$. Thenbe a complete $CAT(O)$ -space. Let$\mu,$$\nu\in \mathcal{P}^{1}(Y)$$d_{Y}(b(\mu), b(\nu))\leq d_{W_{1}}(\mu, \nu)$where$d_{W_{1}}(\mu, \nu)$is the$L^{1}$,-Wasserstein distance on$\mathcal{P}^{1}(Y)$$d_{W_{1}}( \mu, \nu):=\inf_{\pi\in\Pi(\mu\nu)},\int_{YxY}d_{Y}(x, y)\pi(dxdy)$

Here$\Pi(\mu, \nu)$$\nu(B)$for$:=\{\pi\in \mathcal{P}(Y\cross Y)$

defined by.

$\pi(A\cross Y)=\mu(A),$ $\pi(Y\cross B)=$

$A,$ $B\in \mathcal{B}(Y)\}.$

3.

$L^{p}$-MAPS

Let
a completion of
be a -finite measure space andwith respect to . In what follows, we say measumble (resp.
measurable) for -measurable (resp. -measurable). For function :$:=( \int_{E}|f(x)|^{p}\mu(dx))^{1/p},$$Earrow[-\infty, \infty]$ , we set$:= \inf\{\lambda>$$0||f(x)|\leq\lambda\mu-$ a.e. $x\in E\}$ . For two-valued functions $f,$ , they are$f=g\mu-a.e.$said to be -equivalent if$L^{p}(E;\mu)$ denotes the family of
For $p\in]0,$-equivalence class offunctions with finite-norm. Also $L^{0}(E;\mu)$ denotes the family of-equivalence class of functions having finite value $\mu-a.e$ . Fix a metricspace $(Y, d_{Y})$ . For $p\in$ ]: $Earrow Y$ , theand measurable maps
pseudo-distanceis defined by. Moreprecisely, for $p\in$ ]we set$(E, \mathcal{E}, \mu)$$\mathcal{E}^{\mu}$$\sigma$$\mathcal{E}$$\mu-$$\mu$$f$$\mathcal{E}^{\mu}$$\mathcal{E}$$\Vert f\Vert_{\infty}$$\Vert f\Vert_{p}$$\overline{\mathbb{R}}$

$g$

$\mu$

$\infty],$$\mu$$\Vert$$\Vert_{p}$

$\mu$

$0,$

$\infty]$$u,$ $v$$d_{L^{p}}(u, v)$$d_{L^{p}}(u, v)$$0,$$:=\Vert d_{Y}(u, v)\Vert_{p}$$\infty[$$d_{Lp}(u, v) :=( \int_{E}d_{Y}^{p}(u(x), v(x))\mu(dx))^{1/p}$

and for $p=\infty,$
We say that andis the -essentially supremum of $x\mapsto d_{Y}(u(x), v(x))$ .
in short) if
are -equivalent ($d_{\infty}(u, v)$

$u$

$v$

$\mu$

$u\sim v\mu$$\mu$$u(x)=v(x)\mu-a.e.

x\in E.$

93
For a fixed measurable map $h:Earrow Y$ , we set$L_{h}^{p}(E, Y;\mu) :=\{f\in \mathcal{E}/\mathcal{B}(Y)|d_{Y}(f, h)\in L^{p}(E;\mu)\}/\sim\mu$. If $\mu(E)<$
Such a map $h:Earrow Y$ is called a base map of$EarrowY$
is independentis bounded,and the image of :case,assumewecanof the choice of such a base map . In thisfor some fixed point $0\in Y.$
Proposition 3.1 ([23, Proposition 3.3]). Let $(Y, d_{Y})$ be a metric space
and $h:Earrow Y$ a measurable map. Take $p\in[1, \infty]$ . Then we have thefollowing:is so.
(1) If $(Y, d_{Y})$ is complete, then(2) If $(Y, d_{Y})$ is a geodesic space and any pointof the constanttois a continuous
speed geodesic : $[0,1]arrow Y$ joiningis also a, thenmap with respect to$L_{h}^{p}(E, Y;\mu)$$L_{h}^{p}(E, Y;\mu)$$h$$\infty$$h\equiv 0$$h$$(L_{h}^{p}(E, Y;\mu), d_{L^{p}})$$\gamma_{t}$$\gamma_{0}$$\gamma$geodesic$\gamma_{1}$$(L_{h}^{p}(E, Y;\mu), d_{L^{p}})$$(\gamma_{0}, \gamma_{1})$$\mathcal{S}pace.$

In what follows, we assume $m(E)<\infty$ . Let $L^{p}(E, Y;m)$ be thespace of -maps with bounded base maps, that is,$L^{p}$$L^{p}(E, Y;m)$$:=\{u$ : $Earrow Y$$u$is $m$-measurable,$\int_{E}d_{Y}^{p}(u, 0)dm<\infty$

Definition 3.2 (Lipschitz Maps). Let $(Y, d_{Y})$ be a geodesic space and$(E, d)$ a metric space.
For a map $u:Earrow Y$ , we set Lip $(u):=$and$\sup_{x\neq y}\frac{d_{Y}(u(x)(y))}{d(x}$Lip $(E, Y)$ $:=\{u$ : $Earrow Y|$ Lip $(u)<\infty\}.$Lemma 3.3. Let $(Y, d_{Y})$ be a geodesic space and $(E, d)$ a metric space.
Suppose that $m$ has a p-th moment, that is, $\int_{E}d^{p}(x, x_{0})m(dx)<\infty$ forsome/any point $x_{0}\in E$ . Then Lip $(E, Y)\subset L^{p}(E, Y;m)$ .
to . Any
Let $S(E, Y)$ be a space of finite valued maps fromelement of $S(E, Y)$ is called a step map or a simple map. Since $m(E)<$$S(E, Y)$ $(more$ precisely $S(E, Y)/\sim m$ ) is a subset of $L^{p}(E, Y;m)$ .
$E$

$Y$

$\infty,$

Theorem 3.4. Suppose that $(E, d)$ is a Polish space and $(Y, d_{Y})$ is. Then any element ofa separable geodesic space. Take $p\in[1,$$L^{p}(E, Y;m)$ can be-approximated by elements in $S(E, Y)$ . In particular, if $E=supp[m]$ , then $(L^{p}(E, Y;m), d_{L^{p}})$ is a separable metricspace. Moreover, if $m$ has a finite p-th moment, then $L^{p}(E, Y;m)$ canbe -approximated by elements in Lip $(E, Y)$ , if further $E=supp[m],$then Lip $(E, Y)$ is a dense subset of $L^{p}(E, Y;m)$ .
In what follows, $(E, d)$ denotes a Polish space with complete distance$\infty[$$L^{p}$$L^{p}$$d.$94
for Borel Map ). Let X be a conservative Markov
Definition 3.5 (chain on $(E, d)$ . Suppose that $(Y, d_{Y})$ is a complete CAT(0)-space andfor-meaeurable map $u:Earrow Y$ satisfiesa. Then we set.
$P^{\ell}u$$u$$u_{\#}P_{x}^{\ell}\in \mathcal{P}^{1}(Y)$$\mathcal{B}(E)/\mathcal{B}(Y)$$\ell\in \mathbb{N}$$P^{\ell}u(x) :=b(u_{\#}P_{x}^{\ell})$Here$u_{\#}P_{x}^{\ell}$

is a push-forward measure of$P^{\ell}(x, u^{-1}(A)),$ $A\in \mathcal{B}(Y)$$P(x, \cdot)$

by $u;u_{\#}P_{x}^{\ell}(A):=$.

Remark 3.6. Note that any $u\in S(E, Y)$ ULip $(E, Y)$ satisfies. Indeed, for $u\in S(E, Y),$ is a constant on each Borel set
is a finite family of disjoint Borel sets satisfying $E=$
where
, hence. For $u\in Lip(E, Y)$ , we have$u_{\#}P_{x}\in$$A_{i},$$u$$\mathcal{P}^{1}(Y)$$\{A_{i}\}_{i=1}^{l}$$\int_{E}d_{Y}(z_{0}, z)u_{\#}P_{x}(dz)=\sum_{i=1}^{\iota}\Vert d_{Y}(z_{0}, u)\Vert_{\infty,A_{i}}P_{x}(A_{i})<$$\bigcup_{i=1}^{l}A_{i}$$\infty$$\int_{E}d_{Y}(z_{0}, z)u_{\#}P_{x}(dz)=\int_{E}d_{Y}(z_{0}, u(y))P_{x}(dy)$$\leq d_{Y}(z_{0}, u(y_{0}))+Lip(u)\int_{E}d(y_{0}, y)P_{x}(dy)<\infty.$

Lemma 3.7 (Lemma 6.4 in [23]). Let X be a conservative Markov
chain on $(E, d)$ . Suppose that $(Y, d_{Y})$ is a complete sepamble $CAT(O)-$space. Then, for any Borel map $u:Earrow Y$ satisfying-measumble.
for all $x\in E$ , Pu: $Earrow Y$ is$u_{\#}P_{x}\in \mathcal{P}^{1}(Y)$$\mathcal{B}(E)/\mathcal{B}(Y)$Definition 3.8 (Pu for -map ). Fix $p\geq 1$ . Let X be an msymmetric conservative Markov chain on $(E, d)$ . Suppose that $(Y, d_{Y})$is a complete CAT(0)-space and $u\in L^{p}(E, Y, m)$ , we can define $Pu\in$$Iy(E, Y;m)$ in the following way: Let $\{u_{k}\}\subset S(E, Y)$ be an
approximating sequence to . Applying the Jensen’s inequality to theon $Y\cross Y$ and the $m$-symmetry, we have the folconvex function
lowing inequality for any maps $w\in S(E, Y)$ .
$L^{p}$$u$$L^{p_{-}}$$u$$d_{Y}^{p}$

$v,$

(3.1)

$d_{L^{p}}^{p}(Pv, Pw)= \int_{E}d_{Y}^{p}(Pv(x), Pw(x))m(dx)$$\leq\int_{E}Pd_{Y}^{p}(v, w)dm\leq d_{L^{p}}^{p}(v, w)$.

These mean that $\{Pu_{k}\}$ forms an -Cauchy sequence. We set $Pu:=$$\lim_{k}Pu_{k}\in L^{p}(E, Y;m)$ . The well-definedness of $Pu$ is clear from (3.1)
and this is valid for any $w\in L^{p}(E, Y;m)$ .
$L^{p}$$v,$

Definition 3.9 $(P-$ harmonic $Map, [16],[15])$ . (lower or upper) bounded
Borel function :is said to be -subharmonic if $f\leq Pf$ on
and it is said to be -harmonic if both $fand-f$ are -subharmonic.
A Borel map : $Earrow Y$ is said to be -harmonic if $u=Pu$ on
for all $x\in E.$
holds under that
$A$

$f$

$Earrow \mathbb{R}$

$E$

$P$

$P$

$P$

$P$

$u$

$u_{\#}P_{x}\in \mathcal{P}^{1}(Y)$

$E$

95
Lemma 3.10. Let X be a Markov chain on $(E, d)$ . Fix
and$(Y,
d_{Y})$$CAT(O)$
assumeSuppose thatis a complete-space. Then$u\in$ Lip $(E, Y)$ and$(E,Y)$, we have
Lip
and
for
$n\in \mathbb{N}$$\kappa\in \mathbb{R}.$$P^{n\ell}u\in$$\ell\in \mathbb{N}$Lip$(P^{n\ell}u)\leq(1-\kappa_{n})^{\ell}$

Lip $(u)$ ,in particular,

Lip $(P^{\ell}u)\leq(1-\kappa)^{\ell}Lip(u)$ .

Corollary 3.11 (Strong Liouville Property for Lipschitz Maps). Let X
be a Markov chain on $(E, d)$ . Assume thatand there exists
such that. Suppose that $(Y, d_{Y})$ is a complete $CAT(O)$ -space.
Then any-harmonic map $u\in$ Lip $(E, Y)$ is a constant map.
$\kappa\in \mathbb{R}$$n\in \mathbb{N}$$\kappa_{n}>0$$P^{n}$

Definition 3.12 (Variance). Fix $p\geq 1,$$(Y, d_{Y})$ and $u\in L^{p}(E, Y;\mu)$ . The-variance, a metric spaceof is defined by$\mu\in \mathcal{P}(E)$$p$$Var_{\mu}^{p}(u)$$u$$Var_{\mu}^{p}(u):=\inf_{y\in Y}\int_{E}d_{Y}^{p}(u(x), y)\mu(dx)(<\infty)$

The quasi -variance$p$$\overline{Var}_{\mu}^{p}(u)$.

is defined by$\overline{Var}_{\mu}^{p}(u) :=\frac{1}{2}\int_{E}\int_{E}d_{Y}^{p}(u(y), u(x))\mu(dx)\mu(dy)(<\infty)$.

We easily see. When $p=2$ , we write $Var_{\mu}(u)$ $:=$
and
, and call them simply variance, quasi
variance, respectively. Let $(Y, d_{Y})$ be a complete CAT(0)-space. If$u\in L^{2}(E, Y;\mu)$ , then $Var_{\mu}(u)=\int_{E}d_{Y}^{2}(u(x), b(u_{\#}\mu)))\mu(dx)$ holds. For$u\in L^{2}(E, Y;\mu)$ , we have
. If $(Y, d_{Y})$ is a Hilbert
space $H$ , then we have
. In this case we can define
the covariance $Cov_{\mu}(f, g)$ for $f,$ $g\in L^{2}(E, H;\mu)$ by$Var_{\mu}^{p}(u)\leq 2\overline{Var}_{\mu}^{p}(u)$$Var_{\mu}^{2}(u)$$\overline{Var}_{\mu}(u)$$:=\overline{Var}_{\mu}^{2}(u)$$Var_{\mu}(u)\leq\overline{Var}_{\mu}(u)$$Var_{\mu}(u)=\overline{Var}_{\mu}(u)$$Cov_{\mu}(f, g):=\int_{E}\langle f(x)-\langle\mu,$$f\rangle,$$g(x)-\langle\mu,$ $g\rangle\rangle_{H}\mu(dx)$$=\langle\mu, \langle f, g\rangle_{H}\rangle-\langle\langle\mu, f\rangle, \langle\mu, g\rangle\rangle_{H}$$= \frac{1}{2}\int_{E}\int_{E}\langle f(y)-f(x), g(y)-g(x)\rangle_{H}\mu(dx)\mu(dy)$where$\langle\mu,$$f\rangle$$:= \int_{H}f(x)\mu(dx)\in H$

is the barycenter of,$f_{\#}\mu\in \mathcal{P}^{2}(Y)$.

Definition 3.13 (Energy of Maps). Takeand let X be
an $m$-symmetric Markov chain and $(Y, d_{Y})$ is a metric space. For $u\in$$m\in \mathcal{P}(E)$$L^{p}(E, Y;m)$ ,$E^{p}(u):= \frac{1}{2}\int_{E}\int_{E}d_{Y}^{p}(u(y), u(x))P(x, dy)m(dx)$is said to be -energy of$p$$E_{*}^{p}(u)$$u$with respect to X and$:= \frac{1}{2}\int_{E}d_{Y}^{p}(Pu(x), u(x))m(dx)=\frac{1}{2}d_{Lp}^{p}$(Pu, u)96
is said to be quasi -energy of with respect to X for $p\geq 1$ when$(Y, d_{Y})$ is a complete separable CAT(0)-space.
When $p=2$ , we simply say energy (resp. quasi 2-energy) and write$E(u):=E^{2}(u)$ resp. $E_{*}(u):=E_{*}^{2}(u))$ . Since
$u$

$p$

$($

$Vax_{P_{x}}^{p}(u)\leq\int_{E}d_{Y}^{p}(u(y), u(x))P(x, dy)$,

we see(3.2)$\frac{1}{2}\int_{E}Var_{P_{x}}^{p}(u)m(dx)\leq E^{p}(u)$.

We use$\{\begin{array}{ll}D(E^{p}) :=\{u\in L^{p}(E, Y;m)|E^{p}(u)<\infty\} E^{p}(u) :=\frac{1}{2}\int_{E}\int_{E}d_{Y}^{p}(u(y), u(x))P(x, dy)m(dx) , u\in D(E^{p}) .\end{array}$

When $(Y, d_{Y})$ is a Hilbert space $H$ , we use the symbol
for the (2-)energy on $L^{2}(E, H;m)$ and set$\mathcal{E}(f, g)$

for $f,$$:= \frac{1}{2}\int_{E\cross E}\langle f(y)-f(x),$ $g(y)-g(x)\rangle$g\in D(\mathcal{E})$. We$see\mathcal{E}(f)=\mathcal{E}(f, f)$$\mathcal{E}$instead of$E${}_{H}P_{x}(dy)m(dx)$for $f\in L^{2}(E, H;m)$ .

Proposition 3.14. Let X be an $m$ -symmetric Markov chain on $(E, d)$. For measumble maps
and $(Y, d_{Y})$ is a metric space. Fix $p\in[1,$$v:Earrow Y$ , the following inequalities hold:$\infty[$

$u,$

(3.3)

$E^{p}(u)^{\frac{1}{p}}\leq E^{p}(v)^{\frac{1}{p}}+2^{1-\frac{1}{p}}d_{Lp}(u, v)$

,

(3.4)

$Var_{m}^{p}(u)^{\frac{1}{p}}\leq Vair_{m}^{p}(v)^{\frac{1}{p}}+d_{L^{p}}(u, v)$(3.5)$\overline{Var}_{m}^{p}(u)^{\frac{1}{p}}\leq\overline{Var}_{m}^{p}(v)^{\frac{1}{p}}+2^{1-\frac{1}{p}}d_{L^{p}}(u, v)$,.

Corollary 3.15. Let X be an $m$ -symmetric Markov chain on $(E, d)$ .
Suppose that $(Y, d_{Y})$ is a complete separable $CAT(O)$ -space. For $p\geq 1$
and $u\in L^{p}(E, Y;m)$ , the following inequalities hold:
Var

Pu

(3.6)

Var

(3.7)

$\overline{Var}_{m}^{p}(u)^{\frac{1}{p}}\leq\overline{Var}_{m}^{p}(Pu)^{\frac{1}{p}}+2E_{*}^{p}(u)^{\frac{1}{p}}.$$mp(u)^{\frac{1}{p}}\leq$$mp($$)^{\frac{1}{p}}+2^{\frac{1}{p}}E_{*}^{p}(u)^{\frac{1}{p}},$If $u\in L^{2}(E, Y;m)$ , we have
(3.8)

$E_{*}^{2}(u)\leq 4E^{2}(u)$.

Corollary 3.16 (Lower Semi Continuity of Energy). Let X be an mand $(Y, d_{Y})$ a metric space.
symmetric Markov chain withTake $p\geq 1$ and let $(E^{p}, D(E^{p}))$ be the -energy on $L^{p}(E, Y;m)$ associated with X. We set $E^{p}(u)$ $:=\infty foru\in L^{p}(E, Y;m)\backslash D(E^{p})$ . Thenis a-valued lower semi continuous functional on $L^{p}(E, Y;m)$ .
$m\in \mathcal{P}(E)$$p$$E^{p}$$[0, \infty]$

97
Remark 3.17. When $p=2,$, the lower semi continuity of energy
is equivalent to the completeness ofwith respect to the norm$:=\mathcal{E}(f, g)+(f, g)_{m},$defined by. Here$Y=\mathbb{R}$$D(\mathcal{E})$$\Vert\cdot\Vert_{\mathcal{E}_{1}}$$\Vert f\Vert_{\mathcal{E}_{1}}$$f,$ $g\in D(\mathcal{E})$$:=\sqrt{\mathcal{E}_{1}(f,f)}$$\mathcal{E}_{1}(f, g)$.

Lemma 3.18 (Contraction Property). Let X be an $m$ -symmetric Markov
chain on $(E, d)$ with
. Fix $p\geq 1$ . Let $(Y, d_{Y})$ be a complete
separable $CAT(O)$ -space. Then, for any $u\in L^{p}(E, Y;m)$ , we have$m\in \mathcal{P}(E)$(3.9)

Var $mp(Pu)\leq$ Var $mp(u)$ ,(3.10)$\overline{Var}_{m}^{p}(Pu)\leq\overline{Var}_{m}^{p}(u)$.

4. MAIN RESULTS
In this section, we fix $p\geq 1$ and assume$m\in \mathcal{P}(E)$and $supp[m]=E.$

Theorem 4.1 $(Non-$ linear Spectral Radius $of P on L^{p}(E, Y;m)/\{$ const ).
Let X be an $m$ -symmetric Markov chain on $(E, d)$ with
. Let $(Y, d_{Y})$ be a complete sepamble $CAT(O)$ -space.
and assumeThen, we have$\}$$m\in \mathcal{P}^{p}(E)$$\kappa\in \mathbb{R}$(4.1)$\varliminf_{\ellarrow\infty}(\sup_{u\in L^{p}(E,Y;m)}\frac{Var_{m}^{p}(P^{\ell}u)}{Var_{m}^{p}(u)})^{\frac{1}{p\ell}}\leq\inf_{n\in \mathbb{N}}(1-\kappa_{n})^{\frac{1}{n}}\wedge 1,$(4.2)$\varliminf_{\ellarrow\infty}(\sup_{u\in L^{p}(E,Y;m)}\frac{\overline{Var}_{m}^{p}(P^{\ell}u)}{\overline{Var}_{m}^{p}(u)})^{\frac{1}{pl}}\leq\inf_{n\in \mathbb{N}}(1-\kappa_{n})^{\frac{1}{n}}\wedge 1.$

Corollary 4.2 Linear Spectral Radius $of P on L^{2}(E, H;m)/\{$ const ).
Let X be an $m$ -symmetric Markov chain on $(E, d)$ and $H$ a real sepa. Then, we havemble Hilbert space, Assume
and
$\}$

$($

$m\in \mathcal{P}^{2}(E)$(4.3)$\kappa\in \mathbb{R}$$\lim_{\ellarrow\infty}(\sup_{f\in L^{2}(E,H;m)}\frac{Var_{m}(P^{\ell}f)}{Var_{m}(f)})^{\frac{1}{2\ell}}\leq\inf_{n\in \mathbb{N}}(1-\kappa_{n})^{\frac{1}{n}}\wedge 1.$

Consequently, $P$ is an-contraction opemtor on$L^{2}(E, H;m)/$ {const}. In particular, for $f\in L^{2}(E, H;m)/\{const\}$ , thefollowing hold:$\inf_{n\in \mathbb{N}}(1-\kappa_{n})^{\frac{1}{n}}\wedge 1$(4.4)$Var_{m}(Pf)\leq(\inf_{n\in \mathbb{N}}(1-\kappa_{n})^{\frac{2}{n}}\wedge 1)Var_{m}(f)$

,

(4.5)

$| Cov_{m}(Pf, f)|\leq(\inf_{n\in N}(1-\kappa_{n})^{\frac{1}{n}}\wedge 1)Var_{m}(f)$.

The main part of the following theorem is a slight generalization of[22, Corollary 31], and its proof is similar as in [22] based on Corollary 4.2 above.

98
Theorem 4.3 (Poincar\’e Inequality, cf. Corollary 31 in [22]). Assume. Let X be an $m$ -symmetric Markov chain on
and$(E, d)$ and $H$ a real sepamble Hilbert space. Then, for $f\in L^{2}(E, H;m)$$\kappa\in \mathbb{R}$$m\in \mathcal{P}^{2}(E)$

,

(4.6)

$(1- \inf_{n\in \mathbb{N}}(1-\kappa_{n})^{\frac{2}{n}}\wedge 1)Var_{m}(f)\leq\int_{E}Var_{P_{x}}(f)m(dx)$(4.7)$1- \inf_{n\in \mathbb{N}}(1-\kappa_{n})^{\frac{1}{n}}\wedge 1\leq\frac{\mathcal{E}(f)}{Var_{m}(f)}\leq 1+\inf_{n\in \mathbb{N}}(1-\kappa_{n})^{\frac{1}{n}}\wedge 1.$

In particular, if
inequality:$\kappa_{n}>0$

for some$n\in \mathbb{N}$, we have a global Poincar\’e$0<1- \inf_{n\in N}(1-\kappa_{n})^{\frac{1}{n}}\leq\inf_{f\in L^{2}(E,H;m)}\frac{\mathcal{E}(f)}{Vax_{m}(f)}$$\leq\sup_{f\in L^{2}(E,H;m)}\frac{\mathcal{E}(f)}{Var_{m}(f)}\leq 1+\inf_{n\in\mathbb{N}}(1-\kappa_{n})^{\frac{1}{n}}<2.$

Moreover, if X is an even step Markov chain obtained from an msymmetric Markov chain, then
(4.8)$\sup_{f\in L^{2}(E,H;m)}\frac{\mathcal{E}(f)}{Vax_{m}(f)}\leq 1.$

Corollary 4.4 (Estimates of Eigenvalues). Let X be an $m$ -symmetricand theand assume that
Markov chain on $(E, d)$ withis compact. Then any non-zero eigenvalue
embedding$\triangle=I-P$
on $L^{2}(E;m)$ satisfiesof the operator$\kappa\in \mathbb{R}$$m\in \mathcal{P}^{2}(E)$$D(\mathcal{E})\subset L^{2}(E;m)$$\lambda$(4.9)$L^{2}-$-$1- \inf_{n\in N}(1-\kappa_{n})^{\frac{1}{n}}\wedge 1\leq\lambda\leq 1+\inf_{n\in N}(1-\kappa_{n})^{\frac{1}{n}}\wedge 1.$

Moreover, if X is an even step Markov chain obtained from an msymmetric Markov chain, then any eigenvalue satisfies$\lambda$$\lambda\leq 1.$. Let X
andCorollary 4.5 (Recurrence). Assume
$(E,
d)$
$m$
there
exists. Suppose thatbe an -symmetric Markov chain on
anynonis,
recurrent,
that
X
.
Then
is
such thatfortrivial $f\in L_{+}^{1}(E;m)$ , we have $Gf=\infty$ m-a. . Here $Gf:= \sum_{i=0}^{\infty}P^{i}f.$$m\in \mathcal{P}^{2}(E)$$\kappa\in \mathbb{R}$$\kappa_{n}>0$$n\in \mathbb{N}$$e$(1) When X is an $m$-symmetric random walk on a
Remark 4.6.
finite undirected weighted connected graph $G=(V, E)$ with$m(\{x\})$ $:=d_{x}(G)$ , the degree ofat $x\in V$ , Bauer-Jost-Liu [4]proved (4.9) for any(2) If we assume the existence of non-constant Lipschitz eigenfunction$:=I-P$ , then we can directly prove the estimate
for the associated real eigenvalue ;$G$$n\in \mathbb{N}.$$of-\triangle$$\lambda$(4.10)$1-(1-\kappa_{n})^{\frac{1}{n}}\leq\lambda\leq 1+(1-\kappa_{n})^{\frac{1}{n}}$for $(x, y)\in E\cross E\backslash$ diag without$(4.10)$ is equivassuming the $m$-symmetricity of X. If
alent to (4.9). We show (4.10) as mentioned above. Let

under$\kappa_{n}(x, y)\geq\kappa_{n}(\in \mathbb{R})$$\kappa_{n}>0,$

$f$

99
be a non-constant Lipschitz eigenfunction and assume thatis a real eigenvalue of with respect to -A. Then, we have$(I-P)f=\lambda f$ , equivalently, $P^{k}f=(1-\lambda)^{k}f$ for any
By scaling, we may assume that the Lipschitz constant of is
1. Kantorovich-Rubinstein duality formula yields$\lambda$$f$$k\in \mathbb{N}.$$f$$d(x, y)(1-\kappa_{n})\geq d_{W_{1}}(P_{x}^{n}, P_{y}^{n})\geq P^{n}f(x)-P^{n}f(y)$$=(1-\lambda)^{n}(f(x)-f(y))$

for $(x, y)\in E\cross E$ , which implies $(1-\kappa_{n})\geq|1-\lambda|^{n}$ , that is,
we obtain (4.10).
.
Theorem 4.7 (Strong -Liouville Property). Assume
$(E,
d)$
$m$
. Suppose that
Let X be an -symmetric Markov chain on
and there exists
such that $\kappa_{n}>0$ . Let $(Y, d_{Y})$ be a complete$\mathcal{S}epambleCAT(O)$ -space. Suppose that $u\in L^{p}(E, Y;m)$satisfies $Pu=$
m-a.e. on E. Then is a constant map m-a.e. In particular, if$u\in$ Lip $(E, Y)$ is-harmonic, then is a constant map.
$m\in \mathcal{P}^{p}(E)$$L^{p}$$\kappa\in \mathbb{R}$$n\in \mathbb{N}$

$u$

$u$

$P$

$u$

Corollary 4.8 (Ergodicity). Let X be an $m$ -symmetric Markov chain
such thatand there exists
on $(E, d)$ . Suppose that
Then X is ergodic, that is, for any -invariant Borel set $A,$ $m(A)=0$
or $m(A^{c})=0.$$\kappa_{n}>0.$$n\in \mathbb{N}$$\kappa\in \mathbb{R}$

$P$

and
Theorem 4.9 (Poincar\’e Inequality). Assume$(E,d)$. Suppose that there
Let X be an $m$ -symmetric Markov chain on
existssuch that $\kappa_{n}>0$ . Let $(Y, d_{Y})$ be a complete sepamble$CAT(O)$ -space. Then for any[, there exists]$(E, d, m, X)$ and $(Y, d_{Y})$ such that
depending on$m\in \mathcal{P}^{2}(E)$$\kappa\in \mathbb{R}.$$n\in \mathbb{N}$$\epsilon\in$$\epsilon,$$0,1-(1-\kappa_{n})^{\frac{1}{n}}$$\ell_{0}\in \mathbb{N}$$\kappa_{n},$$\inf_{u\in L^{2}(E,Y;m)}\frac{E(u)}{Var_{m}(u)}\geq\frac{(1-(1-\kappa_{n})^{\frac{1}{n}}\wedge 1-\epsilon)^{2}}{8\ell_{0}^{2}}>0.$(1) For the random walk on an undirected weighted
Remark 4.10.
finite graph $G=$ $(V, E)$ with $N$ $:=|V|$ , Bauer-Jost-Liu [4]proved the equivalence among the following:(i)is non-bipartite.
$G$(ii) $\lambda_{N-1}<2.$(iii) There exists
such that)(resp.
. HereSince is connected, we haveof
theis the smallest non-zero (resp. maximum) eigenvalue
Laplace operator on . Under the equivalent conditions $(i)-$(iii), we have a positivity of non-linear spectral gap as in Theorem 4.9. Remark that the positivity of non-linear spectral gap
on the finite connected weighted graph (having no loop andno multi-edges) with graph distance is already proved by IzekiKondo-Nayatani [7] for CAT(0)-space target.
$n\in \mathbb{N}$$G$$\kappa_{n}>0.$$\lambda_{1}>0$$\lambda_{1}$

$G$

$G$

$\lambda_{N-1}$

100(2) Our Theorem 4.9 covers the case for the random walk derived
from the Brownian motion on Riemannian manifolds with positive Ricci curvature as in Example 1.5.
Acknowledgment. The authors thank Professor Kazumasa Kuwada

for his valuable comments.
REFERENCES
1. L. Ambrosio, N. Gigli and G. Savar\’e, Gmdient flows in metnc spaces and in
the spa ce of probability measures, Second edition. Lectures in Mathematics ETH
Z\"urich. Birkh\"auser Verlag, Basel, 2008.
2. K. Ball, E. A. Carlen and E. H. Lieb, Sharp uniform convexity and smoothness
inequalities for trace norms, Invent. Math. 115 (1994), no. 3, 463-482.
3. F. Bauer and J. Jost, Bipartite and neighborhood graphs and the spectrum of thenormalized graph Laplacian, http: //arxiv. org/abs $/0910.3118v3$ , to appear
in Comm. Anal. Geom. 2011.
4. F. Bauer, J. Jost and S. Liu, Ollivier’s Ricci curvature and the spectum of thenormalized graph Laplace opemtor, preprint, 2011.
5. M. R. Bridson and A. Haefliger, Metnc spaces of non-positive cumature,
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles ofMathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999.
6. F.-Z. Gong, Y. Liu and Z.-Y. Wen, Some notes on Ricci-Ollivier curvature,
preprint 2012, to appear in Osaka J. Math.
7. H. Izeki, T. Kondo and S. Nayatani, private communication, (2012).
8. J. Jost and S. Liu, Ollivier’s Ricci curvature, local clustering and curvature
dimention inequality on graphs, preprint, 2011.
9. W. S. Kendall, Probability, convexity, and harmonic maps with small image I:
Uniqueness and fine existence, Proc. London Math. Soc., (3) 61 (1990), no. 2,
371-406.
10. W. S. Kendall, From stochastic parallel transport to harmonic maps, New directions in Dirichlet forms, 49-115, $AMS/IP$ Stud. Adv. Math., 8, Amer. Math.
Soc., Providence, BJ, 1998.
11. Y. Kitabeppu, Lower bound of coarse Ricci curv ature on metnc measure spaces
and eigenvalues of Laplacian, preprint 2012.
12. E. Kokubo and K. Kuwae, On spectral bounds for symmetric Markov processes
with coarse Ricci curvatures, preprint, 2012.
13. K. Kuwae, Jensen’s inequality over $CAT(\kappa)$ -space with small diameter, Proceedings of Potential Theory and Stochastics, Albac Romania, 173-182, Theta
Ser. Adv. Math., 14, Theta, Bucharest, 2009.
14. K. Kuwae, Jensen’s inequality on convex spaces, preprint (2012).
15. K. Kuwae, Vanational convergence over convex spaces, (2012), in preparation.
16. K. Kuwae and K.-Th. Sturm, On a Liouville type theorem for harmonic maps toconvex spaces via Markov chains, Proceedings of German-Japanese symposiumin Kyoto 2006, 177-192, RIMS K\^oky\^uroku Bessatsu B6, 2008.
17. Y. Lin and S.-T. Yau, Ricci curvature and eigenvalue estimate on locally finite
graphs, Math. Res. Lett. 17 (2010), no. 2, 343-356.
18. Y. Lin, L. Lu and S.-T. Yau, Ricci curvature on graphs, Tohoku Math. J. 63(2011), no. 4, 605-627.
19. J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal
tmnsport, Ann. of Math. 169 (2009), no. 3, 903-991.
20. S.-I. Ohta, Convexities of metric spaces, Geom. Dedicata 125, (2007), no. 1,
225-250.

101
21. S.-I. Ohta, Extending Lipschitz and Holder maps between metric spaces, Positivity 13 (2009), no. 2, 407-425.
22. Y. Ollivier, Ricci curvature of Markov chains on metnc spaces, J. Func. Anal.
256 (2009), no. 3, 810-864.
23. K.-Th. Sturm, Nonlinear Markov operators associated with symmetnc Markov
kemels and energy minimizing maps between singular spaces, Calc. Var. Partial
Differential Equations 12 (2001), no. 4, 317-357.
24. K.-Th. Sturm, Nonlinear Markov opemtors, discrete heat flow, and harmonic
maps between singular spaces, Potential Anal. 16 (2002), no. 4, 305-340.
25. K.-Th. Sturm, Pmbability measures on metric spaces of nonpositive curv ature.

Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002),
357-390, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003.
26. K.-T. Sturm, On the geometry of metnc measure spaces. I, Acta Math. 196(2006), no. 1, 65-131.
27. K.-T. Sturm, On the geometry of metnc measure spaces. II, Acta Math. 196(2006), no. 1, 133-177.
28. L. Veysseire, Coarse Ricci curvature for continuous-time Markov processes,
preprint, 2012.
29. C. Villani, Optimal tmnsport, old and new, Grundlehren der Mathematischen
Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338,
Springer-Verlag, Berlin, 2009.
30. M.-K. von Renesse and K.-Th. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math. 58 (2005), no.
7, 923-940.

EIKI KOKUBOHINODE-CHO 7-37, KASUGA CITYFUKUOKA, 816-0873
JAPAN-mail address: city-hunter-xxyyzz@hotmail. co. jp$E$

KAZUHIRO KUWAE
DEPARTMENT OF MATHEMATICS AND ENGINEERINGGRADUATE SCHOOL OF SCIENCE AND TECHNOLOGYKUMAMOTO UNIVERSITYKUMAMOTO, 860-S555
JAPAN-mail address: kuwae@gpo.kumamoto-u.ac. jp$E$