The convergence and uniqueness of a discrete-time nonlinear Markov chain

Ruowei Li and Florentin Münch Ruowei Li: School of Mathematical Sciences, Fudan University, Shanghai 200433, China; Shanghai Center for Mathematical Sciences, Jiangwan Campus, Fudan University, No. 2005 Songhu Road, Shanghai 200438, China; Max Planck Institute for Mathematics in the Sciences, Leipzig 04103, Germany. rwli19@fudan.edu.cn Florentin Münch: Max Planck Institute for Mathematics in the Sciences, Leipzig 04103, Germany cfmuench@gmail.com

Abstract.

In this paper, we prove the convergence and uniqueness of a general discrete-time nonlinear Markov chain with specific conditions. The results have important applications in discrete differential geometry. First, on a general finite weighted graph, we prove the discrete-time Ollivier Ricci curvature flow $d_{n+1}\coloneqq(1-\alpha\kappa_{d_{n}})d_{n}$ converges to a constant curvature metric. Then the author in [34, Theorem 5.1] proved a Laplacian separation principle on a locally finite graph with non-negative Ollivier curvature. Here we prove the Laplacian separation flow converges to the constant Laplacian solution and generalize the result to nonlinear $p$ -Laplace operators. Moreover, our results can also be applied to study the long-time behavior in the nonlinear Dirichlet forms theory and nonlinear Perron–Frobenius theory. At last, we define the Ollivier Ricci curvature of nonlinear Markov chain which is consistent with the classical Ollivier Ricci curvature, sectional curvature [5], coarse Ricci curvature on hypergraphs [17] and the modified Ollivier Ricci curvature for $p$ -Laplace. And we prove the convergence results for the nonlinear Markov chain with nonnegative Ollivier Ricci curvature.

1. Introduction

A nonlinear Markov chain, introduced by McKean [28] to tackle mechanical transport problems, is a discrete space dynamical system generated by a measure-valued operator with the specific feature of preserving positivity. Compared with the linear Markov chain, its transition probability is dependent not only on the state but also on the distribution of the process.

It is quite fundamental to understand the long-time behavior of Markov chains. A classical result is that an irreducible lazy linear Markov chain converges to its unique stationary distribution in the total variation distance [23, 44]. For the nonlinear case, Kolokoltsov [21] and BA Neumann [36] study the long-term behavior of nonlinear Markov chains defined on probability simplex whose transition probabilities are a family of stochastic matrices. Long-term results exist for specific continuous-time Markov chains associated with pressure and resistance games [22] and ergodicity criteria for discrete-time Markov processes [4, 42]. This paper establishes convergence and uniqueness results for a general discrete-time nonlinear Markov chain $P:\Omega\to\Omega$ under some of the following specific conditions:

•
Conditions on domain
- –
  
  (A) $\Omega\subseteq\mathbb{R}^{N}$ is closed.
- –
  
  (B) $\Omega+r\cdot(1,\ldots,1)=\Omega$ for all $r\in\mathbb{R}$ .
•
Basic properties
- –
  Monotonicity
  - *
    
    (1) Monotonicity: $Pf\geq Pg$ if $f\geq g$ .
  - *
    
    (2) Strict monotonicity: $Pf(x)>Pg(x)$ if $f\geq g$ and $f(x)>g(x)$ for some $x$ .
  - *
    
    (3) Uniform strict monotonicity: $Pf\geq Pg+\epsilon_{0}(f-g)$ , if $f\geq g$ for some fixed positive $\epsilon_{0}$ .
- –
  Additivity
  - *
    
    (4) Constant additivity: $P(f+C)=Pf+C$ , where $C$ is a constant.
- –
  Non-expansion
  - *
    
    (5) Non-expansion: $\left\|Pf-Pg\right\|_{\infty}\leq\left\|f-g\right\|_{\infty}$ for all $f,g\in\Omega$ .
•
Connectedness
- –
  
  (6) Connectedness: there exists $n_{0}\in\mathbb{N}_{+}$ such that for every point $x,$ and $f,g\in\Omega$ with $f\geq g$ and $f(x)>g(x)$ , we have $P^{n_{0}}f>P^{n_{0}}g$ , (i.e., strict inequality everywhere).
- –
  
  (7) Uniform connectedness: there exists $n_{0}\in\mathbb{N}_{+}$ , positive $\epsilon_{0}$ such that for every point $x,$ positive $\delta$ and $f,g\in\mathbb{R}^{N}$ with $f\geq g+\delta\cdot 1_{x}$ , we have $P^{n_{0}}f\geq P^{n_{0}}g+\epsilon_{0}\delta$ .
•
Accumulation points
- –
  
  (8) Accumulation point at infinity: $f_{n}\coloneqq P^{n}f-P^{n}f(x_{0})$ has a finite accumulation point $g$ , i.e. for every $n\in\mathbb{N}_{+}$ and positive $\epsilon$ , there exists $N>n$ such that $\left\|f_{N}-g\right\|_{\infty}<\epsilon$ .
- –
  
  (9) Finite accumulation point: $P^{n}f$ has a finite accumulation point $g$ .

Definition 1.

A discrete-time nonlinear Markov chain $P:\Omega\to\Omega$ is a map satisfying monotonicity (1) and non-expansion (5) where $\Omega$ satisfies (A) and (B).

In the theorems, we always reiterate the assumptions (1) and (5), even though the conditions are implicitly given by the definition.

Remark 1.

(a) For a linear Markov chain, strict monotonicity (2) implies that $P$ is lazy, which means it will remain in the same state with positive probabilities.

(b) Uniform strict monotonicity (3) is stronger than monotonicity (1) and strict monotonicity (2), which means that (3) implies (1) and (2).

(c) Since $f\leq g+\left\|f-g\right\|_{\infty}$ for every $f$ and $g$ , monotonicity (1) and constant additivity (4) imply the non-expansion condition (5), which is more natural for nonlinear operators.

(d) A linear Markov chain is called irreducible if for every $x,y$ there exists some $n$ such that its kernel $P^{n}(x,y)>0$ , i.e. every state can be reached from every other state. Saying a discrete Markov chain defined on a graph is irreducible is the same as saying the graph is connected, which is crucial to the uniqueness of the stationary distribution. And (6) is a nonlinear version of the connectedness condition. Moreover, note that $P^{n_{0}}$ also satisfies the strict monotonicity (2).

(e) The assumption of a finite accumulation point for $f_{n}$ (8) is weaker than (9), which allows for cases that $P^{n}f$ goes to infinity at all vertices. Moreover, the assumption (8) is necessary. In subsection 2.2, we provide a counterexample demonstrating that $P^{n}f(y)-P^{n}f(x)$ does not converge, even within the interval $\left[-\infty,\infty\right]$ , without the finite assumption (8).

(f) Consider a Markov chain $Q$ with absorbing states and maximal eigenvalue $0<\lambda<1$ , i.e., $Qf=\lambda f$ . Defining $Pf\coloneqq\text{log}Q\left(\text{exp}f\right)$ , we have $P(f+c)=\text{log}Q\left(\text{exp}\left(f+c\right)\right)=c+Pf$ for all constant $c$ . Then $P\text{log}f=\text{log}\lambda+\text{log}f$ , that is, nonlinear Markov chain $P$ exhibits linear growth by the constant $\text{log}\lambda$ .

We now present our main theorems. We remark that in the theorems, we can also replace $\mathbb{R}^{N}$ by $\Omega$ satisfying (A) and (B). We mainly apply Theorem 2 to applications.

Theorem 1.

If a discrete-time nonlinear Markov chain $P:\mathbb{R}^{N}\to\mathbb{R}^{N}$ satisfies

(1) monotonicity,

(2) strict monotonicity,

(5) non-expansion,

(9) $P^{n}f$ has a finite accumulation point $g$ ,

then $Pg=g$ and $P^{n}f\to g$ as $n\to\infty$ .

Then we give the second convergence result.

Theorem 2.

If a discrete-time nonlinear Markov chain $P:\mathbb{R}^{N}\to\mathbb{R}^{N}$ satisfies

(1) monotonicity,

(2) strict monotonicity,

(4) constant additivity,

(8) accumulation point at infinity, i.e., $f_{n}\coloneqq P^{n}f-P^{n}f(x_{0})$ has a finite accumulation point $g$ , then $f_{n}\to g$ as $n\to\infty$ . Moreover, if $P$ also satisfies

(6) connectedness,

then the convergence limit is unique, i.e., for another sequence $\tilde{f_{n}}$ with finite accumulation point $\tilde{g}$ , we have $\underset{n\to\infty}{\text{lim}}\tilde{f}_{n}=\tilde{g}=g=\underset{n\to\infty}{\text{lim}}f_{n}$ .

Next, we give another convergence result.

Theorem 3.

If a discrete-time nonlinear Markov chain $P:\mathbb{R}^{N}\to\mathbb{R}^{N}$ satisfies

(5) non-expansion,

(7) uniform connectedness,

(8) accumulation point at infinity, i.e., $f_{n}\coloneqq P^{n}f-P^{n}f(x_{0})$ has a finite accumulation point $g$ , then $f_{n}\to g$ as $n\to\infty$ and the limit is unique.

Remark 2.

Theorem 1 proves the convergence of nonlinear Markov chains under the assumption of finite accumulation points (9). But Theorem 2 and Theorem 3 include the case of accumulation points at infinity (8), that is, $P^{n}f$ goes to infinity at all vertices. While Theorem 2 needs a stronger constant additivity condition (4), Theorem 3 needs a stronger uniform connectedness condition (7).

The convergence results have important applications in discrete differential geometry which has become a hot research subject in the last decade. Since Ricci curvature is a fundamental concept in Riemannian Geometry [18], various different discrete analogs on graphs [10, 20, 26, 30, 39, 40, 43, 25, 11] have attracted notable interest. Among them, the idea of discrete Ollivier Ricci curvature $\kappa(x,y)=1-\frac{W(\mu_{x},\mu_{y})}{d(x,y)}$ is based on the Wasserstein distance $W$ between probability measures $\mu_{x},\mu_{y}$ which are defined on the one-step neighborhood of vertices $x,y$ [39, 40]. Lin, Lu, and Yau [25] modified this notion to a limit version that is better suited for graphs:

(1.1)

\kappa_{LLY}(x,y)=\underset{\alpha\to 0}{\text{lim}}\frac{\kappa^{\alpha}(x,y)}{1-\alpha}

and $\kappa^{\alpha}(x,y)=1-\frac{W(\mu_{x}^{\alpha},\mu_{y}^{\alpha})}{d(x,y)}$ , where $0\leq\alpha\leq\underset{x}{\text{max }}deg(x)$ , the probability distribution $\mu_{x}^{\alpha}$ assigns amount $1-\alpha\sum_{y\sim x}\frac{w(x,y)}{m(x)}$ at vertex $x$ and amount $\alpha\frac{w(x,z)}{m(x)}$ to all its neighbors $z$ .

Ricci flow as a powerful method can also be applied to discrete geometry and has drawn significant interest recently. Ollivier [39] suggested defining the continuous time Ricci flow by deforming weights. Ni et al. in [38] define a discrete-time Ricci flow. Bai et al. in [1] adopt Lin-Lu-Yau’s Ollivier Ricci curvature [25] to define a continuous-time Ricci flow and obtain several convergence results on path and star graphs. The authors in [13, 12] introduce a discrete uniformization theorem for polyhedral surfaces and obtain a metric with given curvature using a discrete Yamabe flow. The authors in [38] claim good community detection on networks using discrete Ricci flow. The authors in [7] establish a new algorithm to find circle packings by discrete Ricci flow.

The difficulties of studying the long-time behavior of the Ricci flow are the nonlinearity and accumulation points are not unique. In this paper, we consider the discrete Ricci flow in [38] and study its long-time behavior in the following process. Consider a general finite weighted graph $G=(V,E,w,m,d)$ with $deg(x)\leq 1$ for all $x\in V$ . For an initial metric $d_{0}$ , fix some $C$ as the deletion threshold such that $C>\underset{x\sim y\sim z}{\max}\frac{d_{0}(x,y)}{d_{0}(y,z)}$ . Define a discrete time Ollivier Ricci curvature flow by deforming the metric as

(1.2)

d_{n+1}(x,y)\coloneqq d_{n}(x,y)-\alpha\kappa_{d_{n}}(x,y)d_{n}(x,y),\text{ if }x\sim y,

where $\kappa_{d_{n}}$ is the Ollivier Ricci curvature corresponding to $d_{n}$ . After each Ricci flow step, we do a potential edge deletion step. If $x\sim y\sim z$ and $d_{n+1}(x,y)>Cd_{n+1}(y,z)$ , then update the graph by deleting edges $x\sim y$ from $E$ , starting with the longest edge. Let

d_{n+1}(x,y)=\underset{}{\text{inf}}\left\{\stackrel{{\scriptstyle[}}{{i}}=1]{k}{\sum}d_{n}(x_{i-1},x_{i}):x=x_{0}\sim x_{1}\cdots\sim x_{k}=y\right\},\text{ if }x\nsim y.

Since the graph $G$ is finite and the graph of a single edge can not be deleted, denote the new graph as $\tilde{G}$ after the last edge deletion. Then on each connected component of $\tilde{G}$ , the distance ratios are bounded in $n$ , and hence, $\text{log}d_{n}$ has an accumulation point at infinity. Considering the Ricci flow (1.2) as a nonlinear Markov chain on each connected component of $\tilde{G}$ , by Theorem 2 we can prove that (1.2) converges to a constant curvature metric.

Theorem 4.

For an initial metric $d_{0}$ of a finite weighted graph $G=(V,w,m,d)$ with $deg(x)\leq 1$ for all $x\in V$ , by the discrete Ollivier Ricci curvature flow process defined above, then $\frac{d_{n}(e)}{\text{max }d_{n}(e^{\prime})}$ converges to a metric with constant curvature on every connected component of the final graph $\tilde{G}$ , where the max is taken over all $e^{\prime}$ in the same connected component as $e$ on $\tilde{G}$ .

Remark 3.

In [37], Ni, Lin, Gao and Gu presented a framework to endow a graph with a novel metric through the notion of discrete Ricci flow with an application to network alignment. From the experimental results, they found that the graph Ricci curvature converges through the discrete Ricci flow and mentioned a theoretic proof of the observation was still open. Theorem 4 solved this open problem.

For another application, the author in [15, 34] consider a locally finite graph $G=(V,E,w,m,d)$ with a non-negative Ollivier curvature, where $V=X\cup K\cup Y$ , $K$ is finite and $E(X,Y)=\emptyset$ , that is, there are no edges between $X$ and $Y$ . They want to find a function with a constant gradient on $X\cup Y$ , minimal on $X$ and maximal on $Y$ , and the Laplacian of $f$ should be constant on $K$ . By non-negative Ollivier curvature, it will follow that the cut set $K$ separates the Laplacian $\Delta f$ , i.e., $\Delta f\mid_{X}\geq\text{const.}\geq\Delta f\mid_{Y}$ , which is a Laplacian separation principle [34, Theorem 5.1]. The result is crucial for proving an isoperimetric concentration inequality for Markov chains with non-negative Ollivier curvature [34], a discrete Cheeger-Gromoll splitting theorem [15], and a discrete positive mass theorem [16]. Here we prove a natural parabolic flow converging to the solution $f$ . Now we give the details about the Laplacian separation flow. First define an extremal 1-Lipschitz extension operator $S:\mathbb{R}^{K}\to\mathbb{R}^{V}$ ,

Sf(x)\coloneqq\begin{cases}\begin{array}[]{c}f(x):\\ \underset{y\in K}{\text{min}}\left(f(y)+d(x,y)\right):\\ \underset{y\in K}{\text{max}}\left(f(y)-d(x,y)\right):\end{array}&\begin{array}[]{c}x\in K,\\ x\in Y,\\ x\in X.\end{array}\end{cases}

Let $Lip(1,K)\coloneqq{\left\{f\in\mathbb{R}^{K}:f(y)-f(x)\leq d(x,y),\text{for all }x,y\in K\right\}}$ , where $d$ is the graph distance on $G$ . Then $S(Lip(1,K))$ $\subseteq Lip(1,V)$ . In [15], it is proven via elliptic methods that there exists some $g$ with $\Delta Sg=\text{const}$ . Here we give the parabolic flow $(id+\epsilon\Delta)S$ , and show that this converges to the constant Laplacian solution, assuming non-negative Ollivier Ricci curvature.

Theorem 5.

For a locally finite graph $G$ with non-negative Ollivier curvature, let $x_{0}\in V$ and $P\coloneqq\left((id+\epsilon\Delta)S\right)\mid_{K}$ , where $\epsilon$ is small enough such that $\text{diag}(id+\epsilon\Delta)$ is positive. Then for all $f\in Lip(1,K)$ , there exists $g\in Lip(1,K)$ such that $P^{n}f-P^{n}f(x_{0})$ converges to $g$ and $\Delta Sg\equiv\text{const}$ on $K$ .

Then we want to generalize the result to other nonlinear operators, such as the $p$ -Laplace operator, which can be defined as the subdifferential of the energy functional

\mathscr{E}_{p}(f)=\frac{1}{2}\underset{x,y\in V}{\sum}\frac{w(x,y)}{m(x)}\mid\nabla_{xy}f\mid^{p}.

More explicitly, the $p$ -Laplace operator $\Delta_{p}:\mathbb{R}^{V}\to\mathbb{R}^{V}$ is given by

\Delta_{p}f(x)\coloneqq\frac{1}{m(x)}\underset{y}{\sum}w(x,y)\lvert f(y)-f(x)\rvert^{p-2}\left(f(y)-f(x)\right),\text{if }p>1,

and

\Delta_{1}f(x)\coloneqq\left\{\frac{1}{m(x)}\underset{y}{\sum}w(x,y)f_{xy}:f_{xy}=-f_{yx},f_{xy}\in\textrm{sign}(\nabla_{xy}f)\right\}\text{, }

where $\text{sign}\left(t\right)=\begin{cases}\begin{array}[]{c}1,\\ {}[-1,1]\\ -1,\end{array},&\begin{array}[]{c}t>0,\\ t=0,\\ t<0.\end{array}\end{cases}$ And $p=2$ is the general discrete Laplace operator $\Delta$ .

There are two main difficulties. One is that $\Delta_{p}f$ may jump a lot when $\nabla_{xy}f$ is near zero, for example, the derivative of $\Delta_{1}f$ near $\nabla_{xy}f=0$ is large, which causes the operator $id+\epsilon\Delta_{p}$ to fail to maintain the strict monotonicity condition (2). Our idea is to consider its resolvent $J_{\epsilon}=\left(id-\epsilon\Delta_{p}\right)^{-1}$ instead of the flow $id+\epsilon\Delta_{p}$ . The resolvent operator $J_{\epsilon}$ is single-valued and monotone, and we can check that $J_{\epsilon}$ satisfies the strict monotonicity condition in Lemma 3.

Another difficulty is that we need a new curvature condition to guarantee the Lipschitz decay property, which implies compactness, as well as the existence of accumulation points. Define a new curvature on a graph $G=(V,w,m,d_{0})$ with combinatorial distance $d_{0}$

(1.3)

\hat{k}_{p}(x,y)\coloneqq\underset{\pi_{p}}{\text{sup}}\underset{x^{\prime},y^{\prime}\in B_{1}(x)\times B_{1}(y)}{\sum}\pi_{p}(x^{\prime},y^{\prime})\left(1-\frac{d_{0}(x^{\prime},y^{\prime})}{d_{0}(x,y)}\right),

where $\pi_{p}$ satisfies transport plan conditions and we require $\pi_{p}(x^{\prime},y^{\prime})=0$ if $x^{\prime}=y^{\prime}$ (forbid 3-cycles) for $p>2$ , and $\pi_{p}(x^{\prime},y^{\prime})=0$ if $x^{\prime}\neq x$ and $y^{\prime}\neq y$ and $d_{0}(x^{\prime},y^{\prime})=2$ (forbid 5-cycles) for $1\leq p<2$ , see detailed definition (4.1) in subsection 4.2.

Then we can prove there exists a stationary point $g$ such that $g\in\Delta_{p}Sh$ and $g\equiv\text{const}$ on $K$ .

Theorem 6.

For a locally finite graph $G=(V,w,m,d_{0})$ with a non-negative modified curvature $\hat{k}$ , let $x_{0}\in V$ and $P\coloneqq\left((id+\epsilon\Delta)S\right)\mid_{K}$ , where $\epsilon$ is small enough such that $\text{diag}(id+\epsilon\Delta)$ is positive. Then for all $f\in Lip(1,K)$ , there exists $\tilde{f}\in Lip(1,K)$ such that $P^{n}f-P^{n}f(x_{0})$ converges to $\tilde{f}$ . Moreover, there exist $h,g\in\mathbb{R}^{V}$ such that $g\in\Delta_{p}Sh$ and $g\mid_{X}\geq g\mid_{K}\equiv\text{const}\geq g\mid_{Y}$ .

Moreover, our nonlinear Markov chain settings overlap with the nonlinear Dirichlet forms theory and nonlinear Perron–Frobenius theory, and our theorems can be applied well to them. The theory of Dirichlet forms is conceived as an abstract version of the variational theory of harmonic functions. For many application fields, such as Riemannian geometry [18], it is necessary to generalize Dirichlet forms to a nonlinear version. Since the conditions of our theorems fit well in the nonlinear Dirichlet form theory, with additional accumulation points at infinity assumptions we can obtain the convergence by Theorem 2, see Theorem 8. The classical Perron–Frobenius theory concerns the eigenvalues and eigenvectors of nonnegative coefficients matrices and irreducible matrices. In order to apply the theory to a more general setting, there are a lot of studies on the nonlinear Perron–Frobenius theory. After some replacement of maps, our convergence results can also be applied to the nonlinear Perron–Frobenius theory, see Theorem 9.

In Section 5, we introduce a definition of Ollivier Ricci curvature of nonlinear Markov chains according to the Lipschitz decay property, that is, for a nonlinear Markov chain $P$ with (1) monotonicity, (2) strict monotonicity and (4) constant additivity properties, let $d:V^{2}\to[0,+\infty)$ be the distance function. Then for $r\geq 0$ , define

Ric_{r}(P,d)\coloneqq 1-\underset{Lip(f)\leq r}{\text{sup}}\frac{Lip(Pf)}{r},

i.e., if $Lip(f)=r$ , then $Lip(Pf)\leq(1-Ric_{r})Lip(f)$ . Since the nonnegative Ollivier Ricci curvature ensures the existence of accumulation point at infinity (8), then as a corollary of Theorem 2, we can get the convergence results for the nonlinear Markov chain with a nonnegative Ollivier Ricci curvature. And we can also define the Laplacian separation flow of a nonlinear Markov chain with $Ric_{1}(P,d)\geq 0$ . Then we show that the definition is consistent with the classical Ollivier Ricci curvature (3.1), sectional curvature [5], coarse Ricci curvature on hypergraphs [17] and the modified Ollivier Ricci curvature $\hat{k}_{p}$ for $p$ -Laplace (1.3).

2. Convergence and uniqueness of nonlinear Markov chains

2.1. Proofs of main theorems.

In this section, we give proof ideas and specific proofs of our main theorems. First we summarize the proof ideas for Theorem 1. Let $Lf=Pf-f$ and prove $\lambda(P^{n}f)\coloneqq\left\|LP^{n}f\right\|_{\infty}$ is decreasing in $n$ . Since $g$ is the accumulation point, then $\lambda(P^{k}g)=\lambda(g)$ for any $k$ . As $\eta_{+}(Pg)\subseteq\eta(g)\coloneqq\left\{x:Lg(x)=\lambda(g)\right\}$ , there exists some $x$ such that $P^{k}g(x)=g(x)+k\lambda(g)$ . Taking a subsequence $\left\{k_{i}\right\}$ such that $P^{k_{i}}g$ is also a finite accumulation point, implying $\lambda(g)=0$ . Then $g$ is a fixed point and $P^{n}f$ converges. The proof details are as follows.

Proof of Theorem 1.

Define $\lambda(P^{n}f)\coloneqq\parallel P^{n+1}f-P^{n}f\parallel_{\infty}$ , which is decreasing in $n.$ Since $g$ is a finite accumulation point, then

\lambda(g)=\underset{n\to\infty}{\text{lim}}\lambda(P^{n}f)

and

\lambda(P^{k}g)=\underset{n\to\infty}{\text{lim}}\lambda(P^{n+k}f).

Hence $\lambda(g)=\lambda(P^{k}g)$ for any $k$ . Then define $\eta(f)\coloneqq\left\{x:Lf(x)=\lambda(f)\right\}$ . W.l.o.g., suppose

\lambda(P^{k}g)=\lambda_{+}(P^{k}g)\coloneqq\underset{x}{\text{max}}LP^{k}g(x)

and

x\in\eta_{+}(P^{k}g)\coloneqq\left\{x:LP^{k}g(x)=\lambda_{+}(P^{k}g)\right\},

then we claim that $x\in\eta_{+}(P^{k-1}g)$ . If not, we have

LP^{k-1}g(x)<\lambda_{+}(P^{k-1}g)

and

P^{k}g(x)<P^{k-1}g(x)+\lambda_{+}(P^{k-1}g).

By strict monotonicity (2) and non-expansion condition (5), we know

P^{k+1}g(x)<P\left(P^{k-1}g+\lambda_{+}(P^{k-1}g)\right)(x)\leq P^{k}g(x)+\lambda_{+}(P^{k-1}g).

That implies

LP^{k}g(x)<\lambda_{+}(P^{k-1}g)\leq\lambda(P^{k-1}g)=\lambda(P^{k}g)=\lambda_{+}(P^{k}g),

which induces a contradiction. Hence, we have $x\in\eta_{+}(P^{k}g)$ for all $k$ . Thus,

(2.1)

P^{k}g(x)=g(x)+k\lambda(g).

Since $g$ is a finite accumulation point, by taking a subsequence $\left\{k_{i}\right\}$ such that $P^{k_{i}}g$ is still a finite accumulation point, then $\lambda(g)=0$ by (2.1). Hence $P^{k}g=g$ and $P^{n}f\to g$ as $n\to+\infty$ . ∎

For Theorem 2, without the assumption of finite accumulation points, the argument of $\lambda(P^{n}f)$ is not enough. The proof idea is as follows. We define $\lambda_{+}(P^{n}f)\coloneqq\underset{x}{\text{max}}LP^{n}f(x)$ and prove it is decreasing in $n$ . Since $g$ is the accumulation point, then $\lambda_{+}(P^{k}g)=\lambda_{+}(g)$ for any $k$ . By the conclusion

\eta_{+}(Pg)\subseteq\eta_{+}(g)\coloneqq\left\{x:Lg(x)=\lambda_{+}(g)\right\},

there exists some $x$ such that $LP^{k}g(x)$ attains the maximum, i.e. $P^{k}g(x)=g(x)+k\lambda_{+}(g)$ . And there also exists some $y$ attaining its minimum, i.e. $P^{k}g(y)=g(y)+k\lambda_{-}(g)$ . Taking a subsequence $\left\{k_{i}\right\}$ such that $P^{k_{i}}g$ is also a finite accumulation point, implying $\lambda_{+}(g)=\lambda_{-}(g)$ , that is, $Lg=const$ . Hence $P^{k}g=g+k\lambda_{+}$ , and $g$ is the limit of $f_{n}$ . For the uniqueness, we prove that the linear growth rate $\lambda_{+}$ of different sequences $f_{n}$ is the same by the non-expansion property. Then by the connectedness (6) we know that all accumulation points are the same. The proof details are as follows.

Proof of Theorem 2.

Define $Lf\coloneqq Pf-f$ and $\lambda_{+}(f)\coloneqq\underset{x}{\text{max}}Lf(x)$ , then

Pf\leq f+\lambda_{+}(f).

By monotonicity (1) and constant additivity (4), we have

P^{2}f\leq P(f+\lambda_{+}(f))=Pf+\lambda_{+}(f).

Hence $\lambda_{+}(Pf)\leq\lambda_{+}(f)$ , that is, $\lambda_{+}(P^{n}f)$ is decreasing in $n$ . Since $g$ is a finite accumulation point for $f_{n}$ ,

\lambda_{+}(g)=\underset{n\to\infty}{\text{lim}}\lambda_{+}(P^{n}f).

For any $k$ , since $P^{k}g$ is an accumulation point for $P^{k}f_{n}$ , and $LP^{k}f_{n}=LP^{n+k}f$ by constant additivity (4), we have

\lambda_{+}(P^{k}g)=\underset{n\to\infty}{\text{lim}}\lambda_{+}(P^{n+k}f).

Then $\lambda_{+}(g)=\lambda_{+}(P^{k}g).$

Defining the maximum points set as $\eta_{+}(f)\coloneqq\left\{x:Lf(x)=\lambda_{+}(f)\right\}$ , we claim that $\eta_{+}(Pf)\subseteq\eta_{+}(f)$ if $\lambda_{+}(f)=\lambda_{+}(P^{k}f).$

If $Lf(x)<\lambda_{+}(f)$ , i.e. $Pf(x)<f(x)+\lambda_{+}(f)$ , since $Pf\leq f+\lambda_{+}(f)$ , then by strict monotonicity (2) and constant additivity (4),

P^{2}f(x)<P(f+\lambda_{+}(f))(x)=Pf(x)+\lambda_{+}(f).

That is, we have $LPf(x)<\lambda_{+}(f)=\lambda_{+}(Pf)$ . Hence, $\eta_{+}(Pf)\subseteq\eta_{+}(f)$ .

Then there exists some $x$ such that $x\in\eta_{+}(P^{k}f)$ for any $k$ , that is,

P^{k}g(x)=g(x)+k\lambda_{+}(g)

By the same argument, for $\lambda_{-}(f)\coloneqq\underset{x}{\text{min}}Lf(x)$ , there is $y$ such that

P^{k}g(y)=g(y)+k\lambda_{-}(g).

Since $g$ is a finite accumulation point, then there is a subsequence $\left\{n_{k}\right\}$ such that $f_{n_{k}}\to g$ . Taking a subsequence $\left\{k_{i}\right\}$ with $\left\{n_{k}+k_{i}\right\}$ and $\left\{n_{k}\right\}$ coincide, then $P^{k_{i}}g(x)-P^{k_{i}}g(y)$ must be finite, which implies $\lambda_{+}(g)=\lambda_{-}(g)$ and $Lg=Pg-g\equiv\lambda_{+}$ . Then we get $P^{n}g=g+n\lambda_{+}$ and $P^{n}g-P^{n}g(x_{0})=g-g(x_{0})$ . Since $g$ (resp. $P^{k}g$ ) is an accumulation point for $f_{n}$ (resp. $P^{k}f_{n}$ ), for any $\varepsilon$ , there exists some $n$ such that

\|P^{n}f-P^{n}f(x_{0})-g\|_{\infty}<\varepsilon

and

\|P^{n+k}f-P^{n}f(x_{0})-P^{k}g\|_{\infty}<\varepsilon.

Replacing $P^{k}g$ by $g+k\lambda_{+}$ , we have

\|P^{n+k}f-P^{n}f(x_{0})-g-k\lambda_{+}\|_{\infty}<\varepsilon.

Since $g(x_{0})=0,$ we know $\mid P^{n+k}f(x_{0})-P^{n}f(x_{0})-k\lambda_{+}\mid<\varepsilon.$ Then

		$\displaystyle\\|f_{n+k}-g\\|_{\infty}$
	$\displaystyle\leq$	$\displaystyle\\|P^{n+k}f-P^{n}f(x_{0})-g-k\lambda_{+}\\|_{\infty}+\mid P^{n+k}f(x_{0})-P^{n}f(x_{0})-k\lambda_{+}\mid$
	$\displaystyle<$	$\displaystyle 2\varepsilon,$

which means $f_{n}$ converges to $g$ as $n\to\infty$ .

Then we want to prove the uniqueness with the connectedness assumption (6). By the above argument, for any finite accumulation point $g$ , we know $Pg-g=\text{const}.$ We claim that the constants of any accumulation points are the same. If not, then there exist $c_{1}\neq c_{2}$ such that $P^{n}g^{1}=g^{1}+nc_{1}$ and $P^{n}g^{2}=g^{2}+nc_{2}$ . Hence $\parallel P^{n}g^{1}-P^{n}g^{2}\parallel_{\infty}\to\infty$ , which contradicts to the non-expansion property (5), i.e. $\parallel Pg^{1}-Pg^{2}\parallel_{\infty}\leq\parallel g^{1}-g^{2}\parallel_{\infty}$ . Then we proved the claim.

Next if $g^{1}$ and $g^{2}$ are two different accumulation points, by adding a constant, w.l.o.g., assume $g^{1}\geq g^{2}$ with $g^{1}(y)=g^{2}(y)$ and $g^{1}(x)>g^{2}(x)$ . By the connectedness condition (6), we get $P^{n}g^{1}(y)>P^{n}g^{2}(y)$ , that is, $g^{1}(y)+nc>g^{2}(y)+nc$ which contradicts to $g^{1}(y)=g^{2}(y)$ . Hence all accumulation points are the same, which shows the uniqueness of limits. ∎

For Theorem 3, we remind that there is a naive but fatal idea of considering $\tilde{P}f\coloneqq Pf-Pf(x_{0})$ , which has a finite accumulation point, but may lack the required monotonicity (1) and non-expansion (5) properties.

Our proof idea is as follows. Lemma 1 states that a sequence ${\left\{x_{n}\right\}}$ converges if it has exactly one accumulation point and satisfies $d(x_{n},x_{n+1})\leq C$ for all $n$ . Then we prove Theorem 3 by proving the uniqueness of accumulation points. Let $\tilde{P}=P^{n_{0}}$ , satisfying non-expansion (5) and uniform connectedness (7) with $n_{0}=1$ . Suppose there is a subsequence $\left\{n_{k}=m_{k}n_{0}+i\right\}$ with $0\leq i\leq n_{0}-1$ such that $f_{n_{k}}\to g$ and $P^{n_{k}}f(x_{0})=\tilde{P}^{m_{k}}P^{i}f(x_{0})\to a\in[-\infty,+\infty]$ as $k\to+\infty$ . Divide it into two cases.

In the case of $\mid a\mid<+\infty$ , since $\tilde{P}^{m}P^{i}f$ has a finite accumulation point $\tilde{g}$ , then $\tilde{P}\tilde{g}=\tilde{g}$ by Theorem 1. And accumulation points are the same after adding a constant by the uniform connectedness of $\tilde{P}$ , which implies the uniqueness of accumulation points $\tilde{g}-\tilde{g}(x_{0})$ for $f_{n}=P^{n}f-P^{n}f(x_{0})$ .

In the case of $\mid a\mid=+\infty$ , for example, $a=+\infty$ , define $Qf\coloneqq\underset{r\to+\infty}{\text{lim}}\tilde{P}(f+r)-r$ satisfying non-expansion (5), uniform connectedness (7) with $n_{0}=1$ and constant additivity (4). Then $Q^{k}g$ is of constant linear growth of $k$ , i.e., $Q^{k}g=g+kc$ . Then the constant $c$ of different accumulation points are the same by the non-expansion property. And the uniqueness of accumulation points follows from the uniform connectedness of $Q$ .

First, we give a convergence lemma by the uniqueness of accumulation points.

Lemma 1.

Let $(X,d)$ be a locally compact metric space, if a sequence $\left\{x_{n}\right\}$ with exactly one accumulation point satisfying $d(x_{k},x_{k+1})\leq C$ for some fixed constant $C$ and all $k\in\mathbb{N}_{+}$ , then $\left\{x_{n}\right\}$ converges.

Proof.

Let $x_{0}$ be the accumulation point. Prove by contradiction. For some positive $\epsilon$ , suppose that there is a subsequence $\left\{x_{n_{k}}\right\}$ such that $d(x_{n_{k}},x_{0})\geq\epsilon$ . For $n_{k},$ let $m_{k}\geq n_{k}$ be the smallest number such that $d(x_{m_{k}+1},x_{0})<\epsilon$ , then $d(x_{m_{k}},x_{0})\in[\epsilon,\epsilon+C]$ . Hence $\left\{x_{m_{k}}\right\}$ has an accumulation point different from $x_{0}$ as the local compactness, which contradicts to the uniqueness of accumulation points. ∎

Remark 4.

We give a specific example to illustrate that the non-expansion

d(x_{k},x_{k+1})\leq C

is necessary. For $X=\mathbb{R}$ , let $x_{2k}=k$ , $x_{2k+1}=0$ , which has exactly one accumulation point but does not satisfy $d(x_{k},x_{k+1})\leq C$ for all $k$ , and $\left\{x_{n}\right\}$ does not converge.

Next, we prove Theorem 3 by the uniqueness of accumulation points via the uniform connectedness (7). Recall that if $P$ is uniformly connected, then there exists $n_{0}\in\mathbb{N}_{+}$ , positive $\epsilon_{0}$ such that for some point $x,$ positive $\delta$ and $f,g\in\mathbb{R}^{N}$ with $f\geq g+\delta\cdot 1_{x}$ , we have $P^{n_{0}}f\geq P^{n_{0}}g+\epsilon_{0}\delta$ .

Proof of Theorem 3.

Let $\tilde{P}=P^{n_{0}}$ , then $\tilde{P}$ is uniformly connected with $n_{0}=1$ , implying strict monotonicity (2). As $g$ is an accumulation point, then there is $0\leq i\leq n_{0}-1$ and a subsequence $\left\{n_{k}=m_{k}n_{0}+i\right\}$ such that $f_{n_{k}}\to g$ and $P^{n_{k}}f(x_{0})\to a\in[-\infty,+\infty]$ as $k\to+\infty$ . Divide it into two cases: $\mid a\mid<+\infty$ and $\mid a\mid=+\infty$ .

Case 1. If $\mid a\mid<+\infty$ , w.l.o.g, suppose $a>0$ .

Then $P^{n_{k}}f=\tilde{P}^{m_{k}}P^{i}f\eqqcolon\tilde{P}^{m_{k}}F\to g+a\eqqcolon\tilde{g}$ as $k\to+\infty$ . By Theorem 1, we have $\tilde{P}^{m}F\to\tilde{g}$ as $m\to+\infty$ and $\tilde{P}^{k}\tilde{g}=\tilde{g}$ for all $k$ . Then for different accumulation points of $f_{n}$ , by the non-expansion property (5), their corresponding $a$ must be the same case.

If there are two accumulation points $\tilde{g}_{1}\neq\tilde{g}_{2}$ , suppose $\alpha\coloneqq\underset{x}{\text{max}}\left(\tilde{g}_{1}(x)-\tilde{g}_{2}(x)\right)>0$ . Then $\tilde{g}_{1}\leq\tilde{g}_{2}+\alpha$ . If there exists $x$ such that $\tilde{g}_{1}(x)<\tilde{g}_{2}(x)+\alpha$ , then by the uniform connectedness and non-expansion property of $\tilde{P}$ ,

\tilde{g}_{1}=\tilde{P}\tilde{g}_{1}<\tilde{P}\left(\tilde{g}_{2}+\alpha\right)\leq\tilde{P}\tilde{g}_{2}+\alpha=\tilde{g}_{2}+\alpha,

which implies $\tilde{g}_{1}<\tilde{g}_{2}+\alpha$ and contradicts to the definition of $\alpha$ . Hence $\tilde{g}_{1}=\tilde{g}_{2}+\alpha$ , which means $\tilde{g}_{1}-\tilde{g}_{1}(x_{0})=\tilde{g}_{2}-\tilde{g}_{2}(x_{0})$ . Then the two accumulation points $g_{i}=\tilde{g}_{i}-\tilde{g}_{i}(x_{0})$ for $i=1,2$ of $f_{n}$ are the same. By Lemma 1, we get the convergence of $f_{n}$ .

Case 2. If $\mid a\mid=+\infty$ , w.l.o.g., suppose $a=+\infty$ .

Since $\tilde{P}(f+r)-r$ is decreasing for positive $r$ by the non-expansion condition (5), we define

Qf\coloneqq\underset{r\to+\infty}{\text{lim}}\left(\tilde{P}(f+r)-r\right).

As $Qg-g=\underset{k\to+\infty}{\text{lim}}\left(\tilde{P}^{m_{k}+1}F-\tilde{P}^{m_{k}}F\right)$ is finite by the non-expansion property (5) of $\tilde{P}$ , then for any $f$ , we know $Qf$ is also finite by the non-expansion property. Then $Q$ satisfies the non-expansion property (5) and constant additivity $Q(f+c)=Qf+c$ .

Now we show the connectedness of $Q$ . For some point $x$ , positive $\delta$ and $f,h\in\mathbb{R}^{N}$ with $f\geq h+\delta\cdot 1_{x}$ , since $\tilde{P}$ is uniformly connected with $n_{0}=1$ , we have

\tilde{P}\left(f+r\right)-r>\tilde{P}\left(h+r\right)-r+\epsilon_{0}\delta\geq Qh+\epsilon_{0}\delta.

Take $r\to+\infty$ , then $Qf>Qh+\epsilon_{0}\delta$ and $Q$ is uniformly connected with $n_{0}=1$ and $\epsilon_{0}$ .

Let $\lambda_{+}^{\tilde{P}}(f)\coloneqq\text{max}\left(\tilde{P}f-f\right)$ and $F=P^{i}f$ , then $\lambda_{+}^{\tilde{P}}(\tilde{P}^{m}F)>0$ for all $m$ since $\tilde{P}^{m_{k}}F(x_{0})\to+\infty$ . By the monotonicity and non-expansion property,

\tilde{P}^{m+2}F\leq\tilde{P}\left(\tilde{P}^{m}F+\lambda_{+}^{\tilde{P}}(\tilde{P}^{m}F)\right)\leq\tilde{P}^{m+1}F+\lambda_{+}^{\tilde{P}}(\tilde{P}^{m}F),

which means $\lambda_{+}^{\tilde{P}}(\tilde{P}^{m}F)$ is decreasing in $m$ . Since $g$ is a finite accumulation point, for all $l\in\mathbb{N}_{+}$ , we have

Qg-g=\underset{k\to+\infty}{\text{lim}}\left(\tilde{P}^{m_{k}+1}F-\tilde{P}^{m_{k}}F\right)

and

Q^{l+1}g-Q^{l}g=\underset{k\to+\infty}{\text{lim}}\left(\tilde{P}^{m_{k}+l+1}F-\tilde{P}^{m_{k}+l}F\right).

Then by the monotonicity of $\lambda_{+}^{\tilde{P}}(\tilde{P}^{m}F)$ , we have

\lambda_{+}^{Q}(g)=\text{max}\left(Qg-g\right)=\underset{m\to+\infty}{\text{lim}}\lambda_{+}^{\tilde{P}}(\tilde{P}^{m}F)=\lambda_{+}^{Q}(Q^{l}g).

Since $Qg\leq g+\lambda_{+}^{Q}(g)$ , if there is $x$ such that $Qg(x)<g(x)+\lambda_{+}^{Q}(g)$ , then

Q^{2}g<Qg+\lambda_{+}^{Q}(g)=Qg+\lambda_{+}^{Q}(Qg)

by the uniform connectedness and constant additivity of $Q$ , which contradicts to the definition of $\lambda_{+}^{Q}$ . Hence $Qg=g+\lambda_{+}^{Q}(g)$ and $Q^{l}g=g+l\lambda_{+}^{Q}(g)$ .

Let $f^{(1)},f^{(2)}\in\mathbb{R}^{N}$ and assume $g_{i}$ is an accumulation point of $f_{n}^{(i)}=P^{n}f^{(i)}-P^{n}f^{(i)}(x_{0})$ for $i=1,2$ . Then by the non-expansion property

\left\|g_{1}+l\lambda_{+}^{Q}(g_{1})-g_{2}-l\lambda_{+}^{Q}(g_{2})\right\|_{\infty}=\left\|Q^{l}g_{1}-Q^{l}g_{2}\right\|_{\infty}\leq\left\|g_{1}-g_{2}\right\|_{\infty},

we know $\lambda_{+}^{Q}(g_{1})=\lambda_{+}^{Q}(g_{2})$ . If $g_{1}\neq g_{2}$ , suppose $\alpha\coloneqq\underset{x}{\text{max}}\left(g_{1}(x)-g_{2}(x)\right)>0$ . Then $g_{1}\leq g_{2}+\alpha$ and $0=g_{1}(x_{0})<g_{2}(x_{0})+\alpha=\alpha$ . By the uniform connectedness of $Q$ ,

g_{1}+\lambda_{+}^{Q}(g_{1})=Qg_{1}<Qg_{2}+\alpha=g_{2}+\lambda_{+}^{Q}(g_{2})+\alpha,

which implies $g_{1}<g_{2}+\alpha$ and contradicts to the definition of $\alpha$ . Then the two accumulation points are the same.

By Lemma 1 and letting $f^{(1)}=f^{(2)}=f$ , we get the convergence of $f_{n}$ . Moreover, the case $f^{(1)}\neq f^{(2)}$ shows the uniqueness of the limit. ∎

2.2. An example of non-convergence.

Note that the condition of accumulation points at infinity (8) is necessary for the convergence. Next, we give a concrete example, which shows that without the condition of accumulation points (8), then $P^{n}f(y)-P^{n}f(x)$ does not converge even in $\left[-\infty,\infty\right]$ . First we prove an extension lemma.

Lemma 2.

Suppose that a closed subspace $\Omega\subseteq\mathbb{R}^{N}$ satisfies $a+c\in\Omega$ for any $a\in\Omega$ and $c\in\mathbb{R}$ . If $P:\Omega\to\Omega$ satisfies uniform strict monotonicity (3) and constant additivity (4), then $P$ can be extended to $\mathbb{R}^{N}$ with conditions (3) and (4).

Proof.

For any $f\in\mathbb{R}^{N}$ , define the extension of $P$ as

\bar{P}f\coloneqq\text{inf}\left\{Pg-\epsilon(g-f):g\in\Omega,g\geq f\right\},

where $\epsilon$ is defined in uniform strict monotonicity (3). Then $\bar{P}$ satisfies constant additivity (4). If $f_{1}\geq f_{2},$ we know

	$\displaystyle\bar{P}f_{1}$	$\displaystyle=\inf\left\{Pg-\epsilon(g-f_{1}):g\in\Omega,g\geq f_{1}\right\}$
		$\displaystyle\geq\inf\left\{Pg-\epsilon(g-f_{1}):g\in\Omega,g\geq f_{2}\right\}$
		$\displaystyle=\bar{P}f_{2}+\epsilon(f_{1}-f_{2}).$

Then $\bar{P}$ satisfies uniform strict monotonicity (3). Particularly, $\bar{P}f>-\infty$ for all $f$ . ∎

Now we give a counterexample. Let

\Omega_{e}\coloneqq\left\{\left(n,-n,-\varepsilon_{0},\varepsilon_{0}\right)+\vec{c}\in\mathbb{R}^{4},n\text{ is even},\vec{c}=(c,c,c,c)\in\mathbb{R}^{4},\varepsilon_{0}\ll 1\right\},

\Omega_{o}\coloneqq\left\{\left(n,-n,\varepsilon_{0},-\varepsilon_{0}\right)+\vec{c}\in\mathbb{R}^{4},n\text{ is odd},\vec{c}=(c,c,c,c)\in\mathbb{R}^{4},\varepsilon_{0}\ll 1\right\},

and define $P:\Omega\coloneqq\Omega_{e}\cup\Omega_{o}\to\Omega$ as

\begin{array}[]{cc}P\left(\left(n,-n,-\varepsilon_{0},\varepsilon_{0}\right)+\vec{c}\right)=(n+1,-n-1,\varepsilon_{0},-\varepsilon_{0})+\vec{c},&\text{if }n\text{ is even},\\ P\left(\left(n,-n,\varepsilon_{0},-\varepsilon_{0}\right)+\vec{c}\right)=(n+1,-n-1,-\varepsilon_{0},\varepsilon_{0})+\vec{c},&\text{if }n\text{ is odd}.\end{array}

Then we want to show that $P$ satisfies uniform strict monotonicity (3). Suppose

f=(n,-n,\pm\varepsilon_{0},\mp\varepsilon_{0})+\vec{c_{1}},

g=(m,-m,\pm\varepsilon_{0},\mp\varepsilon_{0})+\overrightarrow{\left(c_{1}-\mid n-m\mid\right)},

then $f\geq g$ . And

	$\displaystyle Pf-Pg-\frac{f-g}{2}$	$\displaystyle\geq\frac{1}{2}\left[\left(n-m,m-n,-2\varepsilon_{0},-2\varepsilon_{0}\right)+\overrightarrow{\mid n-m\mid}\right]$
		$\displaystyle\geq\left(0,0,0,0\right),$

which means that $P$ satisfies uniform strict monotonicity (3) with $\epsilon=\frac{1}{2}$ . By Lemma 2, one can extend it to $\mathbb{R}^{N}$ with uniform strict monotonicity (3) and constant additivity (4). But $P^{n}f(x_{3})-P^{n}f(x_{4})$ always jumps between two values $\pm 2\varepsilon_{0}$ and does not converge.

Figure 1. This figure shows an example of non-convergence.

3. Basic facts of graphs

The main applications of our theorems are parabolic equations on graphs. Now we give a overview of weighted graphs.

3.1. Weighted graphs

A weighted graph $G=(V,w,m,d)$ consists of a countable set $V$ , a symmetric function $w:V\times V\to[0,+\infty)$ called edge weight with $w=0$ on the diagonal, and a function $m:V\to\left(0,+\infty\right)$ called vertex measure or volume. The edge weight $w$ induces a symmetric edge relation $E=\left\{\left(x,y\right):w(x,y)>0\right\}$ . We write $x\sim y$ if $w(x,y)>0$ . In the following, we only consider locally finite graphs, i.e., for every $x\in V$ there are only finitely many $y\in V$ with $w(x,y)>0$ . The degree at $x$ defined as $deg(x)=\underset{y\sim x}{\mathop{\sum}}\frac{w(x,y)}{m(x)}$ . We say a metric $d:V^{2}\rightarrow[0,+\infty)$ is a path metric on a graph $G$ if

d(x,y)=\inf\left\{\stackrel{{\scriptstyle[}}{{i}}=1]{n}{\sum}d(x_{i-1},x_{i}):x=x_{0}\sim\ldots\sim x_{n}=y\right\}.

By assigning each edge of length one, we get the combinatorial distance $d_{0}$ . For a function $f$ defined on vertex set $V,$ denoted as $f\in\mathbb{R}^{V},$ define the difference operator for any $x\sim y$ as

\nabla_{xy}f=f(y)-f(x).

The Laplace operator is defined as

\Delta f(x)\coloneqq\frac{1}{m(x)}\underset{y}{\sum}w(x,y)\nabla_{xy}f.

For $f\in\mathbb{R}^{V}$ , we write $\left\|f\right\|_{\infty}\coloneqq\sup_{x\in V}\mid f(x)\mid$ and $\left\|f\right\|_{2}\coloneqq\underset{x,y}{\sum}\frac{w(x,y)}{m(x)}f(x)^{2}$ .

3.2. Discrete Ricci curvature and curvature flow

Let $G=(V,w,m,d)$ be a locally finite graph with path metric $d$ . A probability distribution over the vertex set $V$ is a mapping $\mu:V\to[0,1]$ satisfying $\sum_{x\in V}\mu(x)=1$ . Suppose that two probability distributions $\mu_{1}$ and $\mu_{2}$ have a finite support. A coupling between $\mu_{1}$ and $\mu_{2}$ is a mapping $\pi:V\times V\to[0,1]$ with finite support such that

\underset{y\in V}{\sum}\pi(x,y)=\mu_{1}(x)\text{ and }\underset{x\in V}{\sum}\pi(x,y)=\mu_{2}(y).

The Wasserstein distance between two probability distributions $\mu_{1}$ and $\mu_{2}$ is defined as

W(\mu_{1},\mu_{2})=\underset{\pi}{\inf}\underset{x,y\in V}{\sum}\pi(x,y)d(x,y),

where the infimum is taken over all coupling $\pi$ between $\mu_{1}$ and $\mu_{2}$ .

For every $x,y\in V$ , the Ollivier Ricci curvature $\kappa(x,y)$ is defined to be

(3.1)

\kappa(x,y)=1-\frac{W(\mu_{x},\mu_{y})}{d(x,y)}.

where

\mu_{x}(z)=\begin{cases}\begin{array}[]{c}\frac{w(x,z)}{m(x)}\\ 0\end{array}&\begin{array}[]{c}:z\sim x,\\ :\text{otherwise}.\end{array}\end{cases}

In this paper, we consider the discrete Ollivier Ricci flow and study its long-time behavior in the following process. Consider a general finite weighted graph $G=(V,w,m,d)$ . For an initial metric $d_{0}$ , fix some $C$ as the deletion threshold such that $C>\underset{x\sim y\sim z}{\max}\frac{d_{0}(x,y)}{d_{0}(y,z)}$ . Define a discrete time Ollivier Ricci curvature flow by deforming the metric as

d_{n+1}(x,y)\coloneqq d_{n}(x,y)-\alpha\kappa_{d_{n}}(x,y)d_{n}(x,y),\text{ if }x\sim y,

d_{n+1}(x,y)=\underset{}{\text{inf}}\left\{\stackrel{{\scriptstyle[}}{{i}}=1]{k}{\sum}d_{n}(x_{i-1},x_{i}):x=x_{0}\sim x_{1}\cdots\sim x_{k}=y\right\},\text{ if }x\nsim y.

Since the graph $G$ is finite and graphs of a single edge can not be deleted, denote the new graph as $\tilde{G}$ after the last edge deletion. Then on each connected component of $\tilde{G}$ , the distance ratios are bounded in $n$ , and hence, $\text{log}d_{n}$ has an accumulation point at infinity. Considering Ricci flow (1.2) as a nonlinear Markov chain on each connected component of $\tilde{G}$ , by Theorem 2 we can prove that (1.2) converges to a constant curvature metric.

3.3. Nonlinear Laplace and resolvent operators

On a locally finite weighted graph $G=(V,w,m,d)$ , for every $p\geq 1$ , define the energy functional of $f\in\mathbb{R}^{V}$ as

\mathscr{E}_{p}(f)=\frac{1}{2}\underset{x,y\in V}{\sum}\frac{w(x,y)}{m(x)}\mid\nabla_{xy}f\mid^{p},

see reference [6, 27]. The $p$ -Laplace operator $\Delta_{p}$ is the subdifferential of energy functional $\mathscr{E}_{p}$ , defined as $\Delta_{p}:\mathbb{R}^{V}\to\mathbb{R}^{V}$

\Delta_{p}f(x)\coloneqq\frac{1}{m(x)}\underset{y}{\sum}w(x,y)\lvert f(y)-f(x)\rvert^{p-2}\left(f(y)-f(x)\right),\text{if }p>1,

and

\Delta_{1}f(x)\coloneqq\left\{\frac{1}{m(x)}\underset{y}{\sum}w(x,y)f_{xy}:f_{xy}=-f_{yx},f_{xy}\in\textrm{sign}(\nabla_{xy}f)\right\}\text{, }

where $sign\left(t\right)=\begin{cases}\begin{array}[]{c}1,\\ {}[-1,1]\\ -1,\end{array},&\begin{array}[]{c}t>0,\\ t=0,\\ t<0.\end{array}\end{cases}$ And $p=2$ is the general discrete Laplace operator $\Delta$ .

The resolvent operator of $\Delta_{p}$ is defined as $J_{\epsilon}=\left(id-\epsilon\Delta_{p}\right)^{-1}$ for $\epsilon>0$ . Since $-\Delta_{p}$ is a monotone operator, i.e., for every $f,g\in\mathbb{R}^{V}$ , we have

\left\langle-\Delta_{p}f+\Delta_{p}g,f-g\right\rangle\geq 0,

then the resolvent $J_{\epsilon}$ is single-valued, monotone, and non-expansion, i.e.,

\parallel J_{\epsilon}f-J_{\epsilon}g\parallel_{\infty}\leq\parallel f-g\parallel_{\infty}.

See details [31, Corollary 2.10] [41, Proposition 12.19]. Moreover, since $\Delta_{p}$ is the subdifferential of convex function $\mathscr{E}_{p}$ , it follows that

J_{\epsilon}f=\underset{g\in\mathbb{R}^{V}}{\text{argmin}}\left\{\mathscr{E}_{p}(g)+\frac{1}{2\epsilon}\left\|g-f\right\|_{2}^{2}\right\}.

3.4. Extremal Lipschitz extension

To study the Laplacian separation flow from the introduction, we give the definition of extremal Lipschitz extension on a specific graph structure. Consider a locally finite graph $G=(V,w,m,d)$ with non-negative Ollivier curvature, where $V=X\sqcup K\sqcup Y$ and $K$ is finite and $E(X,Y)=\emptyset$ . Set $Lip(1,K):=\left\{f\in\mathbb{R}^{K}:\nabla_{xy}f\leq d(x,y)\text{ for all }x,y\in K\right\}$ and define an extremal Lipschitz extension operator $S:\mathbb{R}^{K}\to\mathbb{R}^{V}$ as

Sf(x)\coloneqq\begin{cases}\begin{array}[]{c}f(x):\\ \underset{y\in K}{\text{min}}\left(f(y)+d(x,y)\right):\\ \underset{y\in K}{\text{max}}\left(f(y)-d(x,y)\right):\end{array}&\begin{array}[]{c}x\in K,\\ x\in Y,\\ x\in X,\end{array}\end{cases}

where $d$ is the graph distance on $G$ . Then $S(Lip(1,K))\subseteq Lip(1,V)$ .

4. Applications

In this section, we prove that our convergence and uniqueness results have important applications in the Ollivier Ricci curvature flow, the Laplacian separation flow, the nonlinear Dirichlet form and nonlinear Perron–Frobenius theory.

4.1. The convergence of Ollivier Ricci curvature flow.

Recall the convergence process of the discrete Ollivier Ricci flow. On a general finite weighted graph $G=(V,w,m,d)$ with $deg(x)\leq 1$ for all $x\in V$ . For an initial metric $d_{0}$ , fix some $C$ as the deletion threshold such that $C>\underset{x\sim y\sim z}{\max}\frac{d_{0}(x,y)}{d_{0}(y,z)}$ . For $0<\alpha<1$ , define a discrete time Ollivier Ricci curvature flow by deforming the metric as

d_{n+1}(x,y)\coloneqq d_{n}(x,y)-\alpha\kappa_{d_{n}}(x,y)d_{n}(x,y),\text{ if }x\sim y,

d_{n+1}(x,y)=\underset{}{\text{inf}}\left\{\stackrel{{\scriptstyle[}}{{i}}=1]{k}{\sum}d_{n}(x_{i-1},x_{i}):x=x_{0}\sim x_{1}\cdots\sim x_{k}=y\right\},\text{ if }x\nsim y.

Since the graph $G$ is finite and graphs of a single edge can not be deleted, denote the new graph as $\tilde{G}$ after the last edge deletion. Then on each connected component of $\tilde{G}$ , the distance ratios are bounded in $n$ , and hence, $\text{log}d_{n}$ has an accumulation point at infinity. Considering the Ricci flow as a nonlinear Markov chain on each connected component of $\tilde{G}$ , by Theorem 2 we can prove that (1.2) converges to a constant curvature metric.

Theorem 4. For an initial metric $d_{0}$ of a finite weighted graph $G=(V,w,m,d)$ with $deg(x)\leq 1$ for all $x\in V$ , by the discrete Ollivier Ricci curvature flow process defined above, then $\frac{d_{n}(e)}{\text{max }d_{n}(e^{\prime})}$ converges to a metric with constant curvature on every connected component of the final graph $\tilde{G}$ , where the max is taken over all $e^{\prime}$ in the same connected component as $e$ on $\tilde{G}$ .

Proof.

For $f\in\mathbb{R}_{+}^{E}$ , define $Sf\in\mathbb{R}^{V\times V}$ as

Sf(x,y)=\underset{}{\text{inf}}\left\{\stackrel{{\scriptstyle[}}{{i}}=1]{k}{\sum}f(x_{i-1},x_{i}):x=x_{0}\sim x_{1}\cdots\sim x_{k}=y\right\}.

Note that $Sf$ is a distance on $V$ . By (1.2), for $f\in\mathbb{R}^{E}$ , define

\tilde{P}f(x,y)\coloneqq\alpha W_{Sf}(p(x,\cdot),p(y,\cdot))+(1-\alpha)f(x,y),

where $W_{Sf}$ is the Wasserstein distance corresponding to the distance $Sf$ and $p$ is the Markov kernel. Then $\tilde{P}$ corresponds to one step of the Ollivier Ricci flow, that is,

d_{n+1}\mid_{E}=\tilde{P}(d_{n}\mid_{E}).

Clearly, $\tilde{P}$ satisfies monotonicity (1) and strict monotonicity (2). Since

\tilde{P}(rd)=r\tilde{P}d,\>\forall r>0,

define $Pf\coloneqq\text{log $\tilde{P}$(exp$(f)$)}$ with $f=\text{log}\,d$ . Then for every constant $C\in\mathbb{R}$ ,

P(f+C)=\text{log}(\tilde{P}\left(\text{exp}f\cdot\text{exp}C\right))=\text{log}(\text{exp}C\tilde{\,P}(\text{exp}f))=C+Pf,

which implies that $P$ satisfies constant additivity (4). Since $P$ also satisfies monotonicity (1) and strict monotonicity (2), and as $\frac{d_{n}}{d_{n}(e_{0})}$ has a finite positive accumulation point $d$ on the connected component $\tilde{G}$ containing $e_{0}$ , we obtain that $g=\text{log}\,d$ is a finite accumulation point of $P^{n}f-P^{n}f(e_{0})$ . Then by Theorem 2, we know that $P^{n}f-P^{n}f(e_{0})$ converges to $g$ . Moreover, we know $Pg=g+c$ and $\tilde{P}d=\tilde{c}d$ . That is, its curvature is a constant. ∎

4.2. The gradient estimate for resolvents of nonlinear Laplace.

It is shown in [35] that a lower Ollivier curvature bound is equivalent to a gradient estimate for the continuous time heat equation. In [24, 32, 33, 9, 45, 14], the gradient estimates have been proved under Bakry-Emery curvature bounds. In [5], the authors proved that non-negative sectional curvature implies a logarithmic gradient estimate. Gradient estimates of the discrete random walk $id+\varepsilon\Delta$ have been proved in [3, 25, 15]. In [17, Theorem 5.2.], the authors show a gradient estimate for the coarse Ricci curvature defined on hypergraphs.

Here we modify the definition of Ollivier Ricci curvature and prove the Lipschitz decay for nonlinear parabolic equations. On a locally finite weighted graph $G=(V,w,m,d_{0})$ with the combinatorial distance $d_{0}$ , define

\Delta_{\phi}f(x)\coloneqq\underset{}{\underset{y}{\sum}\frac{w(x,y)}{m(x)}\phi(f(y)-f(x)),}

where $\phi:\mathbb{R}\rightarrow\mathbb{R}$ , is odd, increasing, and either convex or concave on $\mathbb{R}_{+}$ . Recall the transport plan set

\Pi\coloneqq\left\{\pi:B_{1}(x)\times B_{1}(y)\to[0,\infty):\begin{array}[]{c}\underset{x^{\prime}\in B_{1}(x)}{\sum}\pi(x^{\prime},y^{\prime})=\frac{w(y,y^{\prime})}{m(y)}\text{ for all }y^{\prime}\in S_{1}(y),\\ \underset{y^{\prime}\in B_{1}(y)}{\sum}\pi(x^{\prime},y^{\prime})=\frac{w(x,x^{\prime})}{m(x)}\text{ for all }x^{\prime}\in S_{1}(x).\end{array}\right\}.

Then modify the curvature as

(4.1)

\hat{k}_{\phi}(x,y)\coloneqq\underset{\pi_{\phi}\in\Pi_{\phi}}{\text{sup}}\underset{x^{\prime},y^{\prime}\in B_{1}(x)\times B_{1}(y)}{\sum}\pi_{\phi}(x^{\prime},y^{\prime})\left(1-\frac{d_{0}(x^{\prime},y^{\prime})}{d_{0}(x,y)}\right),

where

\Pi_{\phi}\coloneqq\left\{\pi_{\phi}\in\Pi:\begin{array}[]{c}\pi_{\phi}(x^{\prime},y^{\prime})=0\text{ if }x^{\prime}=y^{\prime}\text{ for convex $\phi$},\\ \pi_{\phi}(x^{\prime},y^{\prime})=0\text{ if }x^{\prime}\neq x,y^{\prime}\neq y\text{ and }d_{0}(x^{\prime},y^{\prime})=2\text{ for concave $\phi$.}\end{array}\right\}.

Then we give the gradient estimate for resolvents of nonlinear Laplace.

Theorem 7.

On a locally finite weighted graph $G=(V,w,m,d_{0})$ with combinatorial distance $d_{0}$ , if the modified curvature has a lower bound $\underset{x,y}{\text{inf}}\,\hat{k}(x,y)\geq K$ , then the resolvent $J_{\epsilon}f=\left(id-\epsilon\Delta_{\phi}\right)^{-1}f$ satisfies the Lipschitz decay

Lip(J_{\epsilon}f)\leq Lip(f)\left(1+\epsilon\left(Lip(f)\right)^{-1}\phi(Lip(f))K\right)^{-1},

where $\epsilon$ is small enough such that $1+\epsilon\left(Lip(f)\right)^{-1}\phi(Lip(f))K$ is positive.

Proof.

For any $x\sim y$ , suppose $Lip(f)=C$ , $f(y)=C$ and $f(x)=0$ , then for any $\pi_{\phi}(x^{\prime},y^{\prime})$ satisfying the conditions in (4.1),

\Delta_{\phi}f(x)-\Delta_{\phi}f(y)=\underset{x^{\prime},y^{\prime}}{\sum}\pi_{\phi}(x^{\prime},y^{\prime})\left[\phi(f(x^{\prime})-f(x))-\phi(f(y^{\prime})-f(y))\right].

If $d_{0}(x^{\prime},y^{\prime})=1$ , then

f(x^{\prime})-f(x)\geq f(y^{\prime})-C-(f(y)-C)=f(y^{\prime})-f(y).

Since $\phi$ is increasing, we know

\phi(f(x^{\prime})-f(x))-\phi(f(y^{\prime})-f(y))\geq 0=d_{0}(x,y)-d_{0}(x^{\prime},y^{\prime}).

If $d_{0}(x^{\prime},y^{\prime})=2$ , and $x^{\prime}\neq x$ and $y^{\prime}\neq y$ , then $f(x^{\prime})-f(x)\geq-C$ and $f(y^{\prime})-f(y)\leq C$ , and $f(y^{\prime})-f(x^{\prime})\leq C$ . For convex $\phi$ , such as $p$ -Laplace ( $p\geq 2$ ), we have

\phi(f(x^{\prime})-f(x))-\phi(f(y^{\prime})-f(y))\geq-\phi(C).

If $d_{0}(x^{\prime},y^{\prime})=2$ and either $x^{\prime}=x$ or $y^{\prime}=y$ , then

\phi(f(x^{\prime})-f(x))-\phi(f(y^{\prime})-f(y))\geq-\phi(C).

If $d_{0}(x^{\prime},y^{\prime})=0$ , then $0\leq f(x^{\prime})=f(y^{\prime})\leq C$ . And for concave $\phi$ , such as $p$ -Laplace ( $1<p<2$ ), we have

\phi(f(x^{\prime})-f(x))-\phi(f(y^{\prime})-f(y))=\phi(f(x^{\prime}))-\phi(f(x^{\prime})-C)\geq\phi(C).

Hence,

		$\displaystyle\Delta_{\phi}f(x)-\Delta_{\phi}f(y)$
	$\displaystyle=$	$\displaystyle\underset{x^{\prime},y^{\prime}}{\sum}\pi_{\phi}(x^{\prime},y^{\prime})\left[\phi(f(x^{\prime})-f(x))-\phi(f(y^{\prime})-f(y))\right]$
	$\displaystyle=$	$\displaystyle\left(\underset{d(x^{\prime},y^{\prime})=1}{\sum}+\underset{d(x^{\prime},y^{\prime})=2}{\sum}+\underset{d(x^{\prime},y^{\prime})=0}{\sum}\right)\pi_{\phi}(x^{\prime},y^{\prime})\left[\phi(f(x^{\prime})-f(x))-\phi(f(y^{\prime})-f(y))\right]$
	$\displaystyle\geq$	$\displaystyle\phi(C)\underset{x^{\prime},y^{\prime}}{\sum}\pi_{\phi}(x^{\prime},y^{\prime})\left[d_{0}(x,y)-d_{0}(x^{\prime},y^{\prime})\right].$

That is,

\Delta_{\phi}f(x)-\Delta_{\phi}f(y)\geq\phi(C)\hat{k}(x,y)d_{0}(x,y)\geq\phi(C)K.

And

(id-\epsilon\Delta_{\phi})f(y)-(id-\epsilon\Delta_{\phi})f(x)\geq C+\epsilon\phi(C)K.

For $g\in\mathbb{R}^{V}$ , let $Lip(f)=\mid\nabla f\mid_{\infty}\coloneqq\underset{x\sim y}{\text{sup}}\mid\nabla_{xy}f\mid$ , then by the definition of $J_{\epsilon}$ ,

	$\displaystyle\underset{\mid\nabla g\mid_{\infty}\leq c}{\text{sup}}\mid\nabla J_{\epsilon}g\mid_{\infty}$	$\displaystyle=\underset{\mid\nabla g\mid_{\infty}\leq c}{\text{sup}}\mid\nabla\left(id-\epsilon\Delta_{\phi}\right)^{-1}g\mid_{\infty}$
		$\displaystyle=\underset{\mid\nabla\left(id-\epsilon\Delta_{\phi}\right)h\mid_{\infty}\leq c}{\text{sup}}\mid\nabla h\mid_{\infty}$
		$\displaystyle=\left(\underset{\mid\nabla h\mid_{\infty}\geq c^{-1}}{\text{inf}}\mid\nabla(id-\epsilon\Delta_{\phi})h\mid_{\infty}\right)^{-1}.$

Thus, we know $Lip(J_{\epsilon}f)\leq C^{2}\left(C+\epsilon\phi(C)K\right)^{-1}=C\left(1+\epsilon C^{-1}\phi(C)K\right)^{-1}.$ ∎

Remark 5.

Since $\Delta_{1}$ is a set-valued function, we can not get the Lipschitz decay property by Theorem 7 directly. Since the energy functional $\mathscr{E}_{p}(f)$ is uniformly continuous with respect to $p$ , which means that for any $\delta>0$ , there exists $p$ only depending on $\delta$ such that for all $f_{0}\in\mathbb{R}^{V}$ ,

\underset{f:\left\|f-f_{0}\right\|{}_{\infty}\leq 1}{sup}\mid\mathscr{E}_{p}(f)-\mathscr{E}_{1}(f)\mid\leq\delta.

Then for fixed $f$ and $\epsilon$ , the resolvent $J_{\epsilon}^{p}f=\underset{g\in\mathbb{R}^{V}}{\text{argmin}}\left\{\mathscr{E}_{p}(g)+\frac{1}{2\epsilon}\left\|g-f\right\|_{2}^{2}\right\}$ is also continuous with respect to $p$ . Hence by the Lipschitz decay property of $J_{\epsilon}^{p}$ for $p>1$ , we can get the Lipschitz decay for $J_{\epsilon}^{1}$ .

4.3. The convergence of Laplacian separation flow.

Recall that the extremal 1-Lipschitz extension operator $S$ is defined as $S:\mathbb{R}^{K}\to\mathbb{R}^{V}$ ,

Sf(x)\coloneqq\begin{cases}\begin{array}[]{c}f(x):\\ \underset{y\in K}{\text{min}}\left(f(y)+d(x,y)\right):\\ \underset{y\in K}{\text{max}}\left(f(y)-d(x,y)\right):\end{array}&\begin{array}[]{c}x\in K,\\ x\in Y,\\ x\in X,\end{array}\end{cases}

where $d$ is the graph distance on $G$ . Then $S(Lip(1,K))\subseteq Lip(1,V)$ . In [15], it is proven via elliptic methods that there exists some $g$ with $\Delta Sg=\text{const}$ . Here we give the parabolic flow $(id+\epsilon\Delta)S$ , and show that this converges to the constant Laplacian solution, assuming non-negative Ollivier Ricci curvature.

Theorem 5. For a locally finite graph $G$ with non-negative Ollivier curvature, let $x_{0}\in V$ and $P\coloneqq\left((id+\epsilon\Delta)S\right)\mid_{K}$ , where $\epsilon$ is small enough such that $\text{diag}(id+\epsilon\Delta)$ is positive. Then for all $f\in Lip(1,K)$ , there exists $g\in Lip(1,K)$ such that $P^{n}f-P^{n}f(x_{0})$ converges to $g$ and $\Delta Sg\equiv\text{const}$ on $K$ .

Proof.

We can check that $P$ satisfies monotonicity (1), strict monotonicity (2), and constant additivity (4). Since the non-negative Ollivier Ricci curvature implies Lipschitz decay property of $P$ , i.e., the range of $P$ is $Lip(1,K)$ , then there is a finite accumulation point $g$ of $f_{n}=P^{n}f-P^{n}f(x_{0})$ . By Theorem 2, we can get the convergence and $g$ is a stationary point, that is, $\Delta Sg\equiv\text{const}$ on $K$ . ∎

Next, we want to generalize the result to non-linear cases. For $p\geq 1$ , the resolvent operator of $p$ -Laplace operator $\Delta_{p}$ is defined as $J_{\epsilon}=\left(id-\epsilon\Delta_{p}\right)^{-1}$ for $\epsilon>0$ . Then $J_{\epsilon}$ is monotone [41, Proposition 12.19]. And the following lemma states that the resolvent $J_{\epsilon}$ satisfying the strict monotonicity property (2).

Lemma 3.

If $f\geq g+\delta\mid V\mid 1_{x}$ , where $1_{x}(x)=1$ and $1_{x}(y)=0$ for $y\neq x$ , then $J_{\epsilon}f(x)\geq J_{\epsilon}g(x)+\delta.$

Proof.

Since $J_{\epsilon}$ is monotone, which means $\left\langle J_{\epsilon}f-J_{\epsilon}g,f-g\right\rangle\geq 0.$ Take $f=g+\delta\left(\mid V\mid 1_{x}-1\right)$ , then $\left\langle f-g,1\right\rangle=0$ . By the monotone property,

\left\langle J_{\epsilon}f-J_{\epsilon}g,\delta\left(\mid V\mid 1_{x}-1\right)\right\rangle\geq 0,

which implies $\mid V\mid\left(J_{\epsilon}f-J_{\epsilon}g\right)(x)\geq\left\langle J_{\epsilon}f-J_{\epsilon}g,1\right\rangle=0.$ And set $\tilde{f}=\delta+f=g+\delta\mid V\mid 1_{x}$ which satisfies $\tilde{f}\geq g+\delta\mid V\mid 1_{x}$ , then

J_{\epsilon}\tilde{f}(x)=\delta+J_{\epsilon}f(x)\geq\delta+J_{\epsilon}g(x).

This finishes the proof. ∎

Recall that a new curvature $\hat{k}_{\phi}(x,y)$ is defined in (4.1), whose transport plans forbid 3-cycles for convex $\phi$ on $\mathbb{R}_{+}$ and forbid 5-cycles for concave $\phi$ on $\mathbb{R}_{+}$ .

Then we can prove there exists a stationary point $g$ such that $g\in\Delta_{p}Sh$ and $g\equiv\text{const}$ on $K$ .

Theorem 6. For a locally finite graph $G=(V,w,m,d_{0})$ with a non-negative modified curvature $\hat{k}_{\phi}$ , let $x_{0}\in V$ and $P\coloneqq\left((id+\epsilon\Delta)S\right)\mid_{K}$ , where $\epsilon$ is small enough such that $\text{diag}(id+\epsilon\Delta)$ is positive. Then for all $f\in Lip(1,K)$ , there exists $\tilde{f}\in Lip(1,K)$ such that $P^{n}f-P^{n}f(x_{0})$ converges to $\tilde{f}$ . Moreover, there exist $h,g\in\mathbb{R}^{V}$ such that $g\in\Delta_{p}Sh$ and $g\mid_{X}\geq g\mid_{K}\equiv\text{const}\geq g\mid_{Y}$ .

Proof.

By Lemma 3 we know $J_{\epsilon}$ satisfies strict monotonicity property. It is also constant additive by the definition of resolvent $J_{\epsilon}$ , then $P$ also satisfies the same property. For the non-negative curvature $\hat{k}_{\phi}$ defined as (4.1), by the gradient estimate of Theorem 7, the range of $P$ still is $Lip(1,K)$ . Then there is an accumulation point at infinity. Hence by Theorem 2, there exists $\tilde{f}\in Lip(1,K)$ such that $P\tilde{f}=\tilde{f}+\text{const}$ , which implies that

(4.2)

SJ_{\epsilon}S\tilde{f}=S\tilde{f}+\text{const}.

Define $h_{\epsilon}\coloneqq J_{\epsilon}S\tilde{f}$ and substitute it into the above formula (4.2), then we can get $\Delta_{p}h_{\epsilon}=\text{const}$ . Note that $h_{\epsilon}\neq Sh_{\epsilon}$ , but we claim that $\left\|h_{\epsilon}-Sh_{\epsilon}\right\|_{\infty}\leq c\epsilon$ for some constant $c$ . Since

\left\|h_{\epsilon}-S\tilde{f}\right\|_{\infty}=\left\|J_{\epsilon}S\tilde{f}-S\tilde{f}\right\|_{\infty}\leq\epsilon\left\|\Delta_{p}S\tilde{f}\right\|_{\infty}\leq\epsilon\underset{x}{\text{ max}}deg(x),

and it also holds on $K$ , that is,

\left\|Sh_{\epsilon}-SS\tilde{f}\right\|_{\infty}=\left\|Sh_{\epsilon}-S\tilde{f}\right\|_{\infty}\leq\epsilon.

So by the triangle inequality, we get $\left\|h_{\epsilon}-Sh_{\epsilon}\right\|_{\infty}\leq c\epsilon.$ Then we get $h_{\epsilon}\to h$ for some subsequence as $\epsilon\to 0$ by the compactness, and $h=Sh$ . Take a subsequence $\left\{g_{\epsilon}\right\}$ such that $g_{\epsilon}\in\Delta_{p}h_{\epsilon}$ and $g_{\epsilon}\mid_{X}\geq g_{\epsilon}\mid_{K}\equiv\text{const}\geq g_{\epsilon}\mid_{Y}$ and $g_{\epsilon}\to g.$ By the continuity, we know $g\in\Delta_{p}Sh$ and $g\mid_{X}\geq g\mid_{K}\equiv\text{const}\geq g\mid_{Y}$ . ∎

4.4. The nonlinear Dirichlet form.

In the theory of nonlinear Dirichlet form, one has a correspondence between such forms, semigroups, resolvents, and operators satisfying suitable conditions. Since the assumptions of our theorems fit well in the nonlinear Dirichlet form theory, we can apply our theorems to study the long time behavior of associated continuous semigroups. First, we recall the definition of the nonlinear Dirichlet form [8].

Definition 2.

Let $\mathscr{E}:$ $\mathbb{R}^{N}\to[0,\infty]$ be a convex and lower semicontinuous functional with dense effective domain. Then, the subgradient $-\partial\mathscr{E}$ generates a strongly continuous contraction semigroup $T$ , that is, $u(t)=T_{t}u_{0}$ satisfies

\begin{cases}\begin{array}[]{c}0\in\frac{du}{dt}(t)+\partial\mathscr{E}(u(t)),\\ u(0)=u_{0}\end{array}\end{cases}

pointwise for almost every $t\geq 0$ . We call $\mathscr{E}$ a Dirichlet form if the associated strongly continuous contraction semigroup $T$ is sub-Markovian, which means $T$ is order preserving and $L^{\infty}$ contractive, that is, for every $u,v\in\mathbb{R}^{N}$ and every $t\geq 0$ ,

u\leq v\Rightarrow T_{t}u\leq T_{t}v

and

\left\|T_{t}u-T_{t}v\right\|_{\infty}\leq\left\|u-v\right\|_{\infty}.

For another definition of the nonlinear Dirichlet form in [19], it satisfies the following lemma.

Lemma 4.

[19, Lemma 1.1] For a nonlinear Dirichlet form $\mathscr{E}$ , if $f\in\mathbb{R}^{N}$ and $a,\lambda\in\mathbb{R}$ , then

(4.3)

\begin{array}[]{c}\mathscr{E}(\lambda f)=\lambda^{2}\mathscr{E}(f),\\ \mathscr{E}(f+a)=\mathscr{E}(f).\end{array}

Then we can apply Theorem 2 to obtain the following convergence result with the accumulation point assumption.

Theorem 8.

For a nonlinear Dirichlet form $\mathscr{E}$ with property (4.3), if the semigroup $T_{t}^{n}f$ has an accumulation point at infinity (8) and its associated generator is bounded, then $T_{t}^{n}f$ converges.

Proof.

Define the Markov chain as $P\coloneqq T_{t}$ . By the definition of a Dirichlet form, we know $P$ satisfying monotonicity (1) and non-expansion (5). And the property (4.3) induces the constant additivity (4) of $P$ . Since the associated generator is bounded, then $P$ satisfies strict monotonicity (2). Then by the assumption of accumulation points at infinity, we can get the convergence result by Theorem 2. ∎

Remark 6.

(a) In the nonlinear Dirichlet form setting of [19], if we assume the associated semigroup satisfying sub-Markovian property, then we can also apply Theorem 2 to obtain convergence results with the accumulation point at infinity assumption (8).

(b) As we mentioned before, we can study the long-time behavior of the resolvents of $p$ -Laplace and hypergraph Laplace [17].

(c) Note that our nonlinear Markov chain setting is more general. Since in the nonlinear Dirichlet form setting, the associated generator is required for a kind of reversibility, while our Markov chain does not require it. Moreover, our underlying space is more general than the nonlinear Dirichlet form setting, which requires $L^{2}$ space.

4.5. The nonlinear Perron–Frobenius theory.

The classical Perron–Frobenius theorem shows that a nonnegative matrix has a nonnegative eigenvector associated with its spectral radius, and if the matrix is irreducible then this nonnegative eigenvector can be chosen strictly positive. And there are many nonlinear generalizations.

For example, in [29], the author lets $K$ be a proper cone in $\mathbb{R}^{N}$ , that is, $\alpha K\subset K$ for all $\alpha\in\mathbb{R}_{+}$ , it is closed and convex, $K-K=\mathbb{R}^{N}$ , and $K\cap-K=\left\{0\right\}$ . Then $K$ induces a partial ordering $x\leq y$ on $K$ defined by $x-y\in K$ . Consider maps satisfying:

(M1) $\Lambda:K\to K,K{{}^{\circ}}\to K{{}^{\circ}}.$

(M2) $\Lambda(\alpha x)=\alpha\Lambda(x)$ for all $\alpha\geq 0$ and $x\in K$ .

(M3) $x\leq y$ implies $\Lambda(x)\leq\Lambda(y)$ for all $x,y\in K$ .

(M4) $\Lambda$ is locally Lipschitz continuous near $0$ .

Sufficient conditions for the existence and uniqueness of eigenvectors in the interior of a cone $K$ are developed even when eigenvectors at the boundary of the cone exist [29, Theorem 25, Theorem 28].

Theorem 9.

Let $K$ be the positive function set $\mathbb{R}_{>0}^{N}$ and

Pf\coloneqq\frac{1}{2}\text{log}\left(\text{exp}(f)\cdot\Lambda\left(\text{exp}(f)\right)\right),

where $\Lambda$ is defined as above. If $P^{n}f$ has an accumulation point at infinity (8), then $P^{n}f$ converges.

Proof.

Since $\Lambda$ satisfies (M2) and (M3), then $\tilde{P}f\coloneqq\text{log}\left(\Lambda\left(\text{exp}(f)\right)\right)$ satisfies constant additivity (4) and monotonicity (1). And

Pf=\frac{f+\tilde{P}f}{2}=\frac{1}{2}\text{log}\left(\text{exp}(f)\cdot\Lambda\left(\text{exp}(f)\right)\right)

satisfies constant additivity (4) and strict monotonicity (2). Then we can apply Theorem 2 to get the convergence result with the accumulation points assumption. ∎

Then we introduce another nonlinear generalization that might be applied to a specific case which is relevant to the Ollivier Ricci flow. In [2], the author considers maps $f_{\mathcal{K}}(v)=\text{min}_{A\in\mathcal{K}}Av$ , where $\mathcal{K}$ is a finite set of nonnegative matrices and “min” means component-wise minimum. In particular, he shows the existence of nonnegative generalized eigenvectors of $f_{\mathcal{K}}$ , gives necessary and sufficient conditions for the existence of a strictly positive eigenvector of $f_{\mathcal{K}}$ , which is applicable to our Ollivier Ricci flow (1.2) and it treats the non-connected case. However, it does not give the long-term behavior.

5. The Ollivier Ricci curvature of nonlinear Markov chains

In this section, we introduce a definition of Ollivier Ricci curvature of nonlinear Markov chains according to the Lipschitz decay property. Then we can get the convergence results for the nonlinear Markov chain with a nonnegative Ollivier Ricci curvature. And we can also define the Laplacian separation flow of a nonlinear Markov chain with $Ric_{1}(P,d)\geq 0$ . Then several examples show that the definition is consistent with the classical Ollivier Ricci curvature (3.1), sectional curvature [5], coarse Ricci curvature on hypergraphs [17] and the modified Ollivier Ricci curvature $\hat{k}_{p}$ for $p$ -Laplace (1.3).

Definition 3.

Let $P:\mathbb{R}^{V}\to\mathbb{R}^{V}$ be a nonlinear Markov chain with (1) monotonicity, (2) strict monotonicity and (4) constant additivity, and let $d:V^{2}\to[0,+\infty)$ be the distance function. For $r\geq 0$ , define

Ric_{r}(P,d)\coloneqq 1-\underset{Lip(f)\leq r}{\text{sup}}\frac{Lip(Pf)}{r},

i.e., if $Lip(f)=r$ , then $Lip(Pf)\leq(1-Ric_{r})Lip(f)$ .

By Theorem 2, we can get the following corollary.

Corollary 1.

Let $r\geq 0$ . Assume $(P,d)$ is a nonlinear Markov chain with $Ric_{r}\geq 0$ . Let $x_{0}\in V$ . Then for all $f$ satisfying $Lip(f)\leq r$ , there exists $g\in\mathbb{R}^{V}$ such that $P^{n}f-P^{n}f(x_{0})\to g$ and $P^{n}f-P^{n-1}f\to\text{const}$ as $n\to+\infty$ . Particularly, $Pg=g+\text{const.}$

Proof.

By the definition of $Ric_{r}\geq 0$ , we can get the accumulation point at infinity (8) as $Lip(P^{n}f)\leq r$ for all $n$ , and by compactness. Then apply Theorem 2, we can get the results. ∎

Then we want to define the Laplacian separation flow on a nonlinear Markov chain $(P,d)$ with $Ric_{1}(P,d)\geq 0$ . Assume $V=X\sqcup K\sqcup Y$ such that $d(x,y)=\underset{z\in K}{\text{inf}}d(x,z)+d(z,y)$ for all $x\in X$ and $y\in Y$ . Intuitively that means that $K$ separates $X$ from $Y$ . Recall the extremal 1-Lipschitz extension operator defined as $S:\mathbb{R}^{K}\to\mathbb{R}^{V}$ ,

Sf(x)\coloneqq\begin{cases}\begin{array}[]{c}f(x):\\ \underset{y\in K}{\text{min}}\left(f(y)+d(x,y)\right):\\ \underset{y\in K}{\text{max}}\left(f(y)-d(x,y)\right):\end{array}&\begin{array}[]{c}x\in K,\\ x\in Y,\\ x\in X.\end{array}\end{cases}

Then $S(Lip(1,K))\subseteq Lip(1,V)$ . And we can get the following lemma.

Lemma 5.

Assume $(P,d)$ is a nonlinear Markov chain with $Ric_{1}(P,d)\geq 0$ . Define $\tilde{P}:\mathbb{R}^{K}\to\mathbb{R}^{K}$ as $\tilde{P}f=\left(PSf\right)\mid_{K}$ . Then $Ric_{1}(\tilde{P},d\mid_{K\times K})\geq 0$ .

Proof.

Since $S(Lip(1,K))\subseteq Lip(1,V)$ , i.e., $Sf\in Lip(1,V)$ , and by the definition of $Ric_{1}(P,d)\geq 0$ , we can get $Ric_{1}(\tilde{P},d\mid_{K\times K})\geq 0$ . ∎

Combining Corollary 1, we can get the Laplacian separation result on the nonlinear Markov chain.

Corollary 2.

Let $(P,d)$ is a nonlinear Markov chain with $V=X\sqcup K\sqcup Y$ . Assume $Ric_{1}(P,d)\geq 0$ . Then there exists $f\in\mathbb{R}^{V}$ and $C\in\mathbb{R}$ such that $f=Sf\coloneqq S(f\mid_{K})$ and

\Delta f\begin{cases}\begin{array}[]{c}=C,\>\text{on }K,\\ \leq C,\>\text{on }Y,\\ \geq C,\>\text{on }X,\end{array}\end{cases}

where $\Delta\coloneqq P-id$ .

Proof.

By Corollary 1, there exists $g\in Lip(1,K)$ such that on $K$ ,

PSg=g+\text{const}.

Let $f=Sg$ . Clearly, $f=S(f\mid_{K})$ , and on $K$ ,

Pf=f+\text{const},

i.e., $\Delta f=C$ . Moreover,

SPf=SPSg=Sg+\text{const}=f+\text{const}.

Then on $X$ , we have $SPf\leq Pf$ as $S$ is the minimum Lipschitz extension on $X$ . Hence, $Pf\geq f+C$ , i.e., $\Delta f\geq C$ on $X$ . Similarly, $\Delta f\leq C$ on $Y$ , finishing the proof. ∎

Next, the following examples show that our Ollivier Ricci curvature definition is consistent with other settings.

Example 1.

(a) Let $P$ be a linear Markov chain, then $Ric_{r}$ is the classical Ollivier Ricci curvature $\kappa$ , see definition (3.1).

(b) Let $\tilde{P}$ be a linear Markov chain and define $P(\cdot)=\text{log}\tilde{P}\text{exp}(\cdot)$ , then $Ric_{r}(P,d)\geq 0$ for all $r\geq 0$ if the sectional curvature $\kappa_{sec}\geq 0$ , see [5].

(c) Let $P$ be the resolvent of hypergraph Laplace, then $Ric_{r}\geq 0$ for all $r\geq 0$ if the coarse Ricci curvature of hypergraphs $\kappa\geq 0$ , see [17].

(d) Let $P$ be the resolvent of $p$ -Laplace, then $Ric_{r}\geq 0$ for all $r\geq 0$ if the modified Ollivier Ricci curvature of $p$ -Laplace $\hat{k}_{p}\geq 0$ , see definition (1.3) in introduction.

Remark 7.

For the above examples (a)-(d) with $Ric_{1}(P,d)\geq 0$ , by Corollary 2 the Laplacian separation flow can be defined respectively.

$\mathbf{\boldsymbol{\mathbf{Acknowledgements}\mathbf{}\text{:}}}$ The authors would like to thank Jürgen Jost, Bobo Hua and Tao Wang for helpful discussions and suggestions.

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